Electrostatic field at the centre of a disk

1. The problem statement, all variables and given/known data
A thin conducting disc has radius a thickness b and electrical resistivity ρ. It is
placed in a uniform time-dependent magnetic induction ##B = B_0 sin ωt## directed parallel
to the axis of the disc. Assuming that ρ is large, find E at a distance r < a from the axis
of the disc, in the plane of the disc, and obtain an expression for the induced current density as a function of r

2. Relevant equations
Maxwells equations

3. The attempt at a solution
So there are two ways I can think of doing this question. One is to consider the emf induced in the disk using Faradays law ##emf= -{d\phi}/{dt}## and then calculate the associated current using the resistivity, followed by calculating the associated magnetic field. From this I can then calculate the electric field which is produced as a result of this current using ##integral E.ds= -integral {dB}/{dt} .ds## this can then be subtracted from the initial magnetic field to give the resulting.

However, since the resistivity is large, I'm going to assume that this current is essentially negligible. So can I just skip to the point where I can calculate E using ##\integral E.ds= -\integral {dB}/{dt} ds##? My only concern with this, is this electric field is the same regardless of whether the disk is there or not.

Either way, once I have the electric field I can easily calculate the current density. I just can decide which of these methods is more appropriate..

Many thanks

Edit: Really sorry- Don't know how to get the integral sign, nor can I remember where to find how to do it. If you can help me with either of these, that would be much appreciated! :)

 

Should I take into account the induced current though? Many thanks :)

 

Ah okay. Thank you so much!

Sorry, I should've included this above, but the next part asks for the power dissipated, for which I will need the resistance.


So far I have done
considering ## R=rho * l/A## where L and A are the lengths and areas respectively. The area that is perpendicular to the current is given by ##bdr## where dr is the infinitesimal change in radial distance, and ##l=2*pi*r##. so ##R=rho * 2*pi*r/bdr ## ? I don't feel as though this is right?

Many thanks :)

 

I got the electric field to be ##E=\frac{B_0 \omega}{2 \pi r} cos( \omega t)##
and the Current density to be ##E=\frac{B_0 \omega}{2 \pi r \rho} cos( \omega t)##

However, I'm struggling to find the power dissipated. Many thanks

 

My mistake!

I should have got
##E=\frac{- B_0 \omega r}{2 } cos( \omega t)## (I think this is dimensionally correct)
##J=\frac{- B_0 \omega r}{2 \rho} cos( \omega t)## ?

 

Thank you for your help! Is there any chance you can also help me find the power dissipated?

 

I'm afraid I don't know how to compute the differential power in a thin annulus