Electromagnetics Review

EE140 Review Electrostatics Dr. Ray Kwok

x(0,0)

+Q Q

(a,0) P(x,0)

4. Consider two point charges (+Q, -Q) located along the x-axis as shown.

a. What is the electric potential at point P(2a,0)? (Note: there is no charge at point P.)

b. From this potential, derive an equation for the electric field at that point P.

c. How much work is required to bring another point charge (+q) from infinity to this

point P? What does the sign mean?

d. How much energy is stored in this system of 3-charges? (+Q, -Q, +q).

(a)

(b)

(c)

sign means it doesnt require work to bring the charge to point P. Work is being done by the charge instead.

(d)


5. A conducting cylinder of infinite length and radius a has a total

charge Q uniformly distributed on its surface. A dielectric shell

(with relative dielectric constant K = 2) of inner radius b and

outer radius c is shielding the conducting rod as shown in the

Figure. Calculate the electric field in all regions. Sketch E(r) as a

function of r.

E(r


L x

E=?

x

E=? L

(a)

(b)

r dy

R

6. (a) What is the electric field at a distance x away from the center of a thin line charge with a

uniform linear charge density and length L? (see diagram below). (b) Using this result, write the electric field at a distance x above the center of a square loop of length L.

=

==322x r

dyxk

r

x

r

dykcos

r

dqkE

4/Lxx

Lk

yx

y

x

kE

rx

y

x

sindcos

x

1

x

cosdsecx

r

dy

dydsecx

x

ytan

r

xcos

22

2/L

2/L

22x

2223

32

3

2

+

=

+

=

=

==

=

=

=

=

Apply the equation above to this with 2 changes:

1. the distance x in the equation is from the wire

to the point. So here, it has to be replaced by

2. the E field is at an angle, and only the x-

component add up. So:

22 )2/L(xx +

( ) 2/Lx4/LxLxk4

E

4/Lx

x

2/Lx4/Lx

Lk4E

cos4/L4/Lx4/Lx

Lk4E

2222total,x

222222total,x

22222total,x

++

=

+++

=

+++

=


+Q

Q a

Q +Q

a

x

y

O

7. Consider 4 point-charges (+Q, -Q, +Q, -Q,) located at the corners of a square as shown.

a. List all the locations where electric field is zero.

b. List all the locations where electric potential is zero.

a. Only at center of the square (a/2,a/2) and at infinity (r).

b. On the lines: y = a/2 and x = a/2 and at infinity (r).


a b c

r

E

3

5a

o

2

5

5 b

a

o

5. A non-uniform charge density of = r2 is distributed in an inner sphere of radius a, enclosed by an outer spherical conductor with radius b and c as shown in diagram. The

outer conductor is electrically grounded. Calculate the electric field in ALL regions. Sketch

E(r) as a function of r. What is the voltage between the inner sphere and the outer

conductor? What is the equivalent capacitance of the structure?

For r< a

For a < r < b

E(b < r < c) = 0

(conductor)

E(r > c) = 0

(shielded, net charge =0)

a

b

c

rrE

rrE

drrrdVQEA

o

o

o

r

o

r

o

total

5

5

][4)4(

)4)((

3

52

0

22

0

=

=

===

r

=

==

===

baba

a

a

V

QC

ba

adr

r

ardEV

o

o

o

a

b o

a

b

11

4

11

5

5

4

11

55

5

5

5

2

5

rr

rr

aE

arE

drrrdVQEA

o

o

o

a

o

a

o

total

5

5

][4)4(

)4)((

2

5

52

0

22

0

=

=

===

r


x

E=? r

y

6. The electric field of an infinite line charge at a distance r away is

where is the linear charge density defined by the Q/L (total charge on the line divided by the line length.) Use this result to derive the electric field at a distance x from an infinite

charge plane of surface charge density of .

.2

rr

Eo

=

r

( )

ooo

net

oo

x

ooo

x

x

r

xdyE

r

xdy

r

dydE

r

dy

r

LdA

r

LdQdE

222

2cos

2

22

/

2

/

2

2

===

==

===


+Q Q a

P

7. Consider two point charges (+Q, -Q) located at the base of an equilateral triangle as shown.

a. What is the electric potential at point P (vertex)? (See diagram.)

b. What is the electric field at point P?

c. How much work is required to bring another point charge (+q) from infinity to this

point P?

d. How much energy is stored in this system of 3-charges? (+Q, -Q, +q).

(a)

(b) xa

kQx

a

kQE o )60cos(2

22=

=r

(c) W = U = qV = 0 (from infinity)

(d) U3q = kQ2/a + 0 = kQ

2/a (or Ui )

0==a

kQ

a

kQV


a b c

r

E

or

b

a

or

8. A coaxial cable is made of an inner cylindrical conductor with radius

a, an outer cylindrical conductor with radius b and c as shown in

diagram. The outer conductor is electrically grounded. A dielectric

material (r) is filled between the 2 conductors (a < r < b). Suppose we have an infinite length of such cable, and a surface charge density is introduced onto the inner conductor, calculate the electric field in ALL

regions. Sketch E(r) as a function of r. What would the voltage

between the 2 conductors be? What is the equivalent capacitance per

length (i.e. the total capacitance divided by the total length)?

E(r < a) = 0 (conductor)

For b > r > a

E(c > r > b) = 0 (conductor)

E(r > c) = 0 (grounded conductor) [Think how the charges are distributed.]

a

b

c

rr

aE

aLrLE

aLQEA

or

or

freeor

)2()2(

)2(

=

=

==

r

( )

=

==

===

a

bL

C

a

ba

aL

V

QC

a

badrr

ardEV

or

or

or

a

b or

a

b

ln

2

ln

2

ln

rr

Electromagnetics Review

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