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.
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