I. A hollow box (with sides of length a = 0.19 m. b = 0.29 m and c = 0.39 m) consists of 4 metal plates, which arc welded together and grounded. The top and bottom of the box are made of separate sheets of metal, insulated from the others, and held at a constant potential = 10 V and V2 = 24 V. (i) Derive an expression for the electrostatic potential inside the box. (ii) What should the potential at the (a/2, b/2, c/2) be? (iii) Derive expressions for the charge densities at (a/2, b/2,0), (a/2,b/2,c), and (a/2,0,c/2).
The permittivity of vacuum is equal to e0 = 8.8542 x 10-12 C2/N • m2.
2. The charge configuration shown in figure has three charges qx = — 2q and q2 = c?3 = +q, where q = 2.0 x 10-9C. The three charges are situated at (0,0,0), (0, -a, 4- a/2) and (0,4-a, + a/2). (i) If a = 0.19 mm, derive an expression for the electrostatic potential at P(x, y, z). (ii) For a point P(2.0 m, 2.0 m, 2.0 m) calculate the electrostatic potential using direct substitution in the derived expression, (iii) Using the binomial expansion, find the multipole expansion terms for this configuration and determine the contributions of the monopole, dipole, and Quadrupole terms to the potential, (iii) Show that the potential calculated at point P(2.0 m, 2.0 m, 2.0 m), using the exact expression in part (i), is well approximated by that derived from the multipole expansion, (iv) Show that the potential due to the dipole and quadrupole terms derived in part (iii) arc equivalent to the following two formulas:
Wr)=^fP
Vquadfr) — 4jrfo (r3) E/=l ?i Qij 3
Qi/ = 2 {^(r )n(r/)n — Qn
n=l
Z
3. Consider the shown two concentric metallic spherical surfaces separated by two hemispheres of dielectric materials having relative permittivities erl = 16 and er2 = 32. The radii of the inner conducting sphere and the surrounding conducting spherical shell are equal to R1 = 29 cm, R2 = 59 cm, and /?3 = 69 cm. The inner sphere is maintained at potential V = 0, while the outer shell is connected to a 10-volts battery, (i) Solve Laplace’s equation in the regions of the two dielectric hemispheres (Regions 1 & 2), and in the region outside the shell (Region 3). (ii) Using the obtained result to determine the electric field intensity in the three regions, (ii) Find the free charge densities on the surfaces of the conducting surfaces at /?1; R2 and R3. (iii) Determine the bound charge densities on the surfaces of the dielectric material at Rr and R2. (iv) What is the total charge on the exterior surface of the outer shell at radius R3 = 69 cm? (v) Find the Capacitance C12 of the capacitor formed by the two spherical surfaces having radii and R2. (vi) Determine the stray capacitance Cm due to the free charges distributed on the outer surface of the shell, at R3. (vii) Use the capacitances C12 and Cm to calculate the total electric energy stored in this configuration.
The permittivity of vacuum is equal to 60 = 8.8542 x 10-12 C2/N • m2.
V2 = 10 Volts