ENGR 365 Problem Set #4 Due 4pm on Fri. Oct. 22, 2010 1) Consider a square sheet charge that has dimensions b x b and a uniform ρeS [C/m2]. The sheet is parallel to the x-y plane and is located at the origin. If b = 2 m and ρeS=0.5 C/m2 (a) estimate E (x = 0.1, y = 0.1, z = 50) using the point charge solution. Assume that the charge of the point charge is equal to the total charge of the sheet. (b) estimate E (x = 0.1, y = 0.1, z = 0.01) using the infinite sheet charge solution. (c) determine the contribution d E due to the small charge element located at (x’,y’,z=0) with area dx’dy’. (d) calculate E (x = 0.1, y = 0.1, z =50) and E (x = 0.1, y = 0.1, z = 0.01) using a Matlab program to sum up all the d E ’s caused by the small charge elements. Assume dx’ = dy’ = 0.01. Please hand in a hard copy of you Matlab code. (e) discuss the agreement between the estimates and the calculated results. Give reasons for the agreement/disagreement. 2) Consider a dipole consisting of two charges +Q and –Q separated by a distance d. The dipole is oriented along the x-axis. What is the dipole moment? Find an expression for the potential V at (R,φ,θ) in spherical coordinates. 3) (a) Find the polarization P in a dielectric with εr = 2.8 if D 3.0 10 7 â x [C/m2]. (b) Determine the value of E in a material for which the electric susceptibility is 3.5 and P 2.3 10 7 â x [C/m2]. 4) What is the energy stored in a system of two point charges q1 = 3nC and q2 = 6 nC, separated by a distance of d = 0.2 m? 5) The radius of the core conductor and the inner radius of the outer conductor of a very long co-axial transmission line are a = 1 mm and b = 3mm. The outer conductor is grounded (V = 0) and the inner core is at Vo = 10 V. (a) Find an expression for E in the region a < r < b via Gauss’ law. Assume that the of the Gaussian Gaussian surface encloses a charge Qenc = eLL, where L is the length surface. From E , find an expression for V. Find the value of E and V at r = 0.1 mm and r = 2.5 mm. (b) Find the transmission line’s breakdown voltage if it is air-filled. (c) Find the breakdown voltage if the transmission line is filled with plexiglass. 6) (a) Find the capacitance of two metal surfaces of area A held a distance d apart. Hint: place charges +Q and –Q on the metal surfaces, respectively. Solve for V. (b) Find the capacitance of two concentric metal shells with radius a and b.