Magnetic field generated by a metal plate

[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

 

[OlderDan]

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

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.

 

[thepatientmental]

.

1. The answers are different because they are using different methods to calculate the magnetic field. The first solution is using the Biot-Savart Law, which is a general formula for calculating the magnetic field at a point due to a current-carrying wire. The second solution is using Ampere's Law, which is a special case of the Biot-Savart Law that is applicable when the current is flowing in a closed loop.

2. The second answer is valid at any point outside the metal plate, as long as the point is located along the square path used in the calculation. This is because Ampere's Law states that the magnetic field along a closed loop is equal to the current enclosed by the loop multiplied by the permeability of free space divided by 2 times the distance between the current and the loop. In this case, the square path is enclosing the entire current flowing through the metal plate, so the magnetic field calculated using Ampere's Law is valid at any point along that path.

It is important to note that both solutions are valid and give the same result, but they are just using different methods to arrive at the answer. The first solution is more general and can be applied to any current-carrying wire, while the second solution is specific to a current flowing in a closed loop.