What is Ampere force for a thick/wide conductor?

Question

In simple case in uniform constant Magnetic field, when field is perpendicular to the wire and when wire is very thin Ampere force $F=I B L$

where I - current, B - magnetic field, L - the length of the wire.

What happens if we have not a small wire, where the current goes uniformly?

For example, if I have a metal plate (or aluminium foil) of length L and width d and connect it in such a way, that current goes uniformly in it. and total current is I (when I measure it directly at current source). Is the Ampere force in this simple case (uniform constant Magnetic field, field is perpendicular to the current) will be the same $F=I B L$ or it depends on width d somehow? Or I need to integrate exact equation $dF=I B dl $ to get a full force?

Answer 1

The magnetic force on a general current distribution is $$\vec{F} = \int \vec{j}\times \vec{B} \,dV$$ but in this case the current density has constant magnitude and direction, and the same applies to the magnetic field $\vec{j} = \frac{I}{dh} \hat{x}$, $\vec{B} = B\hat{y}$, where $h$ is the size of the plate in the $z$ direction. Therefore the integral takes the form $$\vec{F} = \frac{IB}{dh}\hat{z} \int \,dV = \frac{IB}{dh}dhL = IBL$$