Magnetic Field inside plate

[deedsy]

Homework Statement

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.

Homework Equations

*see below

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?

 

[RUber]

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

 

[deedsy]

Got it - thanks for the help