2. If B is uniform, show that A = -(r × B), where r is the vector from the origin to the point in question. That is, check that B=V x A and VA = 0.

Determine f (r) for the scaler field f(r)= R sinq cosq for the scaler field F(r). Then confirm that

1. Assuming that the observation point is far away from the antenna (i.e., R » 2h), use Eqs. 9 - 16 to obtain the far-zone electromagnetic fields of the dipole, in spherical coordinates (i.e., in terms of R, θ, and ∅, and the associated unit vectors R, θ, and ∅). Far-zone fields require approximations that take advantage of the fact for am- plitude terms one can say that, for example, R₁ =R. However, this type of approximation is way too crude for the complex exponents, since they are asso- ciated with periodic functions. For the complex exponents one needs instead, for example, R₁ =R-h cos θ.

10.3. Consider the plane interface between a perfect conductor (y <-2x) and a dielectric medium with permittivity & = 480 (y> -2x). A surface charge density in the surface of the conductor is p =-2 nC/m². Determine D, E, and P in the two media close to the interface. Use P in the dielectric to determine the effective surface charge density ps.eff.

10.2. Consider the plane interface between two perfect dielectric media defined by y = -2x. The relative permittivity for y < -2x is &1 = 2, while for y> -2x it is &2=3. The displacement field D for medium 1 at the boundary is D₁ = 4a-6ay (C/m²). Determine the displacement field D₂ in 1 medium 2 at the boundary. Find the angle between Di and the surface normal and the angle 0₂ between D₂ and the surface normal.

10.1. Consider a pair of concentric spherical conductors. The inner sphere is of radius a and contains a uniformly-distributed charge Q; the larger sphere is grounded and of inner radius c. The space between the conductors is filled with two non-conducting, charge-free dielectrics, of permittivity & (a <R<b) and &2 (b<R<c). (a) Use Gauss' Laws to determine the displacement field D in the space between the conductors. (b) Find the electric field E in the two dielectric media. (c) Determine the potential difference Vo between the conductors. (d) Determine the surface charge densities ps on the surfaces of the conductors.

2. Electromagnetic Fields and Waves (a) A charge q is moving at velocity through uniform electric and magnetic fields, E and B. The electric field and magnetic field are per- pendicular to each other and to the velocity of the charge. Write down an expression for the total force on the charge and hence determine an expression for the velocity of the charge for the case where the magnitude of the total force is zero. (b) Two charges of tq and q lie at (-1,0) and (z,0), respectively. Determine an expression for the electric field a distance y along the y-axis. (c) Unpolarised light of intensity I passes through a pair of linear polarisers. The axis of the first polariser is at 0 degrees to the vertical. After the second polariser, the intensity of light is zero. i. Use Malus' law to determine the angle that the axis of the second polariser makes with the vertical. ii. A further linear polariser is inserted between the original two polarisers. The intensity of light passing through all three polarisers is now Io/8. Determine the angle of the axis of the new polariser with respect to the vertical.

9.2. A polarized medium is in the shape of a disc of radius a and thickness t. The polarization field in the medium is P(r) = Po p az. (a) Determine the effective surface and volume charge densities on the disc. (b) Determine the total effective charge of the disc.

5. In the AC circuit depicted in the figure below, the capacitance of the capacitor is C = 0.1 F and the resistance of the resistor, R = 102. The AC generator delivers a voltage &(t) = 20 cos(8t) (V), where t is expressed in seconds, and the natural frequency of the circuit is 1Hz. (a) Calculate the value of the inductance I and the frequency of the AC generator. (b) Calculate the complex reactance of all components. (c) Calculate the complex impedance of the circuit. (d) Demonstrate that the RMS voltage delivered by the AC generator is 20 / √2(V). (e) Derive an expression for the power dissipated on the resistor as a function of time.

4. A capacitor, of capacitance C = 120 μF is charged using a battery of 24 V. The internal resistance of the battery is 100 2. Once charged, the capacitor is connected to the circuit presented in the figure below. It is known that R₂=2R₁, R3=2R2, and R3 = 40052. The value of R₂ is unknown. (a) Calculate how long it takes for the capacitor to be 99% charged and determine the initial charge on the capacitor. (b) Just after connecting the capacitor C, at time t = 0, the current through R₂ is 1A. Calculate the value of R, and the energy stored in the capacitor at t = 0. (c) The resistor network, R₁, R2, R3 and R₂, can be replaced by a single equivalent resistance. Find this equivalent resistance and hence find the time constant of the discharging circuit. (d) Find the time t, after which the electric charge on C drops by a factor of 2 and calculate the energy dissipated in the circuit in time 12. (e) Calculate the energy dissipated on R, after a very long time.