We want to find the magnetic field B due to an infinite slab of current, using Ampere's law. The figure shows a slab of current with current density J→=Jz^, where dimensions of j are amps per square meter. The slab of current is infinite in the x and z directions, and has thickness d in the y-direction.Magnetic field at y<d/2 is nμ0jy. The value of n is….

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We want to find the magnetic field B due to an infinite slab of current, using Ampere's law. The figure shows a slab of current with current density $\vec{J} = J \hat{z}$ , where dimensions of $j$ are amps per square meter. The slab of current is infinite in the x and z directions, and has thickness $d$ in the y-direction.
Magnetic field at $y < d / 2$ is $n μ_{0} j y$ . The value of $n$ is….

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Detailed Solution

Consider a strip of length $h$ along x-axis and thickness dy along y-axis.
Current through the strip,
$I = (area of strip) (current density) = (h d y) j$

Apply Ampere’s circuital law,

$2 (d B) h = μ_{0} h d y j$

$d B = \frac{μ_{0} d y j}{2}$

$B = \int_{0}^{B} d B = \int_{}^{y} \frac{μ_{0} d y j}{2} - \int_{y}^{} \frac{μ_{0} d y j}{2} = \frac{μ_{0} j}{2} [(y + \frac{d}{2} - \frac{d}{2} + y)]$