Find Magnetic Field of Infinite Slab with constant current density

[datran]

Homework Statement

Find the magnetic field from an infinite slab with constant current density, Jo, in the x direction.

ρ(z) = ρ1 x_hat for -b<z<b
ρ(z) = 0 for |z| >= b

Homework Equations

Ampere's Law.

The Attempt at a Solution

I draw a rectangular prism inside the slab with -b<z<b with length 2z.

I use Ampere's Law for the 4 sides and two sides are 0 due to being perpendicular to the current density.

My remaining line integrals give me: 2zB + 2zB = μInet

Inet = Itotal (A of amperian loop / A total slab)
Inet = Itotal (2zxy / 2bxy)
Inet = Jo dot dS (z/b)

I do not know what to do with that integral of Jo dot dS.

Am I on the right track?

Thank you!

 

[mfb]

The magnetic field is constant and [edit:] parallel to your sides, if you chose them properly. This allows to evaluate the integral.

 

[datran]

I don't understand. I've already canceled two due to being perpendicular. Are you saying all 4 sides will be perpendicular?

I've already solved the left side of the Ampere's law. Are you saying the line integrals are wrong?

 

[mfb]

Sorry, that should be parallel.

Are you saying the line integrals are wrong?

I don't understand your notation.

 

[datran]

I am given Jo, the current density.
Inet = Current enclosed by amperian loop
Itotal = Total current in slab.
the slab is infinite in x and y direction but stops at -b< z <b in z direction.
A is area of slab on XY plane
Itotal = integral of J DOT dS

I use ampere's law and create a rectangular prism inside the slab. Two of the lines are 0 due to being perpendicular. The other two with length 2z add up and become

4zB =μ (Inet)

Inet = Itotal (Volume of amperian Loop / Volume of Slab)
Inet = Itotal (2zA / 2bA)
Inet = integral J DOT dS (z/b)
Inet = integral Jo DOT dS (z/b)

I do not know if I am doing the right side correctly.

Thanks for the replies!

 

[mfb]

Can you add a sketch of your rectangle?

Itotal = Total current in slab.
A is area of slab on XY plane

Those are not well-defined.

 

[datran]

z_hat points up. x_hat right. y_hat into the board. slab goes from -infinity< x and y < infinity

-b< z < b. I draw a ampere rectangular prism starting at the origin going to infinity in x and y direction. I go z1 up and -z1 down where -b < z1 < b.

I do the loop in one face since current is pointing in the +x_hat direction. Using Right hand rule, B is coming out in -y direction at z >= b and beyond. B is coming into the board at y_hat direction at z <= -b

 

[mfb]

You have a magnetic field in the slab as well, but that is not relevant here.
I understand the geometry of the problem statement, I just don't understand what you are trying to calculate.

 

[datran]

I am trying to find the magnetic field in and outside of the slab.

 

[mfb]

That is not what I meant. I did understand the problem statement based on the first post, there is no need to repeat.

Apparently you have made some attempt to solve it - but I have no idea what you did there, and I don't think this will change without a sketch or a very clear explanation how you came to those calculations.
If it involves any infinite length, current or whatever: This is wrong.

 

[datran]

I don't understand what you cannot follow. If I'm doing something wrong, I would love to be pointed in the right direction.

I calculated the line integral on the left side of Ampere's Law by calculating the rectangle where two sides cancel and the remaining sides are on the z axis. Since these sides are length 2z, they add up to be 4zB. I do not know how to calculate the right side.

You could throw a dog a bone and show me a little of what to do.