A nonconducting plate of mass m and radius r has a charge q uniformly distributed over it. The plate rotates about its axis with an angular speed . What is the relationship between the magnetic moment and the angular momentum l of the plate?

Step-by-step solution

For a uniform distribution of charge and mass, the ratio of magnetic moment to angular momentum l can be found by considering an elemental ring of radius x and thickness dx on the plate. The charge on this ring is dq = q r^2 (2 x dx) = 2q r^2 x dx The equivalent current due to the rotation of this ring is di = dq T = dq 2 = q r^2 x dx The magnetic moment of this elemental ring is d = di ( x^2) = q r^2 x^3 dx Integrating from x = 0 to x = r , the total magnetic moment is: = ₀^ r q r^2 x^3 dx = q r^2 4 The angular mo