HOMEWORK SET 3 - Due: Tuesday, Feb. 23
1. Magnetic Inductance, Magnetic Energy.
A torus has major radius a (0.1 m), minor radius b (0.02 m), relative permeability mr (10000), and N (100) winds of a wire.
(a) At first the switch is closed, so a steady current of Io (10 A) flows through the wire. Derive an expression and calculate the magnetic energy within the torus, assuming a >> b.
At time, t = 0, the switch is opened. The current flowing in the coil must now flow through the resistor. Derive and expression and calculate the current through the resistor as a function of time I(t). You may follow the steps below or make up your own method.
(b) Derive an expression for the magnetic flux (Y) inside of the torus.
(c) Derive an expression relating the induced voltage V to dI/dt.
(d) Using V = IR (through the resistor) set up a differential equation and solve for I(t).
(e) Integrate I2R from t = 0 to infinity and show that the energy dissipated in the resistor is the same as the stored energy.
(f) Calculate the inductance (L) of the coil.
(g) Show that the stored energy is 1/2LI2.
2. Capacitance, Electric Energy.
(a) A coaxial cable has in inner conductor with radius a (1 mm), an outer conductor with radius b (2 mm), and a dielectric medium with er = 2. Derive an expression and calculate the capacitance of a cable length L (1 km). Use the approximation L >> b.
(b) Derive an expression and calculate the electric field energy stored in the cable if the potential between the two conductors is Vo (100 Volts, switch is open).
(c) Show that the energy is 1/2 CV2.
(d) At t = 0, the switch is closed, and the cable discharges through the resistor. Set up a differential equation and derive and expression for V(t). Hint: dV/dt = I/C.
(e) Integrate V2/R from t = 0 to infinity and show that the energy dissipated in the resistor is the same as the stored energy.
(a) Two plane waves are emitted at the same time, both propagating in the z-direction. One is polarized such that Ex = Eocos(wt-kz) and Ey = 0, the other has Ey = Eosin(wt-kz) and Ex = 0. Show that at any given point in space, the electric field simply rotates.
(b) Calculate the magnetic field.
(c) Calculate the Poynting vector.
4.Coulomb's Law and Poisson's Equation. (This is very hard)
We want to prove that Coulomb's Law and Poisson's Equation are mathematically equivalent. Follow the steps below or do it your own way.
d is the Dirac delta function. Hint: Show for r' = 0. Show that 
except at R = 0 (use spherical coordinates). Then use Gauss's theorem to show that:
(b) Take the divergence of Coulomb's law:
and substitute in the above expression. Hint: remember that:
