Instructors: Dr. Zuhair M. Hejazi & Dr. Ibrahim Elshafiey

King Saud University Mid-Term Exam I

College of Engineering 2nd

Semester 1431-32.

Electrical Engineering Department Date: 26/4/1432 H (31/03/2011G)

EE213: ELECTROMAGNETICS (1) Time Allowed: 90 Minutes

Answer Key

Solve the following two problems.

Problem I (10 Marks: 1 +2+3+4)

(a) State Coulomb’s law.

(b) Point charges of 10 nC each are located at A(1, 0, 0), B(-1,0, 0) and C(0, 1, 0), and D(0, -1, 0) in

free space. Find the total force on the charge at A.

(c) Three infinite uniform sheets of charge are located in free space as follows: nC/m2 at ,

5 nC/m2 at , and 5 nC/m

2 at . A uniform line charge

nC/m is also located

along the z-axis. Find at the points: M(0, 0, 0), N(-5,-5,-5) and P(5,5,5).

(d) A charge density is given as

C/m

3 in the region

, where

and are constants.

Calculate:

i. The total charge included in this region.

ii. The electric field intensity everywhere.

Solution:

(a) The force between two very small objects separated in vacuum or free space by a distance which

is large compared to their size is proportional to the charge on each and inversely proportional to

the square of the distance between them. The force acts along the line joining the two charges and

is repulsive if the charges are alike in sign and attractive if they are of opposite sign.

(b)

Q=10-8

C

nN

(c)

Since M lies on the line charge, EM is infinite.

=5 nC/m2

and

Instructors: Dr. Zuhair M. Hejazi & Dr. Ibrahim Elshafiey

(d)

i. The charge Q is calculated from the charge density as:

ii.

Choose gauss surface to be a sphere of radius r.

For:

For:

For:

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Instructors: Dr. Zuhair M. Hejazi & Dr. Ibrahim Elshafiey

Problem II (10 Marks: 1, 2, 3 + 4)

(a) Given the flux density: in free space. Calculate:

i. The charge density at point

.

ii. The total charge enclosed by the shown cylinder of radius with . Use

the volume side of the divergence theorem.

iii. The total charge enclosed by the same cylinder in (ii) using the surface side of the divergence

theorem (Gauss's law).

(b) Find the total charge within the volume in free space given that the flux density

in this region is defined as:

.

F/m

&

Solution:

(a)

i. The charge density is:

At point

ii. Using the volume side of the divergence theorem, the total charge is:

iii. Using Gauss’s law, it can be seen that has no component along (the sides) and

for the top , for and for , thus:

Flux through Bottom

Flux through Sides

Flux through Top

Instructors: Dr. Zuhair M. Hejazi & Dr. Ibrahim Elshafiey

Which is the same obtained in (ii)

(b) In Cartesian coordinates, we find the total charge within the volume by using

the volume divergence theorem:

where