Question

I. For fixed charge and mass distributions, we expect that the magnetic moment μ of a rotating object is proportional to its angular momentum L, such that μ = aL. (a) Derive the proportionality constant a for a spherical ball whose mass m and charge q are uniformly distributed. (1 points) (b) Does this result hold for any shape of the object? (1 point) (c) List at least one property which a classical object needs to have such that the proportionalilty constant is not a as in (a). (1 point)

Answer 1

The magnetic moment mu of a rotating is to its angular momentum = > mu = aL = > a = mu/L all spherical balls, Let the radius be R and rotating both to, alpha = Q/4 pi R^2 Let w = w k^^By using the spherical coordinate (, phi, theta) both the origin at center of the spherical shell. We use infinitesimal area ds of a circular ring of infinitesimal width of radius r at a distance R from origin. = > r^vector = R sin theta r^^(= R is constant) dq = alpha ds = alpha R^2 sin theta d theta d phi The current in the ring is, dI = dq times f = (alpha R^2 sin theta d theta d phi) times omega/2 pi dI = omega alpha R^2 sin theta d theta d phi/2 pi The magnetic dipole moment of a ring is given by expression in terms of position vector be r^^to