Magnetoresistance and the Corbino disk

[Joseph Rolls]

Homework Statement

Magnetoresistance refers to the direct action of a magnetic field on an electric
current. An example of magnetoresistance can be studied using a Corbino disc (see
figure) consisting of a conducting annulus with electrodes on its inner and outer
rims.
Consider an annular disk with thickness 􏱣 , inner radius 􏱒 outer radius 􏱦 and
conductivity 􏱧 and free charge density 􏱔. A battery connected to the rim electrodes
produces a radial current 􏱙􏰬 flowing from the inner boundary to the outer periphery of the disk. A uniform
magnetic
field 􏱘, constant in time, is applied perpendicular to the plane of the annulus.
(a) Prior to the application of the magnetic field, what is the current density in the disc as a function of radial distance 􏰦 from the centre of the disc?
(b) What is the drift velocity of the electrons in the disc?
(c) What is the magnetic force on these electrons?
(d) Write down an expression for the ‘motional emf’ induced at a distance 􏰦 from the centre of the disc?
(e) Write down an expression for the circular current induced in the disc in terms of parameters defined
above and fundamental constants.

Relevant Equations

J=I/A
J=nqu
F=qv x B

My answers so far are:

a) J=I/(2*pi*r*t)
b) u=J/ne=I/(2*pi*r*t*n*e)
c) F=IB/(2*pi*r*t*n), in theta direction (polar coordinates)
d) This is where I am stuck.
I understand the example for motional emf with a rod moving through a magnetic field but I'm not sure how to apply it to this scenario.
Do we find the work done by the force moving an electron in a full circle round the annulus and get emf from there or something like that?

Any ideas would be greatly appreciated.
:)

 

[TSny]

My answers so far are:

a) J=I/(2*pi*r*t)

OK

b) u=J/ne=I/(2*pi*r*t*n*e)

This would be the drift velocity before B is applied. After B is applied, the drift velocity $\vec u$ will have both a radial component $u_r$ and an azimuthal component $u_{\theta}$.

They might want you to find expressions for $u_r$ and $u_{\theta}$.

c) F=IB/(2*pi*r*t*n), in theta direction (polar coordinates)

I don't think this is correct. The magnetic force will have both radial and azimuthal components.

d) This is where I am stuck.
I understand the example for motional emf with a rod moving through a magnetic field but I'm not sure how to apply it to this scenario.
Do we find the work done by the force moving an electron in a full circle round the annulus and get emf from there or something like that?

I'm not sure what they want here. But your interpretation that it's the work done by the azimuthal component of the magnetic force for a full circle sounds good to me.