Magnetic flux (most often denoted as Φ_m), is a measure of the amount of magnetic B field (also called "magnetic flux density") passing through a given surface (such as a conducting coil). The SI unit of magnetic flux is the weber (Wb) (in derived units: volt-seconds). The CGS unit is the maxwell.

Description

The magnetic flux through a given surface is proportional to the number of magnetic B field lines that pass through the surface. This is the ''net'' number, i.e. the number passing through in one direction, minus the number passing through in the other direction. (See below for how the positive sign is chosen.) For a uniform magnetic field B passing through a perpendicular area the magnetic flux is given by the product of the magnetic field and the area element. The magnetic flux for a uniform B at any angle to a surface is defined by a dot product of the magnetic field and the area element vector a.

:$\backslash displaystyle\; \backslash Phi\_m\; =\; \backslash mathbf\{B\}\; \backslash cdot\; \backslash mathbf\{a\}\; =\; Ba\; \backslash cos\; \backslash theta$    (uniform B with flat area only)

where θ is the angle between B and a vector a that is perpendicular (normal) to the surface.

In the general case, the magnetic flux through a surface ''S'' is defined as the integral of the magnetic field over the area of the surface (See Figures 1 and 2):

:$\backslash Phi\_m\; =\; \backslash iint\backslash limits\_S\; \backslash mathbf\{B\}\; \backslash cdot\; d\backslash mathbf\; S,$

where $\backslash textstyle\; \backslash Phi\_m\; \backslash$ is the magnetic flux, ''B'' is the magnetic field, :''S'' is the surface (area),$\backslash cdot$ denotes dot product, and d''S'' is an infinitesimal vector, whose magnitude is the area of a differential element of ''S'', and whose direction is the surface normal. (See surface integral for more details.)

From the definition of the magnetic vector potential A and the fundamental theorem of the curl the magnetic flux may also be defined as:

:$\backslash Phi\_m\; =\; \backslash oint\backslash limits\_\{\backslash Sigma\}\; \backslash mathbf\{A\}\; \backslash cdot\; d\backslash boldsymbol\{\backslash ell\}$

where the closed line integral is over the boundary of the surface and dℓ is an infinitesimal vector element of that contour Σ.

The magnetic flux is usually measured with a fluxmeter. The fluxmeter contains measuring coils and electronics that evaluates the change of voltage in the measuring coils to calculate the magnetic flux.

Magnetic flux through a closed surface

Gauss's law for magnetism, which is one of the four Maxwell's equations, states that the total magnetic flux through a closed surface is equal to zero. (A "closed surface" is a surface that completely encloses a volume(s) with no holes.) This law is a consequence of the empirical observation that magnetic monopoles have never been found.

In other words, Gauss's law for magnetism is the statement:

:$\backslash Phi\_m=\backslash int\; \backslash !\backslash !\backslash !\; \backslash int\; \backslash mathbf\{B\}\; \backslash cdot\; d\backslash mathbf\; S\; =\; 0,$ for any closed surface ''S''.

Magnetic flux through an open surface

:$\backslash mathcal\{E\}\; =\; \backslash oint\_\{\backslash partial\; \backslash Sigma\; (t)\}\backslash left(\; \backslash mathbf\{E\}(\; \backslash mathbf\{r\},\backslash \; t)\; +\backslash mathbf\{\; v\; \backslash times\; B\}(\backslash mathbf\{r\},\backslash \; t)\backslash right)\; \backslash cdot\; d\backslash boldsymbol\{\backslash ell\}\; =\; -\{d\backslash Phi\_m\; \backslash over\; dt\},$

where (see Figure 3): :''$\backslash mathcal\{E\}$'' is the EMF, :Φ_m is the flux through a surface with an opening bounded by a curve ''∂Σ''(''t''), :''∂Σ''(''t'') is a closed contour that can change with time; the EMF is found around this contour, and the contour is a boundary of the surface over which Φ_m is found, :dℓ is an infinitesimal vector element of the contour ∂Σ(''t''), :''v'' is the velocity of the segment dℓ, :''E'' is the electric field, :''B'' is the magnetic field.

The EMF is determined in this equation in two ways: first, as the work per unit charge done against the Lorentz force in moving a test charge around the (possibly moving) closed curve ''∂Σ''(''t''), and second, as the magnetic flux through the open surface ''Σ''(''t'').

This equation is the principle behind an electrical generator.

Comparison with electric flux

By way of contrast, Gauss's law for electric fields, another of Maxwell's equations, is

:$\backslash Phi\_E\; =\; \backslash int\; \backslash !\backslash !\backslash !\backslash int\_S\; \backslash mathbf\{E\}\backslash cdot\; d\backslash mathbf\{S\}\; =\; \{Q\; \backslash over\; \backslash epsilon\_0\},$

where :''E'' is the electric field, :''S'' is any closed surface, :''Q'' is the total electric charge inside the surface ''S'', :$\backslash epsilon\_0$ is the electric constant (a universal constant, also called the "permittivity of free space").

Note that the flux of ''E'' through a closed surface is ''not'' always zero; this indicates the presence of electric "monopoles", that is, free positive or negative charges.

See also

External articles

Category:Physical quantities Category:Magnetism

"The Christmas Song" (commonly subtitled "Chestnuts Roasting on an Open Fire" or, as it was originally subtitled, "Merry Christmas to You") is a classic Christmas song written in 1944 by musician, composer, and vocalist Mel Tormé (aka The Velvet Fog), and Bob Wells. According to Tormé, the song was written during a blistering hot summer. In an effort to "stay cool by thinking cool", the most-performed (according to BMI) Christmas song was born.

"I saw a spiral pad on his piano with four lines written in pencil", Tormé recalled. "They started, "Chestnuts roasting..., Jack Frost nipping..., Yuletide carols..., Folks dressed up like Eskimos.' Bob (Wells, co-writer) didn't think he was writing a song lyric. He said he thought if he could immerse himself in winter he could cool off. Forty minutes later that song was written. I wrote all the music and some of the lyrics."

The Nat King Cole Trio first recorded the song early in 1946. At Cole's behest and over the objections of his label, Capitol Records a second recording was made the same year utilizing a small string section, this version becoming a massive hit on both the pop and R&B; charts. Cole again recorded the song in 1953, using the same arrangement with a full orchestra arranged and conducted by Nelson Riddle, and once more in 1961, in a stereophonic version with orchestra conducted by Ralph Carmichael. Nat King Cole's 1961 version is generally regarded as definitive, and in 2004 was the most loved seasonal song with women aged 30–49, while Cole's original 1946 recording was inducted into the Grammy Hall of Fame in 1974. Mel Tormé recorded the song himself in 1954, and again in 1961, 1966 and 1992.

The Nat King Cole recordings

Second recording: Recorded at WMCA Radio Studios, New York City, August 19, 1946. First record issue. Label credit: The King Cole Trio with String Choir (Nat King Cole, vocal-pianist, Oscar Moore, guitarist; Johnny Miller, bassist; Charlie Grean, conductor of 4 string players, a harpist and a drummer) Lacquer disc master #981. Issued November 1946 as Capitol 311 (78rpm). This is featured on a CD called ''The Holiday Album'', which has 1940s Christmas songs recorded by Cole and Bing Crosby. In 2005 Capitol restored and re-released it for the 25 bit re-mastered Cole album "The Christmas Song", which also contains tracks from his 1960 and 1963 holiday albums.

Third recording: Recorded at Capitol Studios, Hollywood, August 24, 1953. This was the song,s first magnetic tape recording. Label credit: The King Cole Trio with String Choir (Actual artists: Nat King Cole, vocal; Nelson Riddle, orchestra conductor) Master #11726, take 11. Issued November 1953 as the "new" Capitol 90036(78rpm) / F90036(45rpm) (Capitol first issued 90036 in 1950 with the second recording). Correct label credit issued on October 18, 1954 as Capitol 2955(78rpm) / F2955(45rpm). Label credit: Nat "King" Cole with Orchestra Conducted by Nelson Riddle. This recording is available on the 1990 CD ''Cole, Christmas and Kids,'' as well as the various-artists compilation ''Casey Kasem Presents All Time Christmas Favorites''. It was also included, along with both 1946 recordings, on the 1991 Mosaic Records box set ''The Complete Capitol Recordings of the Nat King Cole Trio''.

Fourth recording: Recorded at Capitol Studios, New York City, March 30, 1961. This rendition, the first recorded in stereo, is widely played on radio stations during the Christmas season, and is probably the most famous version of this song. Label credit: Nat King Cole (Nat King Cole, vocal; Charles Grean and Pete Rugolo, orchestration; Ralph Carmichael, orchestra conductor). The instrumental arrangement is nearly identical with the 1953 version, but the vocals are much deeper and more focused. Originally done for ''The Nat King Cole Story'' (a 1961 LP devoted to stereo re-recordings of Cole's earlier hits), this recording was later appended to a reissue of Cole's 1960 holiday album ''The Magic of Christmas''. Retitled ''The Christmas Song'', the album was issued in 1963 as Capitol W-1967(mono) / SW-1967(stereo) and today is in print on compact disc. This recording of "The Christmas Song" is also available on numerous compilation albums. Some are Capitol pop standards Christmas compilations while others are broader-based. It's available on WCBS-FM's ''Ultimate Christmas Album Volume 3'', for example.

There were several covers of Nat Cole's original record in the 1940s. The first of these was said to be by Dick Haymes on the Decca label, but his was released first not recorded first. The first cover of "The Christmas Song" was performed by pop tenor and bandleader Eddy Howard on Majestic. Howard was a big Cole fan, and also covered Nat's versions of "I Want to Thank Your Folks" and "I Love You for Sentimental Reasons", among others.

Selective list of notable recordings

Parodies

Footnotes

External links

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