# Griffiths 6.5: Find Magnetic Field in Uniform Current Slab

• ehrenfest
In summary: The magnetic field is only in the y-direction because the current is flowing in a uniform direction in all directions.
ehrenfest
[SOLVED] griffiths 6.5

## Homework Statement

A uniform current density $\mathbf{J} = J_0 \hat{\mathbf{z}}$ fills a slab straddling the yz plane x=-a to x=+a. Find the magnetic field inside the slab.
This is part of Griffiths Problem 6.5.

## Homework Equations

$$\nabla \times \mathbf{B} = \mu_0 \mathbf{J}$$
$$\int \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{enc}$$

## The Attempt at a Solution

I first need to find the direction of B. What can I use to do that? The right-hand corkscrew rule does not seem to help here!

ehrenfest said:

## The Attempt at a Solution

I first need to find the direction of B. What can I use to do that? The right-hand corkscrew rule does not seem to help here!

Sure it does. Why do you think it doesn't?

The right-hand corkscrew rule gives you the B-field around a wire. How do you use it to get the B-field of a slab? Why do you think it does help?

I would approach this question by first noting that,

$$\underline{J}=\sigma\underline{E}$$

And then using the differential form of Faraday's law,

$$\text{curl}\left(\underline{E}\right) = -\frac{\partial\underline{B}}{\partial t}$$

Hootenanny said:
I would approach this question by first noting that,

$$\underline{J}=\sigma\underline{E}$$

Wait, but that is not always true. What makes you think Ohm's law applies here?

ehrenfest said:
Wait, but that is not always true. What makes you think Ohm's law applies here?
Sorry, I was looking at the wrong question.

The direction of the magnetic field can definitely be found using the RH 'corkscrew' rule, as you would for a wire. Take your right-hand and place your thumb in the direction of the current density, your curling fingers indicate the direction of the magnetic field.

Hootenanny said:
The direction of the magnetic field can definitely be found using the RH 'corkscrew' rule, as you would for a wire. Take your right-hand and place your thumb in the direction of the current density, your curling fingers indicate the direction of the magnetic field.

OK. But in the case of the wire the B-field is circumferential. Why is it not circumferential here? Obviously there will be some cancellation here...but how do we know that after all the cancellation is done the B-field will only be in the y-direction?

Never mind. In Griffith's Example 5.8, he proved that the magnetic field of an infinite uniform surface current $K\hat{\mathbf{x}}$ flowing over the xy plane produces a magnetic field that is only in the y-direction. This is just a stack of those! We can replace x by z here due to symmetry I think.

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## What is Griffiths 6.5?

Griffiths 6.5 is a popular physics textbook written by David J. Griffiths. It covers topics in classical mechanics, electromagnetism, quantum mechanics, and statistical physics.

## Who is David J. Griffiths?

David J. Griffiths is a retired professor of physics from Reed College. He is also the author of several other physics textbooks and is known for his clear and engaging writing style.

## What topics are covered in Griffiths 6.5?

Griffiths 6.5 covers classical mechanics, electromagnetism, quantum mechanics, and statistical physics. It also includes topics such as relativity, waves, and thermodynamics.

## Is Griffiths 6.5 suitable for beginners?

Griffiths 6.5 is generally considered to be an intermediate-level textbook. It assumes some prior knowledge of calculus and basic physics concepts. However, it is written in a clear and accessible manner, so it can be a good resource for motivated beginners.

## Are there any supplemental materials available for Griffiths 6.5?

Yes, there are several companion materials available for Griffiths 6.5, including a solutions manual, study guides, and online resources such as video lectures and interactive simulations.

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