Electromagnetism
Electricity Magnetism
Electrostatics Electric charge Static electricity Coulomb's law Electric field Electric flux Gauss's law Electric potential energy Electric potential Electrostatic induction Electric dipole moment Polarization density
Magnetostatics Ampère's law Electric current Magnetic field Magnetization Magnetic flux Biot–Savart law Magnetic dipole moment Gauss's law for magnetism
Electrodynamics Lorentz force law emf Electromagnetic induction Faraday's law Lenz's law Displacement current Maxwell's equations EM field Electromagnetic radiation Maxwell tensor Poynting vector Liénard–Wiechert potential Jefimenko's equations Eddy current
Electrical Network Electrical conduction Electrical resistance Capacitance Inductance Impedance Resonant cavities Waveguides
Covariant formulation Electromagnetic tensor EM Stress-energy tensor Four-current Electromagnetic four-potential
Scientists Ampère Coulomb Faraday Gauss Heaviside Henry Hertz Lorentz Maxwell Tesla Volta Weber Ørsted
v t e

Trajectory of a particle with a positive or negative charge q under the influence of a magnetic field B, which is directed perpendicularly out of the screen.

Beam of electrons moving in a circle, due to the presence of a magnetic field. Purple light is emitted along the electron path, due to the electrons colliding with gas molecules in the bulb. Using a Teltron tube.

In physics, particularly electromagnetism, the Lorentz force is the force on a point charge due to electromagnetic fields.

The first derivation of the Lorentz force is commonly attributed to Oliver Heaviside in 1889,^[1] although other historians suggest an earlier origin in an 1865 paper by James Clerk Maxwell.^[2] Lorentz derived it a few years after Heaviside.

Equation (SI units)[link]

One charged particle[link]

The force F acting on a particle of electric charge q with instantaneous velocity v, due to an external electric field E and magnetic field B, is given by:^[3]

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{F} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B})

where × is the vector cross product. All boldface quantities are vectors. More explicitly stated:

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{F}(\mathbf{r},t,q) = q[\mathbf{E}(\mathbf{r},t) + \mathbf{\dot{r}} \times \mathbf{B}(\mathbf{r},t)]

in which r is the position vector of the charged particle, t is time, and the overdot is a time derivative.

A positively charged particle will be accelerated in the same linear orientation as the E field, but will curve perpendicularly to both the instantaneous velocity vector v and the B field according to the right-hand rule (in detail, if the thumb of the right hand points along v and the index finger along B, then the middle finger points along F).

The term qE is called the electric force, while the term qv × B is called the magnetic force.^[4] According to some definitions, the term "Lorentz force" refers specifically to the formula for the magnetic force,^[5] with the total electromagnetic force (including the electric force) given some other (nonstandard) name. This article will not follow this nomenclature: In what follows, the term "Lorentz force" will refer only to the expression for the total force.

The magnetic force component of the Lorentz force manifests itself as the force that acts on a current-carrying wire in a magnetic field. In that context, it is also called the Laplace force.

Continuous charge distribution[link]

For a continuous charge distribution in motion, the Lorentz force equation becomes:

Failed to parse (Missing texvc executable; please see math/README to configure.): d\mathbf{F} = dq\left(\mathbf{E} + \mathbf{v} \times \mathbf{B}\right)\,\!

where dF is the force on a small piece of the charge distribution with charge dq. If both sides of this equation are divided by the volume of this small piece of the charge distribution dV, the result is:

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{f} = \rho\left(\mathbf{E} + \mathbf{v} \times \mathbf{B}\right)\,\!

where f is the force density (force per unit volume) and ρ is the charge density (charge per unit volume). Next, the current density corresponding to the motion of the charge continuum is

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{J} = \rho \mathbf{v} \,\!

so the continuous analogue to the equation is^[6]

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{f} = \rho \mathbf{E} + \mathbf{J} \times \mathbf{B}\,\!

The total force is the volume integral over the charge distribution:

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{F} = \int dV ( \rho \mathbf{E} + \mathbf{J} \times \mathbf{B} ) \,\!

By eliminating ρ and J, using Maxwell's equations, and manipulating using the theorems of vector calculus, this form of the equation can be used to derive the Maxwell stress tensor T, used in General relativity.^[6]

In terms of the tensor field T and the Poynting vector S, another way to write the Lorentz force (per unit volume) is^[6]

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{f} = \nabla\cdot\mathbf{T} - \dfrac{1}{c^2} \dfrac{\partial \mathbf{S}}{\partial t} \,\!

where c is the speed of light and ∇• denotes the divergence of a tensor field. Rather than the amount of charge and its velocity in electric and magnetic fields, this equation relates the energy flux (flow of energy per unit time per unit distance) in the fields to the force exerted on a charge distribution.

History[link]

Early attempts to quantitatively describe the electromagnetic force were made in the mid-18th century. It was proposed that the force on magnetic poles, by Johann Tobias Mayer and others in 1760^{[citation needed]}, and electrically charged objects, by Henry Cavendish in 1762^{[citation needed]}, obeyed an inverse-square law. However, in both cases the experimental proof was neither complete nor conclusive. It was not until 1784 when Charles-Augustin de Coulomb, using a torsion balance, was able to definitively show through experiment that this was true.^[7] Soon after the discovery in 1820 by H. C. Ørsted that a magnetic needle is acted on by a voltaic current, André-Marie Ampère that same year was able to devise through experimentation the formula for the angular dependence of the force between two current elements.^[8][9] In all these descriptions, the force was always given in terms of the properties of the objects involved and the distances between them rather than in terms of electric and magnetic fields.^[10]

The modern concept of electric and magnetic fields first arose in the theories of Michael Faraday, particularly his idea of lines of force, later to be given full mathematical description by Lord Kelvin and James Clerk Maxwell.^[11] From a modern perspective it is possible to identify in Maxwell's 1865 formulation of his field equations a form of the Lorentz force equation in relation to electric currents,^[2] however, in the time of Maxwell it was not evident how his equations related to the forces on moving charged objects. J. J. Thomson was the first to attempt to derive from Maxwell's field equations the electromagnetic forces on a moving charged object in terms of the object's properties and external fields. Interested in determining the electromagnetic behavior of the charged particles in cathode rays, Thomson published a paper in 1881 wherein he gave the force on the particles due to an external magnetic field as^[1]

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{F} = \frac{q}{2}\mathbf{v} \times \mathbf{B}

. Thomson was able to arrive at the correct basic form of the formula, but, because of some miscalculations and an incomplete description of the displacement current, included an incorrect scale-factor of a half in front of the formula. It was Oliver Heaviside, who had invented the modern vector notation and applied them to Maxwell's field equations, that in 1885 and 1889 fixed the mistakes of Thomson's derivation and arrived at the correct form of the magnetic force on a moving charged object.^[1][12][13] Finally, in 1892, Hendrik Lorentz derived the modern day form of the formula for the electromagnetic force which includes the contributions to the total force from both the electric and the magnetic fields. Lorentz began by abandoning the Maxwellian descriptions of the ether and conduction. Instead, Lorentz made a distinction between matter and the luminiferous aether and sought to apply the Maxwell equations at a microscopic scale. Using the Heaviside's version of the Maxwell equations for a stationary ether and applying Lagrangian mechanics (see below), Lorentz arrived at the correct and complete form of the force law that now bears his name.^[14][15]

Trajectories of particles due to the Lorentz force[link]

Charged particle drifts in a homogeneous magnetic field. (A) No disturbing force (B) With an electric field, E (C) With an independent force, F (e.g. gravity) (D) In an inhomogeneous magnetic field, grad H

In many cases of practical interest, the motion in a magnetic field of an electrically charged particle (such as an electron or ion in a plasma) can be treated as the superposition of a relatively fast circular motion around a point called the guiding center and a relatively slow drift of this point. The drift speeds may differ for various species depending on their charge states, masses, or temperatures, possibly resulting in electric currents or chemical separation.

Significance of the Lorentz force[link]

While the modern Maxwell's equations describe how electrically charged particles and currents or moving charged particles give rise to electric and magnetic fields, the Lorentz force law completes that picture by describing the force acting on a moving point charge q in the presence of electromagnetic fields.^[3][16] The Lorentz force law describes the effect of E and B upon a point charge, but such electromagnetic forces are not the entire picture. Charged particles are possibly coupled to other forces, notably gravity and nuclear forces. Thus, Maxwell's equations do not stand separate from other physical laws, but are coupled to them via the charge and current densities. The response of a point charge to the Lorentz law is one aspect; the generation of E and B by currents and charges is another.

In real materials the Lorentz force is inadequate to describe the behavior of charged particles, both in principle and as a matter of computation. The charged particles in a material medium both respond to the E and B fields and generate these fields. Complex transport equations must be solved to determine the time and spatial response of charges, for example, the Boltzmann equation or the Fokker–Planck equation or the Navier–Stokes equations. For example, see magnetohydrodynamics, fluid dynamics, electrohydrodynamics, superconductivity, stellar evolution. An entire physical apparatus for dealing with these matters has developed. See for example, Green–Kubo relations and Green's function (many-body theory).

[edit] Lorentz force law as the definition of E and B

In many textbook treatments of classical electromagnetism, the Lorentz Force Law is used as the definition of the electric and magnetic fields E and B.^[17] To be specific, the Lorentz Force is understood to be the following empirical statement:

The electromagnetic force F on a test charge at a given point and time is a certain function of its charge q and velocity v, which can be parameterized by exactly two vectors E and B, in the functional form:

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{F}=q(\mathbf{E}+\mathbf{v}\times\mathbf{B}).

If this empirical statement is valid (countless experiments have shown that it is^{[citation needed]}), then two vector fields E and B are thereby defined throughout space and time, and these are called the "electric field" and "magnetic field". Note that the fields are defined everywhere in space and time with respect to what force a test charge would receive regardless of whether a charge is present to experience the force.

Note also that as a definition of E and B, the Lorentz force is only a definition in principle because a real particle (as opposed to the hypothetical "test charge" of infinitesimally-small mass and charge) would generate its own finite E and B fields, which would alter the electromagnetic force that it experiences. In addition, if the charge experiences acceleration, as if forced into a curved trajectory by some external agency, it emits radiation that causes braking of its motion. See for example Bremsstrahlung and synchrotron light. These effects occur through both a direct effect (called the radiation reaction force) and indirectly (by affecting the motion of nearby charges and currents). Moreover, net force must include gravity, electroweak, and any other forces aside from electromagnetic force.

Force on a current-carrying wire[link]

Right-hand rule for a current-carrying wire in a magnetic field B

When a wire carrying an electrical current is placed in a magnetic field, each of the moving charges, which comprise the current, experiences the Lorentz force, and together they can create a macroscopic force on the wire (sometimes called the Laplace force). By combining the Lorentz force law above with the definition of electrical current, the following equation results, in the case of a straight, stationary wire:

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{F} = I \boldsymbol{\ell} \times \mathbf{B} \,\!

where ℓ is a vector whose magnitude is the length of wire, and whose direction is along the wire, aligned with the direction of conventional current flow I.

If the wire is not straight but curved, the force on it can be computed by applying this formula to each infinitesimal segment of wire dℓ, then adding up all these forces by integration. Formally, the net force on a stationary, rigid wire carrying a steady current I is

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{F} = I\int d\boldsymbol{\ell}\times \mathbf{B}

This is the net force. In addition, there will usually be torque, plus other effects if the wire is not perfectly rigid.

One application of this is Ampère's force law, which describes how two current-carrying wires can attract or repel each other, since each experiences a Lorentz force from the other's magnetic field. For more information, see the article: Ampère's force law.

EMF[link]

The magnetic force (q v × B) component of the Lorentz force is responsible for motional electromotive force (or motional EMF), the phenomenon underlying many electrical generators. When a conductor is moved through a magnetic field, the magnetic force tries to push electrons through the wire, and this creates the EMF. The term "motional EMF" is applied to this phenomenon, since the EMF is due to the motion of the wire.

In other electrical generators, the magnets move, while the conductors do not. In this case, the EMF is due to the electric force (qE) term in the Lorentz Force equation. The electric field in question is created by the changing magnetic field, resulting in an induced EMF, as described by the Maxwell-Faraday equation (one of the four modern Maxwell's equations).^[18]

Both of these EMF's, despite their different origins, can be described by the same equation, namely, the EMF is the rate of change of magnetic flux through the wire. (This is Faraday's law of induction, see above.) Einstein's theory of special relativity was partially motivated by the desire to better understand this link between the two effects.^[18] In fact, the electric and magnetic fields are different faces of the same electromagnetic field, and in moving from one inertial frame to another, the solenoidal vector field portion of the E-field can change in whole or in part to a B-field or vice versa.^[19]

Lorentz force and Faraday's law of induction[link]

Given a loop of wire in a magnetic field, Faraday's law of induction states the induced electromotive force (EMF) in the wire is:

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{E} = -\frac{d\Phi_B}{dt}

where

Failed to parse (Missing texvc executable; please see math/README to configure.): \Phi_B = \iint_{\Sigma(t)} d \bold{A} \cdot \mathbf{B}(\mathbf{r}, t)

is the magnetic flux through the loop, B is the magnetic field, Σ(t) is a surface bounded by the closed contour ∂Σ(t), at all at time t, dA is an infinitesimal vector area element of Σ(t) (magnitude is the area of an infinitesimal patch of surface, direction is orthogonal to that surface patch).

The sign of the EMF is determined by Lenz's law. Note that this is valid for not only a stationary wire - but also for a moving wire.

From Faraday's law of induction (that is valid for a moving wire, for instance in a motor) and the Maxwell Equations, the Lorentz Force can be deduced. The reverse is also true, the Lorentz force and the Maxwell Equations can be used to derive the Faraday Law.

Let Σ(t) be the moving wire, moving together without rotation and with constant velocity v and Σ(t) be the internal surface of the wire. The EMF around the closed path ∂Σ(t) is given by:^[20]

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{E} =\oint_{\part \Sigma (t)} d \boldsymbol{\ell} \cdot \mathbf{F} / q

where

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{E} = \mathbf{F} / q

is the electric field and dℓ is an infinitesimal vector element of the contour ∂Σ(t).

NB: Both dℓ and dA have a sign ambiguity; to get the correct sign, the right-hand rule is used, as explained in the article Kelvin-Stokes theorem.

The above result can be compared with the version of Faraday's law of induction that appears in the modern Maxwell's equations, called here the Maxwell-Faraday equation:

Failed to parse (Missing texvc executable; please see math/README to configure.): \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \ .

The Maxwell-Faraday equation also can be written in an integral form using the Kelvin-Stokes theorem:.^[21]

So we have, the Maxwell Faraday equation:

Failed to parse (Missing texvc executable; please see math/README to configure.): \oint_{\partial \Sigma(t)}d \boldsymbol{\ell} \cdot \mathbf{E}(\mathbf{r},\ t) = - \ \iint_{\Sigma(t)} d \bold {A} \cdot {{d \,\mathbf {B}(\mathbf{r},\ t)} \over dt }

and the Faraday Law,

Failed to parse (Missing texvc executable; please see math/README to configure.): \oint_{\partial \Sigma(t)}d \boldsymbol{\ell} \cdot \mathbf{F}/q(\mathbf{r},\ t) = - \frac{d}{dt} \iint_{\Sigma(t)} d \bold {A} \cdot \mathbf{B}(\mathbf{r},\ t)

The two are equivalent if the wire is not moving. Using the Leibniz integral rule and that div B = 0, results in,

Failed to parse (Missing texvc executable; please see math/README to configure.): \oint_{\partial \Sigma(t)} d \boldsymbol{\ell} \cdot \mathbf{F}/q(\mathbf{r}, t) = - \iint_{\Sigma(t)} d \bold{A} \cdot \frac{d}{dt} \mathbf{B}(\mathbf{r}, t) + \oint_{\partial \Sigma(t)} \mathbf{v} \times \mathbf{B} d \boldsymbol{\ell}

and using the Maxwell Faraday equation,

Failed to parse (Missing texvc executable; please see math/README to configure.): \oint_{\partial \Sigma(t)} d \boldsymbol{\ell} \cdot \mathbf{F}/q(\mathbf{r},\ t) = \oint_{\partial \Sigma(t)} d \boldsymbol{\ell} \cdot \mathbf{E}(\mathbf{r},\ t) + \oint_{\partial \Sigma(t)} \mathbf{v} \times \mathbf{B}(\mathbf{r},\ t) d \boldsymbol{\ell}

since this is valid for any wire position it implies that,

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{F}= q\,\mathbf{E}(\mathbf{r},\ t) + q\,\mathbf{v} \times \mathbf{B}(\mathbf{r},\ t)

Faraday's law of induction holds whether the loop of wire is rigid and stationary, or in motion or in process of deformation, and it holds whether the magnetic field is constant in time or changing. However, there are cases where Faraday's law is either inadequate or difficult to use, and application of the underlying Lorentz force law is necessary. See inapplicability of Faraday's law.

If the magnetic field is fixed in time and the conducting loop moves through the field, the flux magnetic flux Φ_B linking the loop can change in several ways. For example, if the B-field varies with position, and the loop moves to a location with different B-field, Φ_B will change. Alternatively, if the loop changes orientation with respect to the B-field, the B • dA differential element will change because of the different angle between B and dA, also changing Φ_B. As a third example, if a portion of the circuit is swept through a uniform, time-independent B-field, and another portion of the circuit is held stationary, the flux linking the entire closed circuit can change due to the shift in relative position of the circuit's component parts with time (surface ∂Σ(t) time-dependent). In all three cases, Faraday's law of induction then predicts the EMF generated by the change in Φ_B.

Note that the Maxwell Faraday's equation implies that the Electric Field E is non conservative when the Magnetic Field B varies in time, and is not expressible as the gradient of a scalar field, and not subject to the gradient theorem since its rotational is not zero. See also.^[20][22]

Lorentz force in terms of potentials[link]

The E and B fields can be replaced by the magnetic vector potential A and (scalar) electrostatic potential ϕ by

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{E} = - \nabla \phi - \frac { \partial \mathbf{A} } { \partial t }

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{B} = \nabla \times \mathbf{A}

where ∇ is the gradient, ∇• is the divergence, ∇ × is the curl.

The force becomes

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{F} = q\left[-\nabla \phi- \frac{\partial \mathbf{A}}{\partial t}+\mathbf{v}\times(\nabla\times\mathbf{A})\right]

and using an identity for the triple product simplifies to

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{F} = q\left[-\nabla \phi- \frac{\partial \mathbf{A}}{\partial t}+ \nabla(\mathbf{v}\cdot\mathbf{A})-(\mathbf{v}\cdot\nabla)\mathbf{A} \right]

Lorentz force and Lagrangian mechanics[link]

The Lagrangian for a charged particle of mass m and charge q in an electromagnetic field equivalently describes the dynamics of the particle in terms of its energy, rather than the force exerted on it. The classical expression is given by:^[23]

Failed to parse (Missing texvc executable; please see math/README to configure.): L=\frac{m}{2}\mathbf{\dot{r}}\cdot\mathbf{\dot{r}}+q\mathbf{A}\cdot\mathbf{\dot{r}}-q\phi

where A and ϕ are the potential fields as above. Using Lagrange's equations, the equation for the Lorentz force can be obtained.

Derivation of Lorentz force (SI units)

For an A field, a particle moving with velocity v = ṙ has potential momentum Failed to parse (Missing texvc executable; please see math/README to configure.): q\mathbf{A}(\mathbf{r},t) , so its potential energy is Failed to parse (Missing texvc executable; please see math/README to configure.): q\mathbf{A}(\mathbf{r},t)\cdot\mathbf{\dot{r}} . For a ϕ field, the particle's potential energy is Failed to parse (Missing texvc executable; please see math/README to configure.): q\phi(\mathbf{r},t) . The total potential energy is then: Failed to parse (Missing texvc executable; please see math/README to configure.): V=q\phi-q\mathbf{A}\cdot\mathbf{\dot{r}} and the kinetic energy is: Failed to parse (Missing texvc executable; please see math/README to configure.): T=\frac{m}{2}\mathbf{\dot{r}}\cdot\mathbf{\dot{r}} hence the Lagrangian: Failed to parse (Missing texvc executable; please see math/README to configure.): L=T-V=\frac{m}{2}\mathbf{\dot{r}}\cdot\mathbf{\dot{r}}+q\mathbf{A}\cdot\mathbf{\dot{r}}-q\phi Failed to parse (Missing texvc executable; please see math/README to configure.): L=\frac{m}{2}(\dot{x}^2+\dot{y}^2+\dot{z}^2) + q(\dot{x}A_x+\dot{y}A_y+\dot{z}A_z) - q\phi Lagrange's equations are Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{d}{dt}\frac{\partial L}{\partial \dot{x}}=\frac{\partial L}{\partial x} (same for y and z). So calculating the partial derivatives: Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align}\frac{d}{dt}\frac{\partial L}{\partial \dot{x}} & =m\ddot{x}+q\frac{d A_x}{dt} \\ & = m\ddot{x}+ \frac{q}{dt}\left(\frac{\partial A_x}{\partial t}dt+\frac{\partial A_x}{\partial x}dx+\frac{\partial A_x}{\partial y}dy+\frac{\partial A_x}{\partial z}dz\right) \\ & = m\ddot{x}+ q\left(\frac{\partial A_x}{\partial t}+\frac{\partial A_x}{\partial x}\dot{x}+\frac{\partial A_x}{\partial y}\dot{y}+\frac{\partial A_x}{\partial z}\dot{z}\right)\\ \end{align} Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\partial L}{\partial x}= -q\frac{\partial \phi}{\partial x}+ q\left(\frac{\partial A_x}{\partial x}\dot{x}+\frac{\partial A_y}{\partial x}\dot{y}+\frac{\partial A_z}{\partial x}\dot{z}\right) equating and simplifying: Failed to parse (Missing texvc executable; please see math/README to configure.): m\ddot{x}+ q\left(\frac{\partial A_x}{\partial t}+\frac{\partial A_x}{\partial x}\dot{x}+\frac{\partial A_x}{\partial y}\dot{y}+\frac{\partial A_x}{\partial z}\dot{z}\right)= -q\frac{\partial \phi}{\partial x}+ q\left(\frac{\partial A_x}{\partial x}\dot{x}+\frac{\partial A_y}{\partial x}\dot{y}+\frac{\partial A_z}{\partial x}\dot{z}\right) Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} F_x & = -q\left(\frac{\partial \phi}{\partial x}+\frac{\partial A_x}{\partial t}\right) + q\left[\dot{y}\left(\frac{\partial A_y}{\partial x} - \frac{\partial A_x}{\partial y}\right)+\dot{z}\left(\frac{\partial A_z}{\partial x}-\frac{\partial A_x}{\partial z}\right)\right] \\ & = qE_x + q[\dot{y}(\nabla\times\mathbf{A})_z-\dot{z}(\nabla\times\mathbf{A})_y] \\ & = qE_x + q[\mathbf{\dot{r}}\times(\nabla\times\mathbf{A})]_x \\ & = qE_x + q(\mathbf{\dot{r}}\times\mathbf{B})_x \end{align} and similarly for the y and z directions. Hence the force equation is: Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{F}= q(\mathbf{E} + \mathbf{\dot{r}}\times\mathbf{B})

The potential energy depends on the velocity of the particle, so the force is velocity dependent, so it is not conservative.

Equation (cgs units)[link]

The above-mentioned formulae use SI units which are the most common among experimentalists, technicians, and engineers. In cgs-Gaussian units, which are somewhat more common among theoretical physicists, one has instead

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{F} = q_\mathrm{cgs} \left(\mathbf{E}_\mathrm{cgs} + \frac{\mathbf{v}}{c} \times \mathbf{B}_\mathrm{cgs}\right).

where c is the speed of light. Although this equation looks slightly different, it is completely equivalent, since one has the following relations:

Failed to parse (Missing texvc executable; please see math/README to configure.): q_\mathrm{cgs}=\frac{q_\mathrm{SI}}{\sqrt{4\pi \epsilon_0}},\quad \mathbf E_\mathrm{cgs} =\sqrt{4\pi\epsilon_0}\,\mathbf E_\mathrm{SI},\quad \mathbf B_\mathrm{cgs} ={\sqrt{4\pi /\mu_0}}\,{\mathbf B_\mathrm{SI}}

where ε₀ is the vacuum permittivity and μ₀ the vacuum permeability. In practice, the subscripts "cgs" and "SI" are always omitted, and the unit system has to be assessed from context.

Relativistic form of the Lorentz force[link]

Because the electric and magnetic fields are dependent on the velocity of an observer, the relativistic form of the Lorentz force law can best be exhibited starting from a coordinate-independent expression for the electromagnetic and magnetic fields,^[24] Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{F} , and an arbitrary time-direction, Failed to parse (Missing texvc executable; please see math/README to configure.): \gamma_0 , where

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{E} = (\mathcal{F}\cdot\gamma_0)

and

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{B} = (\mathcal{F}\wedge\gamma_0)^* .

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal F

is a space-time plane (bivector), which has six degrees of freedom corresponding to boosts (rotations in space-time planes) and rotations (rotations in space-space planes).  The dot product with the vector Failed to parse (Missing texvc executable; please see math/README to configure.): \gamma_0
pulls a vector from the translational part, while the wedge-product creates a space-time trivector, whose dot product with the volume element (the dual above) creates the magnetic field vector from the spatial rotation part.  Only the parts of the above two formulas perpendicular to gamma are relevant.

The relativistic velocity is given by the (time-like) changes in a time-position vector Failed to parse (Missing texvc executable; please see math/README to configure.): \dot x , where

Failed to parse (Missing texvc executable; please see math/README to configure.): \dot x^2 = 1,

(which shows our choice for the metric) and the velocity is

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{v} = \dot x \wedge \gamma_0 / (\dot x \cdot \gamma_0).

Then the Lorentz force law is simply (note that the order is important)

Failed to parse (Missing texvc executable; please see math/README to configure.): m\ddot x = q \mathcal{F}\cdot {\dot x} .

Covariant form of the Lorentz force[link]

Field tensor[link]

Using the metric signature (-1,1,1,1), The Lorentz force for a charge q can be written in covariant form:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{d p^\alpha}{d \tau} = q U_\beta F^{\alpha \beta}

where p^α is the four-momentum, defined as:

Failed to parse (Missing texvc executable; please see math/README to configure.): p^\alpha = \left(p_0, p_1, p_2, p_3 \right) = \left(E/c, p_x, p_y, p_z \right) \, ,

Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle \tau

the proper time of the particle, Fαβ the contravariant electromagnetic tensor

Failed to parse (Missing texvc executable; please see math/README to configure.): F^{\alpha \beta} = \begin{pmatrix} 0 & E_x/c & E_y/c & E_z/c \\ -E_x/c & 0 & B_z & -B_y \\ -E_y/c & -B_z & 0 & B_x \\ -E_z/c & B_y & -B_x & 0 \end{pmatrix}

and U is the covariant 4-velocity of the particle, defined as:

Failed to parse (Missing texvc executable; please see math/README to configure.): U_\beta = \left(U_0, U_1, U_2, U_3 \right) = \gamma \left(-c, u_x, u_y, u_z \right) \, ,

where Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle \gamma

is the Lorentz factor defined above.  

The fields are transformed to a frame moving with constant relative velocity by:

Failed to parse (Missing texvc executable; please see math/README to configure.): \acute{F}^{\mu \nu} = {\Lambda^{\mu}}_{\alpha} {\Lambda^{\nu}}_{\beta} F^{\alpha \beta} \, ,

where Λ^μ_α is the Lorentz transformation tensor.

Translation to vector notation[link]

The α = 1 component (x-component) of the force is

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{d p^1}{d \tau} = q U_\beta F^{1 \beta} = q\left(U_0 F^{10} + U_1 F^{11} + U_2 F^{12} + U_3 F^{13} \right) .\,

Substituting the components of the covariant electromagnetic tensor F yields

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{d p^1}{d \tau} = q \left[U_0 \left(\frac{-E_x}{c} \right) + U_2 (B_z) + U_3 (-B_y) \right] \,

Using the components of covariant four-velocity yields

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} \frac{d p^1}{d \tau} & = q \gamma \left[-c \left(\frac{-E_x}{c} \right) + u_y B_z + u_z (-B_y) \right] \\ &= q \gamma \left(E_x + u_y B_z - u_z B_y \right) \\ & = q \gamma \left[ E_x + \left( \mathbf{u} \times \mathbf{B} \right)_x \right] \, . \end{align}

The calculation for α = 2, 3 (force components in the y and z directions) yields similar results, so collecting the 3 equations into one:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{d \mathbf{p} }{d \tau} = q \gamma\left( \mathbf{E} + \mathbf{u} \times \mathbf{B} \right) \, ,

which is the Lorentz force.

Applications[link]

The Lorentz force occurs in many devices, including:

• Cyclotrons and other circular path particle accelerators
• Mass spectrometers
• Velocity Filters
• Magnetrons

In its manifestation as the Laplace force on an electric current in a conductor, this force occurs in many devices including:

See also[link]

Wikimedia Commons has media related to: Lorentz force

References[link]

The numbered references refer in part to the list immediately below.

• Feynman, Richard Phillips; Leighton, Robert B.; Sands, Matthew L. (2006). The Feynman lectures on physics (3 vol.). Pearson / Addison-Wesley. ISBN 0-8053-9047-2 : volume 2.

• Griffiths, David J. (1999). Introduction to electrodynamics (3rd ed.). Upper Saddle River, [NJ.]: Prentice-Hall. ISBN 0-13-805326-X 

• Jackson, John David (1999). Classical electrodynamics (3rd ed.). New York, [NY.]: Wiley. ISBN 0-471-30932-X 

• Serway, Raymond A.; Jewett, John W., Jr. (2004). Physics for scientists and engineers, with modern physics. Belmont, [CA.]: Thomson Brooks/Cole. ISBN 0-534-40846-X 

• Srednicki, Mark A. (2007). Quantum field theory. Cambridge, [England] ; New York [NY.]: Cambridge University Press. ISBN 978-0-521-86449-7. http://books.google.com/?id=5OepxIG42B4C&pg=PA315&dq=isbn=9780521864497 

Footnotes[link]

1. ^ ^{a b c} Oliver Heaviside By Paul J. Nahin, p120
2. ^ ^{a b} Huray, Paul G. (2009). Maxwell's Equations. Wiley-IEEE. p. 22. ISBN 0-470-54276-4. http://books.google.com/books?id=0QsDgdd0MhMC&pg=PA22#v=onepage&q&f=false. 
3. ^ ^{a b} See Jackson page 2. The book lists the four modern Maxwell's equations, and then states, "Also essential for consideration of charged particle motion is the Lorentz force equation, F = q ( E+ v × B ), which gives the force acting on a point charge q in the presence of electromagnetic fields."
4. ^ See Griffiths page 204.
5. ^ For example, see the website of the "Lorentz Institute": \[1], or Griffiths.
6. ^ ^{a b c} Introduction to Electrodynamics (3rd Edition), D.J. Griffiths, Pearson Education, Dorling Kindersley, 2007, ISBN 81-7758-293-3
7. ^ Meyer, Herbert W. (1972). A History of Electricity and Magnetism. Norwalk, CT: Burndy Library. pp. 30–31. ISBN 0-262-13070-X. 
8. ^ Verschuur, Gerrit L. (1993). Hidden Attraction : The History And Mystery Of Magnetism. New York: Oxford University Press. pp. 78–79. ISBN 0-19-506488-7. 
9. ^ Darrigol, Olivier (2000). Electrodynamics from Ampère to Einstein. Oxford, [England]: Oxford University Press. pp. 9, 25. ISBN 0-19-850593-0 
10. ^ Verschuur, Gerrit L. (1993). Hidden Attraction : The History And Mystery Of Magnetism. New York: Oxford University Press. p. 76. ISBN 0-19-506488-7. 
11. ^ Darrigol, Olivier (2000). Electrodynamics from Ampère to Einstein. Oxford, [England]: Oxford University Press. pp. 126–131, 139–144. ISBN 0-19-850593-0 
12. ^ Darrigol, Olivier (2000). Electrodynamics from Ampère to Einstein. Oxford, [England]: Oxford University Press. pp. 200, 429–430. ISBN 0-19-850593-0 
13. ^ Heaviside, Oliver. "On the Electromagnetic Effects due to the Motion of Electrification through a Dielectric". Philosophical Magazine, April 1889, p. 324. http://en.wikisource.org/wiki/Motion_of_Electrification_through_a_Dielectric. 
14. ^ Darrigol, Olivier (2000). Electrodynamics from Ampère to Einstein. Oxford, [England]: Oxford University Press. p. 327. ISBN 0-19-850593-0 
15. ^ Whittaker, E. T. (1910). A History of the Theories of Aether and Electricity: From the Age of Descartes to the Close of the Nineteenth Century. Longmans, Green and Co.. pp. 420–423. ISBN 1-143-01208-9. http://books.google.com/books?id=CGJDAAAAIAAJ&printsec=frontcover#v=onepage&q&f=false. 
16. ^ See Griffiths page 326, which states that Maxwell's equations, "together with the [Lorentz] force law...summarize the entire theoretical content of classical electrodynamics".
17. ^ See, for example, Jackson p777-8.
18. ^ ^{a b} See Griffiths pages 301–3.
19. ^ Tai L. Chow (2006). Electromagnetic theory. Sudbury MA: Jones and Bartlett. p. 395. ISBN 0-7637-3827-1. http://books.google.com/?id=dpnpMhw1zo8C&pg=PA153&dq=isbn=0763738271. 
20. ^ ^{a b} Landau, L. D., Lifshit︠s︡, E. M., & Pitaevskiĭ, L. P. (1984). Electrodynamics of continuous media; Volume 8 Course of Theoretical Physics (Second ed.). Oxford: Butterworth-Heinemann. p. §63 (§49 pp. 205–207 in 1960 edition). ISBN 0-7506-2634-8. http://worldcat.org/search?q=0750626348&qt=owc_search. 
21. ^ Roger F Harrington (2003). Introduction to electromagnetic engineering. Mineola, NY: Dover Publications. p. 56. ISBN 0-486-43241-6. http://books.google.com/?id=ZlC2EV8zvX8C&pg=PA57&dq=%22faraday%27s+law+of+induction%22. 
22. ^ M N O Sadiku (2007). Elements of elctromagnetics (Fourth ed.). NY/Oxford: Oxford University Press. p. 391. ISBN 0-19-530048-3. http://books.google.com/?id=w2ITHQAACAAJ&dq=isbn:0-19-530048-3. 
23. ^ Classical Mechanics (2nd Edition), T.W.B. Kibble, European Physics Series, Mc Graw Hill (UK), 1973, ISBN 07-084018-0.
24. ^ Hestenes, David. "SpaceTime Calculus". http://geocalc.clas.asu.edu/html/STC.html. 

External links[link]

• Interactive Java tutorial on the Lorentz force National High Magnetic Field Laboratory
• Lorentz force (demonstration)
• Faraday's law: Tankersley and Mosca
• Notes from Physics and Astronomy HyperPhysics at Georgia State University; see also home page
• Interactive Java applet on the magnetic deflection of a particle beam in a homogeneous magnetic field by Wolfgang Bauer

Nikola Tesla
Tesla, aged 37, 1893, photo by Napoleon Sarony
Born	(1856-07-10)10 July 1856 Smiljan, Austrian Empire (Croatian Military Frontier)
Died	7 January 1943(1943-01-07) (aged 86) Manhattan, New York, USA
Residence	Manhattan, New York, USA Karlovac, Croatia Budapest, Hungary
Citizenship	Austrian Empire (10 July 1856 – 29 October 1918)[citation needed] United States (30 July 1891 – 7 January 1943)
Fields	Electrical engineering Mechanical engineering
Institutions	Edison Machine Works Tesla Electric Light & Manufacturing Westinghouse Electric & Manufacturing Co.
Alma mater	Graz University of Technology Charles University in Prague
Known for	Alternating current Bifilar coil Death ray Electrogravitics Induction motor Particle beam weapon Rotating magnetic field Teleforce Tesla coil Tesla electric car Tesla principle Tesla turbine Tesla's Egg of Columbus Tesla's oscillator Telegeodynamics Terrestrial stationary waves Wireless technology
Influences	Ernst Mach
Influenced	Gano Dunn
Notable awards	Order of St. Sava (1892) Elliott Cresson Medal (1894) Edison Medal (1916) John Scott Medal (1934)
Signature

Nikola Tesla (Serbian Cyrillic: Никола Тесла; 10 July 1856 – 7 January 1943) was a Serbian-American inventor, physicist, mechanical engineer, electrical engineer, and futurist. He was an important contributor to the use of commercial electricity, and is best known for developing the modern alternating current (AC) electrical supply system. His many revolutionary developments in the field of electromagnetism in the late 19th and early 20th centuries were based on the theories of electromagnetic technology discovered by Michael Faraday. Tesla's patents and theoretical work also formed the basis of wireless communication and the radio.

Born in the village of Smiljan (now part of Gospić, present day Croatia), Tesla was a subject of the Austrian Empire by birth and later became an American citizen.^[1] Because of his 1894 demonstration of short range wireless communication through radio^[2] and as the eventual victor in the "War of Currents", he was widely respected as one of the greatest electrical engineers who worked in America.^[3] He pioneered modern electrical engineering and many of his discoveries were of groundbreaking importance. In the United States during this time, Tesla's fame rivaled that of any other inventor or scientist in history or popular culture.^[4] Tesla demonstrated wireless energy transfer to power electronic devices in 1891,^[5] and aspired to intercontinental wireless transmission of industrial power in his unfinished Wardenclyffe Tower project.

Tesla became reclusive towards the end of his life, living alone in a New York City hotel room and only appearing occasionally to make unusual statements to the press.^[6][7] Because of his pronouncements and the nature of his work over the years Tesla gained a reputation in popular culture as the archetypal "mad scientist".^[8][9] He died with little money to his name.^[10]

Tesla's work fell into relative obscurity after his death but since the 1990s his reputation has experienced a popular culture comeback.^[11] In 2005 he was listed amongst the top 100 nominees in the TV show "The Greatest American", an open access popularity poll conducted by AOL and The Discovery Channel.

The SI unit measuring magnetic field B (also referred to as the magnetic flux density and magnetic induction), the tesla, was named in his honor (at the CGPM, Paris, 1960).

Early years[link]

Nikola Tesla was born to Serbian parents in the village of Smiljan, Austrian Empire near the town of Gospić, in the territory of modern-day Croatia. His baptismal certificate reports that he was born on 28 June (N.S. 10 July) 1856 to father Milutin Tesla, a priest in the Serbian Orthodox Church.^[12] His mother was Đuka Tesla, née Mandić, whose father was also a Serbian Orthodox priest.^[13] She was talented in making home craft tools and memorized many Serbian epic poems, but never learned to read.^[14] Tesla's biographer John O'Neill relates that "the Tesla and Mandić families originally came from the western part of Serbia near Montenegro."^[15]

Nikola was the fourth of five children, having one older brother (Danilo, who was killed in a horse-riding accident when Nikola was five) and three sisters (Milka, Angelina and Marica).^[16] His family moved to Gospić in 1862. Tesla attended school at Higher Real Gymnasium in Karlovac.^[17] He finished a four-year term in the span of three years.^[18]

Nikola Tesla's house (parish hall) in village Smiljan where he was born and the church where his father served (present day Croatia).

Tesla went on to study electrical engineering at the Austrian Polytechnic in Graz (1875). While there, he studied the uses of alternating current. Some sources say he received Baccalaureate degrees from the university at Graz.^[19][20][21] However, the university says that he did not receive a degree and did not continue beyond the first semester of his third year, during which he stopped attending lectures.^{[22][23][24][25]} In December 1878, Tesla left Graz and broke all relations with his family. His friends thought that he had drowned in the Mur River. He went to Marburg, (today's Maribor, in Slovenia), where he was first employed as an assistant engineer for a year. He suffered a nervous breakdown during this time. Tesla was later persuaded by his father to attend the Charles-Ferdinand University in Prague, which he attended for the summer term of 1880. Here, he was influenced by Ernst Mach. However, after his father died, he left the university, having completed only one term.^[26]

Nikola Tesla c. 1879 at age 23, and his passport from 1883

Tesla engaged in reading many works, memorizing complete books, supposedly having a photographic memory.^[27] Tesla related in his autobiography that he experienced detailed moments of inspiration. During his early life, Tesla was stricken with illness time and time again. He suffered a peculiar affliction in which blinding flashes of light would appear before his eyes, often accompanied by visions. Much of the time the visions were linked to a word or idea he might have come across, at other times they would provide the solution to a particular problem he had been encountering; just by hearing the name of an item, he would be able to envision it in realistic detail. Modern-day synesthetes report similar symptoms. Tesla would visualize an invention in his mind with extreme precision, including all dimensions, before moving to the construction stage; a technique sometimes known as picture thinking. He typically did not make drawings by hand, instead just conceiving all ideas with his mind. Tesla also often had flashbacks to events that had happened previously in his life; these began during his childhood.^[27]

In 1880, he moved to Budapest to work under Tivadar Puskás in a telegraph company,^[28] the National Telephone Company. There, he met Nebojša Petrović, a young, Serbian inventor who lived in Austria. Although their encounter was brief, they did work on a project together using twin turbines to create continual power. On the opening of the telephone exchange in Budapest, 1881, Tesla became the chief electrician to the company, and was later engineer for the country's first telephone system. He also developed a device that, according to some, was a telephone repeater or amplifier, but according to others could have been the first loudspeaker.^[29]

Drawing from U.S. Patent 381968, illustrating principle of Tesla's alternating current motor

Working for Edison in France and the U.S[link]

In 1882 he moved to Paris, to work as an engineer for the Continental Edison Company, designing improvements to electric equipment brought overseas from Edison's ideas. According to Tesla's autobiography, in the same year he conceived of his induction motor and began developing various devices that use rotating magnetic fields for which he received patents in 1888. The paternity of the invention remains controversial since a prototype induction motor was demonstrated in Europe in 1885 by Galileo Ferraris.^{[30][31][32][33]} Ferraris published his findings in 1888.

On 6 June 1884, Tesla first arrived in the United States, in New York City^[34] with little besides a letter of recommendation from Charles Batchelor, a former employer. In the letter of recommendation to Thomas Edison, it is claimed that Batchelor wrote, 'I know two great men and you are one of them; the other is this young man', but the exact contents of the letter is disputed in McNichol's book. Edison hired Tesla to work for his Edison Machine Works. Tesla's work for Edison began with simple electrical engineering and quickly progressed to solving some of the company's most difficult problems. Tesla was even offered the task of completely redesigning the Edison company's direct current generators.^[35]

In 1885 Tesla claimed he could redesign Edison's inefficient motor and generators, making an improvement in both service and economy. According to Tesla, Edison remarked "There's fifty thousand dollars in it for you - if you can do it".^[36] This has been noted as an odd statement from an Edison whose company was stingy with pay and did not have that sort of cash on hand.^[37] After months of work when Tesla finished the task and inquired about payment Edison claimed he was only joking replying, "Tesla, you don't understand our American humor".^[38][39] Edison offered a $10 a week raise over Tesla's US$18 per week salary, but Tesla refused it and immediately resigned.^[36]

Tesla, in need of work, eventually found himself digging ditches for a short period of time for the Edison company. He used this time to focus on his AC polyphase system.^[40]

Middle years[link]

Wireless transmission of power and energy demonstration during his high frequency and potential lecture of 1891 Electromechanical devices and principles developed by Nikola Tesla: Various devices that use rotating magnetic fields (1882) The Induction motor, rotary transformers, and "high" frequency alternators The Tesla coil,[41] his magnifying transmitter, and other means for increasing the intensity of electrical oscillations (including condenser discharge transformations and the Tesla oscillators[42][43]) Alternating current long-distance electrical transmission system[44] (1888) and other methods and devices for power transmission Systems for wireless communication (prior art for the invention of radio) and radio frequency oscillators[45] Robotics and the electronic logic gate[46] Electrotherapy Tesla currents[47][48][49] Wireless transfer of electricity and the Tesla effect[50][51] Tesla impedance phenonomena[52] Tesla electro-static field Tesla principle Bifilar coil Telegeodynamics Tesla insulation Tesla impulses[53] Tesla frequencies[41] Tesla discharge[41] Forms of commutators and methods of regulating third brushes Tesla turbines (e.g., bladeless turbines) for water, steam and gas and the Tesla pumps Tesla igniter Corona discharge ozone generator Tesla compressor X-rays Tubes using the Bremsstrahlung process Devices for ionized gases and "Hot Saint Elmo's Fire".[54] Devices for high field emission Devices for charged particle beams Phantom streaming devices[55] Arc light systems Methods for providing extremely low level of resistance to the passage of electric current (predecessor to superconductivity) Voltage multiplication circuitry Devices for high voltage discharges Devices for lightning protection VTOL aircraft Dynamic theory of gravity Concepts for electric vehicles Polyphase systems

In 1886, Tesla formed his own company, Tesla Electric Light & Manufacturing. The initial financial investors disagreed with Tesla on his plan for an alternating current motor and eventually relieved him of his duties at the company. Tesla worked in New York as a laborer from 1886 to 1887 to feed himself and raise capital for his next project. In 1887, he constructed a brushless alternating current induction motor, which he demonstrated to the American Institute of Electrical Engineers (now IEEE) in 1888. In the same year, he developed the principles of his Tesla coil, and began working with George Westinghouse at Westinghouse Electric & Manufacturing Company's Pittsburgh labs. Westinghouse listened to his ideas for polyphase systems which would allow transmission of alternating current electricity over long distances.

In April 1887, Tesla began investigating what would later be called X-rays using his own single terminal vacuum tubes (similar to his patent #514,170). This device differed from other early X-ray tubes in that it had no target electrode. The modern term for the phenomenon produced by this device is bremsstrahlung (or braking radiation). It is now known that this device operated by emitting electrons from the single electrode through a combination of field electron emission and thermionic emission. Once liberated, electrons are strongly repelled by the high electric field near the electrode during negative voltage peaks from the oscillating HV output of the Tesla Coil, generating X rays as they collide with the glass envelope. He also used Geissler tubes. By 1892, Tesla became aware of the skin damage that Wilhelm Röntgen later identified as an effect of X rays.

In the early research, Tesla devised several experimental setups to produce X-rays. Tesla held that, with his circuits, the "instrument will [... enable one to] generate Roentgen rays of much greater power than obtainable with ordinary apparatus".^[56]

He also commented on the hazards of working with his circuit and single-node X-ray-producing devices. Of his many notes in the early investigation of this phenomenon, he attributed the skin damage to various causes. He believed early on that damage to the skin was not caused by the Roentgen rays, but the ozone generated in contact with the skin, and to a lesser extent, nitrous acid. Tesla incorrectly held that x-rays were longitudinal waves, such as those produced in waves in plasma. There are known examples of this and these plasma waves can occur in the situation of force-free magnetic fields.^[57][58] His hypotheses and experiments were confirmed by others.^[59]

Tesla continued research in the field. He performed several experiments prior to Roentgen's discovery (including photographing the bones of his hand; later, he sent these images to Roentgen) but did not make his findings widely known; much of his research was lost in the 5th Avenue laboratory fire of March 1895.

Tesla demonstrated wireless energy transmission as early as 1891. The Tesla effect is a term for an application of this type of electrical conduction (that is, the movement of energy through space and matter, not just the production of voltage across a conductor).^[50][60]

American citizenship[link]

Tesla demonstrated a radio-controlled boat on 1898 U.S. Patent 613,809, named Method of an Apparatus for Controlling Mechanism of Moving Vehicle or Vehicles

On 30 July 1891, at the age of 35, he became a naturalized citizen of the United States. Tesla established his South Fifth Avenue laboratory in New York in the same year. Later, Tesla established his Houston Street laboratory in New York at 46 E. Houston Street. He lit electric lamps wirelessly at both of the New York locations, providing evidence for the potential of wireless power transmission.^[61]

Soon thereafter, in 1892,^[62] Tesla was awakened from a dream in which his mother had died.^[63] He returned to Europe for her funeral. After her death, Tesla fell ill. He spent two to three weeks recuperating in Gospić and the village of Tomingaj near Gračac, his mother's birthplace.

Some of Tesla's closest friends were artists. He befriended Century Magazine editor Robert Underwood Johnson, who adapted several Serbian poems of Jovan Jovanović Zmaj (which Tesla translated). Also during this time, Tesla was influenced by the Vedic philosophy (i.e., Hinduism) teachings of the Swami Vivekananda; so much so that, after his exposure to Hindu-Vedic thought, Tesla started using Sanskrit words to name some of his fundamental concepts regarding matter and energy.^[64]

Nikola Tesla's AC dynamo-electric machine (Electric generator) used to generate AC which is used to transport electricity across great distances. It is contained in U.S. Patent 390,721.

When Tesla was 36 years old, the first patents concerning the polyphase power system were granted. He continued research of the system and rotating magnetic field principles. Tesla served, from 1892 to 1894, as the vice president of the American Institute of Electrical Engineers, the forerunner (along with the Institute of Radio Engineers) of the modern-day IEEE. From 1893 to 1895, he investigated high frequency alternating currents. He generated AC of one million volts using a conical Tesla coil and investigated the skin effect in conductors, designed tuned circuits, invented a machine for inducing sleep, invented a cordless gas discharge lamp, and transmitted electromagnetic energy without wires, building the first radio transmitter.^{[citation needed]} In St. Louis, Missouri, Tesla made a demonstration related to radio communication in 1893.^{[citation needed]} Addressing the Franklin Institute in Philadelphia, Pennsylvania and the National Electric Light Association, he described and demonstrated in detail its principles. Tesla's demonstrations were written about widely through various media outlets. Tesla also investigated harvesting energy that is present throughout space. He believed that it was merely a question of time when men would succeed in attaching their machinery to the very wheelwork of nature, stating: "Ere many generations pass, our machinery will be driven by a power obtainable at any point of the universe."^[65]

At the 1893 World's Fair, the World's Columbian Exposition in Chicago, an international exposition was held which, for the first time, devoted a building to electrical exhibits. It was a historic event as Tesla and George Westinghouse introduced visitors to AC power by using it to illuminate the Exposition. On display were fluorescent lamps developed by Westinghouse^[66] and single node bulbs. An observer noted:

Within the room was suspended two hard-rubber plates covered with tin foil. These were about fifteen feet apart, and served as terminals of the wires leading from the transformers. When the current was turned on, the lamps or tubes, which had no wires connected to them, but lay on a table between the suspended plates, or which might be held in the hand in almost any part of the room, were made luminous. These were the same experiments and the same apparatus shown by Tesla in London about two years previous, "where they produced so much wonder and astonishment".^[67]

Tesla also explained the principles of the rotating magnetic field and induction motor by demonstrating how to make an egg made of copper stand on end in his demonstration of the device he constructed known as the "Egg of Columbus".

The Tesla generator was developed by Tesla in 1895, in conjunction with his developments concerning the liquefaction of air. Tesla knew from Lord Kelvin's discoveries that more heat is absorbed by liquefied air when it is re-gasified and used to drive something than is required by theory; in other words, that the liquefaction process is somewhat anomalous or 'over unity'.^[68] Just before Tesla's completion of his work and the filing of a patent application, Tesla's laboratory burned down, destroying all his equipment, models and inventions. Immediately after the fire, Carl von Linde, in Germany, filed a patent application for the same process.^[69]

A "world system" for "the transmission of electrical energy without wires" that depends upon the electrical conductivity of the earth was proposed, in which transmission in various natural media with current that passes between the two points are used to power devices. In a practical wireless energy transmission system using this principle, a high-power ultraviolet beam might be used to form a vertical ionized channel in the air directly above the transmitter-receiver stations.^{[citation needed]} The same concept is used in virtual lightning rods and the electrolaser electroshock weapon,^[70] and has been proposed for disabling vehicles.

Also in the late 1880s, Tesla and Thomas Edison became adversaries in part because of Edison's promotion of direct current (DC) for electric power distribution over the more efficient alternating current advocated by Tesla and Westinghouse. Until the development of the induction motor, AC's advantages for long distance high voltage transmission were counterbalanced by the inability to operate motors on AC.^{[citation needed]} As a result of the "War of Currents", Edison and Westinghouse went nearly bankrupt, so in 1897, Tesla released Westinghouse from contract, providing Westinghouse a break from Tesla's patent royalties. Also in 1897, Tesla researched radiation, which led to setting up the basic formulation of cosmic rays.^[71]

In 1897, at age 41, Tesla filed the first radio patent (U.S. Patent 645,576). A year later, he demonstrated a radio-controlled boat to the US military, believing that the military would want things such as radio-controlled torpedoes. Tesla claimed to have developed the "Art of Telautomatics", a form of robotics, as well as the technology of remote control.^[72] In 1898, he demonstrated a radio-controlled boat to the public during an electrical exhibition at Madison Square Garden. Tesla called his boat a "teleautomaton".^[73] In the same year, Tesla devised an "electric igniter" or spark plug for Internal combustion gasoline engines. He gained U.S. Patent 609,250, "Electrical Igniter for Gas Engines", on this mechanical ignition system. Tesla lived in the former Gerlach Hotel, renamed The Radio Wave building, at 49 W 27th St. (between Broadway and Sixth Avenue), Lower Manhattan, before the end of the century where he conducted the radio wave experiments. A commemorative plaque was placed on the building in 1977 to honor his work. Remote radio control remained a novelty until World War I and afterward, when a number of countries used it in military programs.

Colorado Springs[link]

Publicity picture of Nikola Tesla sitting in his laboratory in Colorado Springs with his "Magnifying transmitter" generating millions of volts. The arcs are about 7 meters (23 ft) long. (Tesla's notes identify this as a multiple exposure photograph.)

An experiment in Colorado Springs. This bank of lights is receiving power by means of electrodynamic induction from a nearby transmitter

A Colorado Springs experiment: here a grounded tuned coil in resonance with a distant transmitter illuminates a light near the bottom of the picture.

In 1899, Tesla decided to move and began research in Colorado Springs, Colorado in a lab located near Foote Ave. and Kiowa St.,^[74] where he would have room for his high-voltage, high-frequency experiments. Upon his arrival he told reporters that he was conducting wireless telegraphy experiments transmitting signals from Pikes Peak to Paris. Tesla's diary contains explanations of his experiments concerning the ionosphere and the ground's telluric currents via transverse waves and longitudinal waves.^[75] At his lab, Tesla proved that the earth was a conductor, and he produced artificial lightning (with discharges consisting of millions of volts, and up to 135 feet long).^[76] Tesla also investigated atmospheric electricity, observing lightning signals via his receivers. Reproductions of Tesla's receivers and coherer circuits show an unpredicted level of complexity (e.g., distributed high-Q helical resonators, radio frequency feedback, crude heterodyne effects, and regeneration techniques).^[77] Tesla stated that he observed stationary waves during this time.^[78]

Tesla researched ways to transmit power and energy wirelessly over long distances (via transverse waves, to a lesser extent, and, more readily, longitudinal waves). He transmitted extremely low frequencies through the ground as well as between the Earth's surface and the Kennelly–Heaviside layer. He received patents on wireless transceivers that developed standing waves by this method. In his experiments, he made mathematical calculations and computations based on his experiments and discovered that the resonant frequency of the Earth was approximately 8 hertz (Hz). In the 1950s, researchers confirmed that the resonant frequency of the Earth's ionospheric cavity was in this range (later named the Schumann resonance).

In Colorado Springs Tesla carried out various long distance wireless transmission-reception experiments. Tesla effect is the application of a type of electrical conduction (that is, the movement of energy through space and matter; not just the production of voltage across a conductor). Through longitudinal waves,^{[citation needed]} Tesla transferred energy to receiving devices. He sent electrostatic forces through natural media across a conductor situated in the changing magnetic flux and transferred electrical energy to a wireless receiver.

In the Colorado Springs lab, Tesla observed unusual signals that he later thought may have been evidence of extraterrestrial radio wave communications coming from Venus or Mars.^[79] He noticed repetitive signals from his receiver which were substantially different from the signals he had noted from storms and earth noise. Specifically, he later recalled that the signals appeared in groups of one, two, three, and four clicks together. Tesla had mentioned that he thought his inventions could be used to talk with other planets. There have even been claims that he invented a "Teslascope" for just such a purpose. It is debatable what type of signals Tesla received or whether he picked up anything at all. Research has suggested that Tesla may have had a misunderstanding of the new technology he was working with,^[80] or that the signals Tesla observed may have been non-terrestrial natural radio source such as the Jovian plasma torus signals.^[81]

Tesla left Colorado Springs on 7 January 1900. The lab was torn down ca. 1905 and its contents sold to pay debts. The Colorado experiments prepared Tesla for the establishment of the trans-Atlantic wireless telecommunications facility known as Wardenclyffe near Shoreham, Long Island.

Wardenclyffe years[link]

In 1900, with US$150,000 (51% from J. Pierpont Morgan), Tesla began planning the Wardenclyffe Tower facility. In June 1902, Tesla's lab operations were moved to Wardenclyffe from Houston Street. The tower was dismantled for scrap during World War I.^[82]

On his 50th birthday in 1906, Tesla demonstrated his 200 hp (150 kW) 16,000 rpm bladeless turbine. During 1910–1911 at the Waterside Power Station in New York, several of his bladeless turbine engines were tested at 100–5000 hp.

Nobel prize and Tesla[link]

Since the Nobel Prize in Physics was awarded to Guglielmo Marconi for radio in 1909, Thomas Edison and Tesla were mentioned in a press dispatch as potential laureates to share the Nobel Prize of 1915, leading to one of several Nobel Prize controversies. Some sources have claimed that because of their animosity toward each other neither was given the award, despite their scientific contributions; that each sought to minimize the other's achievements and right to win the award; that both refused ever to accept the award if the other received it first; and that both rejected any possibility of sharing it.^[14][83]

In the years after these rumors, neither Tesla nor Edison won the Prize (although Edison did receive one of 38 possible bids in 1915, and Tesla did receive one bid out of 38 in 1937).^[84] Earlier, Tesla alone was rumored to have been nominated for the Nobel Prize of 1912. The rumored nomination was primarily for his experiments with tuned circuits using high-voltage high-frequency resonant transformers.

Later years[link]

In 1915, Tesla filed a lawsuit against Marconi attempting, unsuccessfully, to obtain a court injunction against Marconi's claims. After Wardenclyffe, Tesla built the Telefunken Wireless Station in Sayville, Long Island. Some of what he wanted to achieve at Wardenclyffe was accomplished with the Telefunken Wireless. In 1917, the facility was seized and torn down by the Marines, because it was suspected that it could be used by German spies.

The Wardenclyffe Tower facility

Before World War I (1914-1918), Tesla looked overseas for investors to fund his research. When the war started, Tesla lost the funding he was receiving from his patents in European countries. After the war ended, Tesla made predictions regarding the relevant issues of the post-World War I environment, in a printed article (20 December 1914). Tesla believed that the League of Nations was not a remedy for the times and issues. Tesla started to exhibit pronounced symptoms of obsessive-compulsive disorder in the years following. He became obsessed with the number three; he often felt compelled to walk around a block three times before entering a building, demanded a stack of three folded cloth napkins beside his plate at every meal, etc. The nature of OCD was little understood at the time and no treatments were available, so his symptoms were considered by some to be evidence of partial insanity, and this undoubtedly hurt what was left of his reputation.

At this time, he was staying at The Waldorf-Astoria Hotel, renting in an arrangement for deferred payments. Eventually, the Wardenclyffe deed was turned over to George Boldt, proprietor of the Waldorf-Astoria, to pay a US$20,000 debt. In 1917, around the time that the Wardenclyffe Tower was demolished by Boldt to make the land a more viable real estate asset, Tesla received AIEE's highest honor, the Edison Medal.

Tesla, in August 1917, first established principles regarding frequency and power level for the first primitive radar units.^[85]

In 1934, Émile Girardeau, working with the first French radar systems, stated he was building them "according to the principles stated by Tesla". By the 1920s, Tesla was reportedly negotiating with the United Kingdom government about a ray system.^{[citation needed]} Tesla had also stated that efforts had been made to steal the so called "death ray".^{[citation needed]} It is suggested that the removal of the Chamberlain government ended negotiations.^{[citation needed]}

On Tesla's 75th birthday in 1931, Time magazine put him on its cover. The cover caption noted his contribution to electrical power generation. Tesla received his last patent in 1928 for an apparatus for aerial transportation which was the first instance of VTOL aircraft. By the end of 1931, Tesla released "On Future Motive Power" which covered an ocean thermal energy conversion system. In 1934, Tesla wrote to consul Janković of his homeland. The letter contained a message of gratitude to Mihajlo Pupin who had initiated a donation scheme by which American companies could support Tesla. Tesla refused the assistance, choosing instead to live on a modest pension received from Yugoslavia, and to continue his research.

Tesla's famous telegram exchange with Vladko Maček is preserved in the Technical Museum in Zagreb, Croatia

In 1936, replying to a birthday telegram from Vladko Maček, Tesla said he was "equally proud" of his "Serbian origin and Croatian homeland",^[86] a phrase often paraphrased in conciliatory context at modern-day joint Croatian-Serbian Tesla celebrations.^[87] In addition, in the same telegram, Tesla wrote "Long live all Yugoslavs."^{[citation needed]} When others tried to co-opt him into ethnic and other conflicts in Yugoslavia, Tesla once replied: "If your hate could be turned into electricity, it would light up the whole world."^[86]

Field theories[link]

When he was 81, Tesla stated he had completed a "dynamic theory of gravity".^[88] He stated that it was "worked out in all details" and that he hoped to soon give it to the world.^[89] The theory was never published.

The bulk of the theory was developed between 1892 and 1894, during the period that he was conducting experiments with high frequency and high potential electromagnetism and patenting devices for their use. Reminiscent of Mach's principle, Tesla stated in 1925 that:

Nikola Tesla, with Ruđer Bošković's book Theoria Philosophiae Naturalis, sits in front of the spiral coil of his high-frequency transformer at East Houston Street, New York.

There is no thing endowed with life—from man, who is enslaving the elements, to the nimblest creature—in all this world that does not sway in its turn. Whenever action is born from force, though it be infinitesimal, the cosmic balance is upset and the universal motion results.

Tesla was critical of Einstein's relativity work, calling it:

...[a] magnificent mathematical garb which fascinates, dazzles and makes people blind to the underlying errors. The theory is like a beggar clothed in purple whom ignorant people take for a king ... its exponents are brilliant men but they are metaphysicists rather than scientists ...^[90]

Tesla also argued:

I hold that space cannot be curved, for the simple reason that it can have no properties. It might as well be said that God has properties. He has not, but only attributes and these are of our own making. Of properties we can only speak when dealing with matter filling the space. To say that in the presence of large bodies space becomes curved is equivalent to stating that something can act upon nothing. I, for one, refuse to subscribe to such a view.^[91]

Tesla also believed that much of Albert Einstein's relativity theory had already been proposed by Ruđer Bošković, stating in an unpublished interview:

...the relativity theory, by the way, is much older than its present proponents. It was advanced over 200 years ago by my illustrious countryman Ruđer Bošković, the great philosopher, who, notwithstanding other and multifold obligations, wrote a thousand volumes of excellent literature on a vast variety of subjects. Bošković dealt with relativity, including the so-called time-space continuum ...'.^[92]

Directed-energy weapon[link]

Later in life, Tesla made remarkable claims concerning a "teleforce" weapon.^[93] The press called it a "peace ray" or death ray.^[94][95] In total, the components and methods included:^{[96][97][dubious – discuss]}

• An apparatus for producing manifestations of energy in free air instead of in a high vacuum as in the past. This, according to Tesla in 1934, was accomplished.
• A mechanism for generating tremendous electrical force. This, according to Tesla, was also accomplished.
• A means of intensifying and amplifying the force developed by the second mechanism.
• A new method for producing a tremendous electrical repelling force. This would be the projector, or gun, of the invention.

Tesla worked on plans for a directed-energy weapon from the early 1900s until his death.^{[dubious – discuss]} In 1937, Tesla wrote a treatise entitled "The Art of Projecting Concentrated Non-dispersive Energy through the Natural Media", which concerned charged particle beams.^{[98][dubious – discuss]} Tesla published the document in an attempt to expound on the technical description of a "superweapon that would put an end to all war." This treatise describing the particle beam is currently in the Nikola Tesla Museum archive in Belgrade. It describes an open-ended vacuum tube with a gas jet seal that allows particles to exit, a method of charging particles to millions of volts, and a method of creating and directing nondispersive particle streams (through electrostatic repulsion).^{[98][dubious – discuss]}

His records indicate that the device is based on a narrow stream of atomic clusters of liquid mercury or tungsten accelerated via high voltage (by means akin to his magnifying transformer). Tesla gives the following description concerning the particle gun's operation:

[The nozzle would] send concentrated beams of particles through the free air, of such tremendous energy that they will bring down a fleet of 10,000 enemy airplanes at a distance of 200 miles from a defending nation's border and will cause armies to drop dead in their tracks.^[99]

Tesla described the weapon as being able to be used against ground based infantry or for antiaircraft purposes.^[100] Tesla tried to interest the US War Department in the device.^[101] He also offered this invention to European countries.^[102] None of the governments purchased a contract to build the device. He was unable to act on his plans.^[103]

Theoretical inventions[link]

Another of Tesla's theorized inventions is commonly referred to as Tesla's Flying Machine, which appears to resemble an ion-propelled aircraft.^[104] Tesla claimed that one of his life goals was to create a flying machine that would run without the use of an airplane engine, wings, ailerons, propellers, or an onboard fuel source. Initially, Tesla pondered about the idea of a flying craft that would fly using an electric motor powered by grounded base stations. As time progressed, Tesla suggested that perhaps such an aircraft could be run entirely electro-mechanically. The theorized appearance would typically take the form of a cigar or saucer.^[105]

Personal life[link]

Nikola Tesla's father Milutin, Serbian Orthodox priest in the village of Smiljan.

Possessing an eidetic memory, Nikola Tesla was a polyglot, and along with his native tongue he also spoke Czech, English, French, German, Hungarian, Italian, and Latin.^[106]

During his second year of study at Graumltz, Tesla developed a passion for (and became very proficient at) billiards, chess and card-playing, sometimes spending more than 48 hours in a stretch at a gaming table.^[107][108] Tesla by nature required little sleep, claiming to never sleep more than two hours.^[109] On one occasion at his laboratory Tesla worked for a period of 84 hours without sleep or rest.^[110]

Tesla may have suffered from obsessive-compulsive disorder, and had many unusual quirks and phobias. He did things in threes, and was adamant about staying in a hotel room with a number divisible by three. Tesla was physically revolted by jewelry, notably pearl earrings.^[111] He was fastidious about cleanliness and hygiene, and was by all accounts mysophobic.

Tesla was obsessed with pigeons, ordering special seeds for the pigeons he fed in Central Park and even bringing injured ones into his hotel room to nurse them back to health. Tesla was an animal-lover, often reflecting contentedly about a childhood cat, "The Magnificent Mačak." Tesla never married. He was celibate and claimed that his chastity was very helpful to his scientific abilities.^[27] Nonetheless there have been numerous accounts of women vying for Tesla's affection, even some madly in love with him. Tesla, though polite, behaved rather ambivalently to these women in the romantic sense.

Tesla was prone to alienating himself and was generally soft-spoken. However, when he did engage in a social life, many people spoke very positively and admiringly of him. Robert Underwood Johnson described him as attaining a "distinguished sweetness, sincerity, modesty, refinement, generosity, and force." His loyal secretary, Dorothy Skerrit, wrote: "his genial smile and nobility of bearing always denoted the gentlemanly characteristics that were so ingrained in his soul." Tesla's friend Hawthorne wrote that "seldom did one meet a scientist or engineer who was also a poet, a philosopher, an appreciator of fine music, a linguist, and a connoisseur of food and drink."

Nevertheless, Tesla could be harsh at times; he openly expressed his disgust for overweight people, once firing a secretary because of her weight.^[112] He was quick to criticize others' clothing as well, on several occasions directing a subordinate to go home and change her dress.^[27]

Tesla was widely known for his great showmanship, presenting his innovations and demonstrations to the public as an artform, almost like a magician. This seems to conflict with his observed reclusiveness; Tesla was a complicated figure. He refused to hold conventions without his Tesla coil blasting electricity throughout the room, despite the audience often being terrified, though he assured them everything was perfectly safe.

Mark Twain in Tesla's lab, early 1894

In middle age, Tesla became close friends with Mark Twain. They spent a lot of time together in his lab and elsewhere.

Tesla remained bitter in the aftermath of his dispute with Edison. The day after Edison died the New York Times contained extensive coverage of Edison's life, with the only negative opinion coming from Tesla, who was quoted as saying:

He had no hobby, cared for no sort of amusement of any kind and lived in utter disregard of the most elementary rules of hygiene  ... His method was inefficient in the extreme, for an immense ground had to be covered to get anything at all unless blind chance intervened and, at first, I was almost a sorry witness of his doings, knowing that just a little theory and calculation would have saved him 90 percent of the labor. But he had a veritable contempt for book learning and mathematical knowledge, trusting himself entirely to his inventor's instinct and practical American sense.^[113]

Shortly before he died, Edison said that his biggest mistake had been in trying to develop direct current, rather than the superior alternating current system that Tesla had put within his grasp.^[114]

Tesla was good friends with Robert Underwood Johnson. He had amicable relations with Francis Marion Crawford, Stanford White, Fritz Lowenstein, George Scherff, and Kenneth Swezey. He ripped up a Westinghouse contract that would have made him the world's first billionaire, in part because of the implications it would have on his future vision of free power, and in part because it would run Westinghouse out of business, and Tesla had no desire to deal with the creditors.^{[citation needed]}

Tesla lived the last ten years of his life in a two-room suite on the 33rd floor of the Hotel New Yorker, room 3327. There, near the end of his life, Tesla showed signs of encroaching senility, claiming to be visited by a specific white pigeon daily. Several biographers note that Tesla viewed the death of the pigeon as a "final blow" to himself and his work.

Tesla believed that war could not be avoided until the cause for its recurrence was removed, but was opposed to wars in general. However, Tesla came to find exceptions in which he thought certain situations and wars were justifiable.^[115] Tesla sought to reduce distance, such as in communication for better understanding, transportation, and transmission of energy, as a means to ensure friendly international relations.^[116]

Tesla was a life-long bachelor. Like many of his era, he became a proponent of an imposed selective breeding version of eugenics. His opinions stemmed from a belief that humans already interfered with the natural "ruthless workings of nature", rather than conceptions of a "master race" or inherent superiority of one person over another. His advocacy of it was however to push it further. In a 1937 interview, he stated:

... man's new sense of pity began to interfere with the ruthless workings of nature. The only method compatible with our notions of civilization and the race is to prevent the breeding of the unfit by sterilization and the deliberate guidance of the mating instinct .... The trend of opinion among eugenists is that we must make marriage more difficult. Certainly no one who is not a desirable parent should be permitted to produce progeny. A century from now it will no more occur to a normal person to mate with a person eugenically unfit than to marry a habitual criminal.^[117]

In 1926, Tesla commented on the ills of the social subservience of women and the struggle of women toward gender equality, indicated that humanity's future would be run by "Queen Bees". He believed that women would become the dominant sex in the future.^[118]

In his later years Tesla became a vegetarian. In an article for Century Illustrated Magazine he wrote: "It is certainly preferable to raise vegetables, and I think, therefore, that vegetarianism is a commendable departure from the established barbarous habit." Tesla argued that it is wrong to eat uneconomic meat when large numbers of people are starving; he also believed that plant food was "superior to [meat] in regard to both mechanical and mental performance". He also argued that animal slaughter was "wanton and cruel".^[119]

In his final years he suffered from extreme sensitivity to light, sound and other influences.^[120]

Death[link]

This article's references may not meet Wikipedia's guidelines for reliable sources. Please help by checking whether the references meet the criteria for reliable sources. (January 2012)