Alternating magnetic field

[FourierX]

Homework Statement

A conducting disk (very thin) of thickness h, diameter D, and conductivity $$\sigma$$ is placed in a uniform alternating magnetic field B = B_osin$$\omega$$t parallel to the axis of the disk.
Determine the induced current density as a function of the distance from the axis of the disk. Also mention the direction of the current.

Homework Equations

$$\oint$$B.da = $$\mu_o$$ I_enc

The Attempt at a Solution

From ampere's law, i derirved $$\nabla$$xB = $$\mu_o$$J. Is my approach correct ? How do i make use of thickness, diameter and conductivity of the disk?

 

[wdednam]

Hi FourierX,

Your relevant equation, Ampere's law, might not be very helpful here. I suggest you use Faraday's law of induction instead. The relation between the electric field and current density, J = sigma*E might also come in handy.

Good luck,

Wynand.

 

[FourierX]

Thank for the reply. But i wonder how?

 

[wdednam]

Hi,

You've got to ask yourself the following questions: Which phenomenon am I dealing with here? What is the CHANGING (clue) magnetic field going to do to the conducting disk? In general, what do changing magnetic fields do?

Hope this helps,

Wynand.

 

[FourierX]

Thanks bud!

Ok, here is what i did.

-calculated the flux (area of the disk times the magnetic field)
- used Faraday's law (emf = negative time derivative of magnetic flux)
- I = emf/R = emf (conductivity)

does it seem correct? So far the, the current is expressed as a function of time. Any idea on how to express it as a function of distance from the axis of the disk?

 

[wdednam]

No problem!

Spot on with steps 1 and 2 of your calculations.

Okay, according to the problem statement we should calculate the current density J as a function of the distance from the disk's axis, so I don't think you need to worry about finding the current I.

So, from step 2 of your calculations we need to get to J as a function of the time derivative of the flux and the distance from the disk's axis.

Now, a changing magnetic field not only induces an emf, but also an Electric field, E. If you can find the relationship between those (hint: it involves a line integral around a closed path), and use the relationship between E and J, i.e. J = conductivity*E, then your problem is basically solved.

You'll notice that the thickness h of the disk won't appear in your answer for J. If you were required to calculate I, the current, instead, then it would appear in your answer for that.

Hope this helps,

Cheers,

Wynand.