What is the Surface Current and Magnetic Field for a Rotating Charged Disc?

[lovinuz]

Homework Statement

There is a disc with radius R which has a uniformly-distributed total charge Q, rotating with a constant angular velocity w.

(a) in a coordinate system arranged so that the disc lies in the xy plane with its center at the origin, and so that the angular momentum point in the positive z direction, the local current density can be written J(x,y,z) = K(x,y) d(z). determine the surface current K(x,y) in terms of Q, w, and R.

(b) using the law of Biot and Savart, determine the magnetic field at point r=sk, k is the vector direction. find the same for r=-sk.

Homework Equations

The Attempt at a Solution

 

[lovinuz]

i might add that we can use cylindrical coordinates, expressing this as K(r,phi) where r=sqrt(x square + y square) and phi = tan inverse (y/x). this is for part (a).

 

[thomate1]

(a) The surface current density can be written as K(x,y) = QωR/πR^2, where Q is the total charge, ω is the angular velocity, and R is the radius of the disc. This can be derived by considering the current flowing in a circular path on the disc and using the definition of current density.

(b) Using the law of Biot and Savart, the magnetic field at point r=sk can be written as B = μ0I/4πr^3, where μ0 is the permeability of free space, I is the current flowing in the circular path on the disc, and r is the distance between the point and the current element. In this case, r=sk, so the magnetic field can be written as B = μ0K(x,y)/4πr^3. Similarly, for r=-sk, the magnetic field can be written as B = -μ0K(x,y)/4πr^3. This shows that the magnetic field at points sk and -sk have opposite directions, as expected for a rotating charge distribution.