Magnetic field formulation!

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Hi,
Why the magnetic field's formula is B=k2i/r.Where does come from?Please prove to me.Thanks.
 
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collinsmark
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Hi,
Why the magnetic field's formula is B=k2i/r.Where does come from?Please prove to me.Thanks.
There's a couple of ways to derive the equation for the magnetic field of a long wire. Ampère's Law is probably the easiest way.

Ampère's Law states

[tex] \oint _C \vec B \cdot \vec {dl} = \mu _0 I _{enc} [/tex]

Where where the integral involves any arbitrary closed path (meaning the path must start and end at the same point -- in other words a loop). [tex] \mu _0 [/tex] is the permeability of free space, and Ienc is the current flowing through the loop.

Consider a very long wire with current flowing through it. Now imagine tracing out a hoop around the wire, such that the wire passes through the middle of the hoop. The hoop as a radius r.

Now we can solve Ampère's Law in cylindrical coordinates. Note that in spherical coordinates, [tex] dl = rd \phi [/tex]. Putting this together gives us

[tex] \int _0 ^{2 \pi} \vec B \cdot r\vec {d \phi} = \mu _0 I _{enc} [/tex]

Note that r is a constant, here. Also note that the magnitude of B is also a constant due to cylindrical symmetry. So if we work with just the magnitudes, we can pull B and r out from under the integral.

[tex] Br \int _0 ^{2 \pi} d \phi = \mu _0 I _{enc} [/tex]

[tex] = 2 \pi r B = \mu _0 I _{enc} [/tex]

[tex] B = \frac{\mu _0}{2 \pi}\frac{I}{ r} [/tex]
 
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