1. Given a round disc of radius R rotating at angular velocity. The disc carries

Question

1. Given a round disc of radius R rotating at angular velocity. The disc carries a surface charge density given by -as. ("a" is a constant, "s" the radius term.) (a.) (5 pts) Determine the total charge Q in coulombs. ANS WER In terms of Q, find: (b.) (5 pts) the surface current density, K. ANSWER: (c.) (5 pts) the magnetic dipole moment, m. ANSWER: (d.) (5 pts) the vector potential A at points rR (Note: using a spherical coordinate based T") ANSWER: (e.) (5 pts) the magnetic field B at poins r»R. (Note: using a spherical coordinate based "r") ANSWER:

Explanation / Answer

1. for the disc, radius = R

angular velocity = w

surface charge density, sigma = as^2

a is a constat, s is radius

a. total charge = integral(dQ)

dQ = 2*pi*s*ds*sigma = 2*pi*s*a*s^2*ds = 2*pi*a*s^3ds

integrating

Q = pi*a*R^4/2

b. at radius s

dQ = 2*pi*a*s^3*ds

now, surface current density, = K

now, at radius s

speed , v = ws

then

dK = sigma*w = w*as^2 = was^2

hence average surface current density

K = integrate(dk*ds)/integrate(ds) = waR^2/3

c. magnetic dipole moment dm = i(r)*dA

i(r) = was^2*2*pi*s*ds = 2*pi*was^3ds

dm = 2*pi*was^3ds(pi*s^2) = 2pi^2*was^5ds

m = pi^2*waR^6/3

d. vector magnetic field = dB

dB = 2ksi(s)/(z^2 + s^2)

dB = 2ks*2pi*was^3*ds/(z^2 + s^2)

dB = 4pi*k*w*a*s^4ds/(z^2 + s^2)

integrating

B = 4*pi*k*w*a*[z^3*arctan(R/z) - z^2*R + R^3/3]

where R is radius of the disc and z is perpendiculr distance from the center of the disc on its axis

hence

A = B/mu = w*a*[z^3*arctan(R/z) - z^2*R + R^3/3]

e. B = 4*pi*w*a[z^3*arctan(R/z) - z^2*R + R^3/3]/mu

where mu is permeability of free space

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