(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A thick slab in the region [itex] 0 \leq z \leq a [/itex], and infinite in [itex] xy [/itex] plane carries a current density [itex] \vec{J} = Jz\hat{x} [/itex]. Find the magnetic field as a function of [itex]z[/itex], both inside and outside the slab.

2. Relevant equations

Ampere's Law: [tex]\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enc}}=\mu_0\int_S\vec{J}\cdot d\vec{a}[/tex]

Biot-Savart Law: [tex]\vec{B}(\vec{r}) = \frac{\mu_0}{4\pi}I\int_C d\vec{l}'\times\frac{\vec{r}-\vec{r}'}{\left|\vec{r}-\vec{r}'\right|^3}[/tex]

3. The attempt at a solution

My first approach was to start with the Ampere's Law. If the current density was uniform ([itex]\vec{J}=J\hat{x}[/itex]), by symmetry we would have [itex]B=0[/itex] at [itex]z=\frac{a}{2}[/itex] and find the magnetic field:

[tex]Bl=\mu_0l(z-a/2)J \quad \Rightarrow \quad \vec{B_{\text{in}}} = -\mu_0J(z-a/2)\hat{y}[/tex]

[itex]\dots[/itex]

In this case though, with a non-uniform current density, I can't see any symmetry or a way to find a proper Amperian loop to use. Am I missing something here?

Next I thought about using Biot-Savart Law,

[tex]\vec{B}(\vec{r})\stackrel{?}{=}\frac{\mu_0}{4\pi}\int_V \vec{J}d\tau'\times\frac{\vec{r}-\vec{r}'}{\left|\vec{r}-\vec{r}'\right|^3}[/tex]

which I think is a bit of a ...stretch, since [itex]V[/itex] is infinite.

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# Homework Help: Magnetostatics - Magnetic field of a nonuniform current slab

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