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deedsy
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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
[itex] \oint \vec B \cdot d \vec l = \mu_0 I[/itex]
So outside the plate,
[itex] 2Bl = \mu_0 \int J_0(1-az)ldz[/itex] *The integral is taken from from z=0 to z=1/a (the whole plate)
[itex]B=\frac{\mu_0 J_0}{4a} [/itex]
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
[itex] \oint \vec B \cdot d \vec l = \mu_0 I[/itex]
[itex] 2Bl = \mu_0 \int J_0(1-az)ldz[/itex] *taking the integral from 0 to z this time
[itex] B=\frac{\mu_o J_0}{2} (z-\frac{az^2}{2})[/itex]
but the back of the book says the answer is [itex] B = \mu_0 J_0 (\frac{az^2}{2} - z +\frac{1}{4a}) [/itex]
So I'm not sure where the [itex] \frac{1}{4a} [/itex] came from... Does anyone know how i should go about finding the magnetic field inside the plate?
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