Assignment 4

Problem 2:

A thin wire of radius b is used to form a circular wire loop of radius a (a >> b) and total resistance R.
The loop is rotating about the z-axis with constant angular velocity ωk in a region with constant magnetic field B = B₀i. 
At t = 0 the loop lies in the y-z plane and the point A at the center of the wire crosses the y-axis.

 

Let the (θ/θ) direction be tangential to the loop and be equal to the positive z direction at point A.
Let the (φ/φ) direction be tangential to the wire and be equal to the direction indicated in the figure.
(a)  Find the current flowing in the loop.  Neglect the self-inductance of the loop.  What is current density J as a function of time?
(b)  Find the thermal energy generated per unit time, averaged over one revolution.
(c)  Write down an expression for the the Poynting vector S on the surface of the wire.
(d)  Use S to find the field energy per unit time flowing into the wire, averaged over one revolution.

Problem 3:

(a)  From Maxwell's equations, derive the conservation of energy equation,
(∂u/∂t) + ∇∙S = -E∙j,
where u is the energy density and S is the Poynting vector.  Rewrite this equation in integral form and explain why it is a statement of energy conservation.
(b)  A long, cylindrical conductor of radius a and conductivity σ carries a constant current I.  Find S at the surface of the cylinder and interpret your result in terms of conservation of energy.

Problem 4:

Show that a radially oscillating spherically symmetric charge distribution does not radiate.
Assume a harmonic time variation of frequency ω and use the complex formalism,
namely ρ(r,t) = Re[ρ(r)exp(-iωt)], D(r,t) = Re[D(r)exp(-iωt)], and so forth.

(a)  First write down Maxwell's equations and the continuity equation using the complex formalism to eliminate time derivatives.
(b)  Note that one can write j(r) = rf(r).  Compare ∇∙D and ∇∙j to relate D and j.
(c)  Show that ∫ρ(r) d³r = 0.
(d)  What are D and B outside the source?  (Hint: Consider ∇∙B and ∇×B.)
(e)  What do you conclude concerning radial oscillations of a spherically symmetric source with an arbitrary time variation?