GRADUATE PROGRAM IN ELECTRICAL ENGINEERING – UFPE

PGEE936 – ADVANCED ELECTROMAGNETICS 2021.01

Instructor: Eduardo Fontana

HOMEWORK # 3 – 04/30/2021

COMPLETION DEADLINE – 05/21/2021

Remarks:

· Homework must be handwritten, and solved clearly and concisely.

· Clear reasoning should be demonstrated in the solution development

1. Based on lecture notes #9:

a) Obtain the Legendre differential (a) on page 5.

b) Insert the proposed series expansion and obtain the relationships (c), (d) and (e) on page 5.

c) Demonstrate that conditions (f) and (g) on page 5 are equivalent.

2. Using Rodrigues formula, obtain P₅(x).

3. Assume that in a sphere of radius a, the potential is given by the condition:

a) Determine the potential in the regions R < a and R > a.

b) Determine the electric field vector in these regions.

4. Assume that the potential is specified on the spherical surfaces of radii a and b (b > a) by:

a) Determine the potential in the region

b) Determine the corresponding electric field vector.

5. Consider a charged disk located on the z = 0 plane, with a superficial charge density .

a) Determine the potential on the z axis.

b) From this solution, obtain the potential at a point having coordinates .

HInt: Use the concept of obtaining the solution from its behavior on a subdomain.

6. Consider a charge q located at the origin. Assume that surfaces are grounded. Use the method of images and show that:

a) There are infinite images, periodically spaced, on the z axis, located at . Determine the values and coordinates of the image charges.

b) Determine the potential function for .

c) Determine the superficial charge densities at .

d) Determine the total induced charges on the two grounded surfaces.

e) What is the potential in the region ? Explain your response.

7. Consider a grounded sphere of radius a, with center at the origin. There is a charged ring having linear density λ, centered at the origin, having radius b > a, and located on the plane z = 0.

a) Determine the image of the charged ring, that is, both its radius and its charge density.

b) Determine the potential function at a point on the z axis.

(iii) Based on the concept obtaining the general solution from its behavior in a a sub-domain, determine the potential, outside the sphere, at a point having coordinates .

8. Do the following problems from ref.[2]: 2.2, 2.7, 3.9, 3.10

References:

[1] Fontana, "Advanced Electromagnetics", Lecture notes #9 to #12

[2] D. Jackson, "Classical Electrodynamics", Chapter 2 (Sections 2.1–2.7), Chapter 3 (Sections 31–3.3 and 3.5–3.8)

[3] Fontana, e-book, Chapter 3, Section 3.5