# Magnetic field generated by a metal plate

1. May 19, 2005

### tiagobt

How do I calculate the magnetic field genereated by a very long metal plate with width w and current i flowing along the direction of the largest dimension? If I calculate the intesity of the magnetic field in a point with distance b from the border of the metal plate, I get:

$$B = \frac {\mu_0 i} {2 \pi w} \ln \left(1 + \frac w b \right)$$

(using Biot Savart Law to a long wire integrated over the area)

But I saw another solution for this problem with a different answer. This time, Ampere's Law was used along a square path where current flows, getting the following answer:

$$B = \frac {\mu_0 i} {2w}$$

My questions are:

1. Why are the answers different?
2. What does the second answer mean, i.e., in which point of space is that field intensity valid? I don't understand how he could not have used a point to calculate the field.

Thanks for the help

2. May 19, 2005

### OlderDan

I think the second result is only valid at the surfaces of the metal plate, and relatively near the middle of the surface. Construct a rectangular Ampere's law path with legs parallel to the plate just outside the two surfaces, with the area of the loop perpendicular to the current. By symmetry, the field in the middle of the plate, half way between the two surfaces will be zero, and it will be symmetrical with respect to a plane through the middle, parallel to the wide surfaces. The only net contributions to the Ampere's Law path integral will be from the outer legs of length L << w. The contained current will be the linear current density i/2w times the two external legs (2L), and the path integral of B will be 2LB. By Ampere's law that will give

$$2LB = \mu_0 \frac{i}{2w}2L$$

$$B = \frac{\mu_0 i}{2w}$$

If you preserve the legs of length L and extend to loop to distances far removed from the surface you will have long legs of path being cut by B fields that are nearly, but not quite, perpendicular to the path. Those very small components parallel to the path can be ignored very close to the surface, but not as you move far away. Those small contributions over a large path length must reduce the contributions from the legs of length L because the current through the loop is constant. At large distances, the shape of the current distribution should be irrelevant and the field should reduce to the field from a current carrying wire.

If your more general solution is correct, I think it should reduce to the second solution as b --> 0, and to the field from a wire at large distances. That does not appear to be the case. I'd like to see your solution if you would care to post it.