# Magnetic Field inside plate

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1. Mar 12, 2015

### deedsy

1. The problem statement, all variables and given/known data
What is the magnetic field outside and inside of a large, flat conducting plate if the current density decreases linearly with depth z inside the plate. J = J0(1-az). The plate thickness is 1/a.

2. Relevant equations
*see below

3. The attempt at a solution
$\oint \vec B \cdot d \vec l = \mu_0 I$

So outside the plate,
$2Bl = \mu_0 \int J_0(1-az)ldz$ *The integral is taken from from z=0 to z=1/a (the whole plate)
$B=\frac{\mu_0 J_0}{4a}$

Which agrees with the back of the book solution.

Now inside is where I am having trouble...
I tried applying Ampere's Law again but I just got
$\oint \vec B \cdot d \vec l = \mu_0 I$
$2Bl = \mu_0 \int J_0(1-az)ldz$ *taking the integral from 0 to z this time
$B=\frac{\mu_o J_0}{2} (z-\frac{az^2}{2})$

but the back of the book says the answer is $B = \mu_0 J_0 (\frac{az^2}{2} - z +\frac{1}{4a})$

So I'm not sure where the $\frac{1}{4a}$ came from... Does anyone know how i should go about finding the magnetic field inside the plate?

Last edited: Mar 12, 2015
2. Mar 12, 2015

### RUber

At some point z, you have a field above and a field below. You should account for both of them.

3. Mar 13, 2015

### deedsy

Got it - thanks for the help