# Find Magnetic Field of Infinite Slab with constant current density

• datran
In summary, the magnetic field is constant and parallel to the sides you chose if you calculated it correctly.
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

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!

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

Last edited:
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?

Sorry, that should be parallel.
Are you saying the line integrals are wrong?

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!

Itotal = Total current in slab.
A is area of slab on XY plane
Those are not well-defined.

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

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.

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

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.

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.

## 1. What is an infinite slab with constant current density?

An infinite slab with constant current density refers to a theoretical scenario in which there is an infinitely large and flat slab of material with a constant flow of electric current passing through it. This allows for simplified calculations and analysis of the magnetic field produced by the current.

## 2. How is the magnetic field of an infinite slab with constant current density calculated?

The magnetic field of an infinite slab with constant current density can be calculated using the formula B = μ0 * I, where B is the magnetic field strength, μ0 is the permeability of free space, and I is the current density. This formula assumes that the slab is infinitely long and has a constant current density throughout its volume.

## 3. What is the direction of the magnetic field in an infinite slab with constant current density?

The direction of the magnetic field in an infinite slab with constant current density is perpendicular to the plane of the slab, and its direction is determined by the right-hand rule. If you point your thumb in the direction of the current flow, your fingers will curl in the direction of the magnetic field.

## 4. How does the magnetic field change as you move away from the slab?

The magnetic field strength decreases as you move away from the infinite slab with constant current density. This is because the magnetic field lines spread out as they move away from the source, resulting in a weaker field further away from the slab.

## 5. What are some real-world applications of an infinite slab with constant current density?

An infinite slab with constant current density is a theoretical concept used in physics and engineering to simplify calculations and understand the behavior of magnetic fields in certain scenarios. It can also be used as a model for certain materials, such as thin metal sheets or conductive films, that exhibit similar properties in terms of their current flow and magnetic field production.

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