CALIFORNIA STATE UNIVERSITY, BAKERSFIELD (CSUB)

DEPARTMENT OF ELECTRICAL & COMPUTER ENGINEERING & COMPUTER SCIENCE

ECE 3320: FIELDS AND WAVES

Homework 3

Solution

QUESTION 1:(25 POINTS)

For each of the following vector fields, determine ∇ · A analytically and then compare the result with your

expectations on the basis of the displayed pattern.

Answer:

1

2

3

QUESTION 2:(25 POINTS)

Vector field E is characterized by the following properties: (a) E points along R, (b) the magnitude of E is a

function of only the distance from the origin, (c) E vanishes at the origin, and (d) ∇ · E = 12, everywhere. Find

an expression for E that satisfies these properties.

Answer:

4

QUESTION 3:(25 POINTS)

A vector field D = rr3 exists in the region between two concentric cylindrical surfaces defined by r = 1 and

r = 2, with both cylinders extending between z = 0 and z = 5. Verify the divergence theorem by evaluating:

a)∮S D · ds

b)∫ν ∇ ·D dν

Answer:

5

QUESTION 4:(25 POINTS)

For the vector field E = xxy − y(x2 + 2y2), calculate

a)∮C E · dl around the triangular contour shown in the following figure.

b)∫S(∇×E) · ds over the area of the triangle.

Answer:

6