Chapter 28 Problem 57

Problem

A very large flat conducting sheet of thickness $t$ carries a uniform current density ${\vec {j}}$ throughout. Determine the magnetic field (magnitude and direction) at a distance $y$ above the plane. (Assume the plane is infinitely long and wide.)

Solution

The sheet may be treated as an infinite number of parallel wires.

The magnetic field from two wires placed symmetrically on either side of where we are measuring the magnetic field has its vertical component cancelled out.

Therefore, the field above the wire must be horizontal and to the left with magnitude $B_{\parallel }$.

By symmetry, the field a location y below the wire must have the same magnitude, but point in the opposite direction.

$\oint {\vec {B}}\cdot d{\vec {l}}=\int _{\textrm {sides}}{\vec {B}}\cdot d{\vec {l}}+\int _{\textrm {top}}{\vec {B}}\cdot d{\vec {l}}+\int _{\textrm {bottom}}{\vec {B}}\cdot d{\vec {l}}$ $=0+2B_{\parallel }D$

$2B_{\parallel }D=\mu _{0}I_{\textrm {encl}}=\mu _{0}(jtD)$

$\rightarrow B_{\parallel }={\frac {1}{2}}\mu _{0}jt$