Lecture_7

Dr. Ahmed Said Eltrass

Electrical Engineering Department

Alexandria University, Alexandria, Egypt

Fall 2015

Office hours: Sunday (10:00 to 12:00 a.m )

4th floor, Electrical Engineering Building

ELECTROMAGNETICS

Lecture 7

r

QV

4

Recall: For Point Charge Q Q

Absolute Potential due to Multiple Point Charges

addition)(Scalar 4

.........444

...........

3

3

2

2

1

1

321

N

Np

Np

R

Q

R

Q

R

Q

R

QV

VVVVV

1Q

2Q3Q

NQ

1r

2r 3rNr

P

Chapter 4

Energy and Potential (continued)

Absolute Potential due to a distributed charge

If the charge is distributed over a line or a surface or a volume and the required

is to calculate V at certain point, we will make the following steps:

dQ

Charged Body Observation point

VE

dVV

r

dQV

dVdQ

dSdQ

dldQ

dvdsdl

v

s

field electric get thecan You -5

4

4d -3

Charge) (Volume

Charge) (Surface

charge) (Line

charge)point a asit (consider element thisof charge theFind-2

) , ,(body charged thefromelement an Choose -1

rP

This is only a single integration because V is scalar

Examples:

4- Find the absolute potential at the

point P(0,0,b) due to the uniformly

charged ring shown in the figure.

Deduce the electric field intensity.

(0,0,b)

5- Find the absolute potential at the

point (0,0,b) due to the uniformly

charged disc shown in the figure.

Deduce the electric field intensity.

),z,(ρ

ρl

000Ppoint aat potential absolute the

find ,density with charge line finiteshown For the-6

x

y

z

),z,(ρ 000P

a

b

Ideas

a- Uniform surface charge distributions of 6, 4, and 2 nC/m2 are present

at r = 2, 4, and 6 cm, respectively, in free space. Assuming V = 0 at

infinity, calculate V at r = 1, 3, 5, and 7 cm.

Ideas

b- Using Gauss’s Law, find the electric flux density everywhere.

Then find the absolute potential for each surface.

x

y

z

a

b

1v

2v1s

ar

brr

ρ

ρ

s

v

v

at uniform is

a , 1

uniform is

1

2

1

x

y

z

a

b

b

a

s

s

at - Uniform

at Uniform

Ideas

c- Given the two infinite cylinders shown, find Vab

s

s

Notes

given in exams

1- You can find E by finding first the potential (V) by a single

integration because V is scalar. Then

2- Recall

VE

Examples:

vρD

EE

),,(-zyx

density charge volume theand ,density flux electric the

, ofdirection the,density field electric theV, potential the:Find

634Ppoint a and ,52V field, potential Given the -7 2

dr with ,Ppoint at V and :Find . distanceby

separated and charges of consistsshown dipole electric The -8

)(r,θEd

–QQ

Energy

1- The energy stored (work done) in a system of N point

charges:

Example:

9- Four 0.8 nC point charges are located in free space at the corners

of a square 4 cm on a side. Find the total potential energy stored.

2- The energy density stored in a region of continuous

charge distribution:

3

vol

2

0 J/m 2

1 dvEWE

Example:

a b

ba

E s

a , a whereL,length of

cable coaxial a ofsection a of field ticelectrosta in the storedenergy theFind -10