What Determines the Formula for a Magnetic Field Around a Wire?

[coki2000]

Hi,
Why the magnetic field's formula is B=k2i/r.Where does come from?Please prove to me.Thanks.

 

[collinsmark]

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

$$\oint _C \vec B \cdot \vec {dl} = \mu _0 I _{enc}$$

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). $$\mu _0$$ is the permeability of free space, and I_enc 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, $$dl = rd \phi$$. Putting this together gives us

$$\int _0 ^{2 \pi} \vec B \cdot r\vec {d \phi} = \mu _0 I _{enc}$$

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.

$$Br \int _0 ^{2 \pi} d \phi = \mu _0 I _{enc}$$

$$= 2 \pi r B = \mu _0 I _{enc}$$

$$B = \frac{\mu _0}{2 \pi}\frac{I}{ r}$$