Assignment 2

Problem:

(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 2:

Consider the circuit shown.

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Let V = 50 V, R1 = 10 Ω, R2 = 100 Ω, and L = 50 H.
All circuit elements are ideal and no current flows before the switch is closed.
(a)  After the switch is closed at t = 0, find the current I flowing through the switch as a function of time.
(b)  After 8 s the switch is opened again.  Right after the switch is opened, what is the voltage across R2 and across the switch?

Problem 3:

A thin disc with radius R and uniform charge density σ is centered at the origin.  Its normal points along the z-axis.  It is rotating with angular velocity ω k about the z-axis.
(a)  Find the potential and the electric field due to the disc on the z-axis.
(b)  Show that the potential reduces to that of a point charge for large distances from the origin.
(c)  Find the magnetic moment of the disk.
(d)  At large distances from the origin, find the magnitude and direction of the Poynting vector S.
Does the rotating disk produce a radiation field?  Explain.

Problem 4:

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.
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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 pointing 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 5:

A finite spherically symmetrical charge distribution disperses under the influence of mutually repulsive forces. 
Suppose that the charge density ρ(r, t), as a function of the distance r from the center of symmetry and of time, is known. 

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(a)  Prove that the curl of the magnetic field is zero at any point.
(b)  Use this to show that B is zero at any point.