A solid conducting sphere of radius a is held at a constant potential V0 which is not necessarily zero. It is surrounded by a thin concentric spherical shell of radius b, as shown in Figure 1 , on which a surface charge density exists, given by σ(θ) = k cosθ where k is a constant and θ is the polar angle in spherical coordinates. The regions in-between the conductor and the shell, and outside the shell, are vacuum. Figure 1 (a) State the boundary conditions on the potential and its radial derivative at r = b. (b) You are given the general solution to Laplace’s equation with azimuthal symmetry, in the two regions, as follows: V(r,θ) = (P l Alr l + Bl r l+1 Pl(cos θ), for r ≥ b P l Clr l + Dl r l+1 Pl(cos θ), for a ≤ r ≤ b Determine the coefficients Al , Bl , Cl and Dl for all l. (c) Hence, show that the induced surface charge density on the inner conductor is given by σi(θ) = -kcosθ + V0 ϵ0/a The first two Legendre polynomials are given for this question: P0(x) = 1, P1(x) = x.
A long, cylindrical rod of radius a carries a uniformly distributed free current I. The rod is made of a paramagnetic material, which has a relative magnetic permeability µr slightly greater than 1. The resistance per unit length of this wire is R. Figure 2 (a) For the regions inside and outside the rod, find the a. magnetic field strength H, b. magnetic flux density B, c. magnetization M. (b) Write down Poynting’s theorem in words, and also in equation form. (Derivation is not needed, but you should explain each of the symbols used.) (c) By integration of the Poynting vector, show that the power flow per unit length into the wire from the Poynting vector is equal to the ohmic power dissipated per unit length in the wire.
(a) Derive the boundary condition on the tangential component of E, starting from Faraday's law in integral form. Assume time-independent fields. A microwave of frequency f = 3.5 MHz is propagating in the +z direction in a vacuum. It is normally incident upon the surface of a perfect conducting plane located at z = 0. The surface of the conductor lies on the xy-plane. The amplitude of the incident electric field is E0 and it is polarized in the x-direction. (b) State the vector relationship between the electric field E, magnetic field B and the propagation vector k. (c) Write down the incident and reflected electric and magnetic fields in complex phasor notation. (d) What is the value of the reflection coefficient R, given by the ratio of the complex amplitudes, R = E˜0r/E˜0i? (e) Given the attenuation coefficient, κ = ω√(μϵ/2 √(1+(σ/ϵω)^2 )-1). show that the skin depth is approximately inversely proportional to f^(1/2) for real conductors. State any assumptions used.