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Advanced Electromagnetic

Engineering

Potential and Electric Potential

Energy

Lecture 3

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Comparison of Gravitational and

Electrical Potential Energy

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Energy Considerations

When a force, F, acts on a particle, work is done on theparticle in moving from point a to point b

b

abaldFW

If the force is a conservative, then the work done can beexpressed in terms of a change in potential energy

UUUW abba

Also if the force is conservative, the total energy of the

particle remainsconstant

bbaa PEKEPEKE

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Work Done by Uniform Electric Field

Force on charge is

EqF

0

Work is done on the

charge by field

EdqFdW ba 0

The work done is independentof path taken frompoint a to point b because

The Electric Force is a conservative force

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Electric Potential Energy

UUUW abba

abuniformb

a

ab yyqEsdFUU

The work done only depends upon the

changein position

The work done by the force is the same as

the change in the particles potential energy

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Electric Potential Energy

General Points

1) Potential Energy increases if the particlemoves in the direction oppositeto the force on it

Work will have to be done by anexternal agent for this to occur

and

2) Potential Energy decreases if the particlemoves in the samedirection as the force on it

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Potential Energy of Two Point Charges

Suppose we have two charges q and q0separated by a distance r

The force between the two charges is

given by Coulombs Law

20

04

1

r

qqF

We now displace charge q0 along a

radial line from point a to point b

The force is not constant during this displacement

ba

r

r

r

r

rbarr

qqdr

r

qqdrFW

b

a

b

a

11

44

1

0

020

0

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The work done is notdependent upon the path

taken in getting from

point a to point b

rdF

Potential Energy of Two Point Charges

The work done is related to

the component of the force

along the displacement

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Potential Energy

Looking at the work done we notice thatthere is the same functional at points a andb and that we are taking the difference

ba

barr

qqW

11

4 0

0

We define this functional to be the potential energy

r

qqU 0

04

1

The signs of the charges areincluded in the calculation

The potential energy is taken to be zero when the two

charges are infinitely separated

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A System of Point Charges

Suppose we have more than two charges

Have to be careful of the question being asked

Two possible questions:

1) Total Potential energy of one of the chargeswith respect to remaining charges

or

2) Total Potential Energy of the System

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Case 1: Potential Energy of one charge

with respect to others

Given several charges, q1qn, in place

Now a test charge, q0, is brought into

position

Work must be done against theelectric fields of the original charges

This work goes into the potential energy of q0

We calculate the potential energy of q0 with respect to each of

the other charges and then

Just sum the individual potential energies i i

iq

r

qqPE 0

04

10

Remember - Potential Energy is a Scalar

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Case 2: Potential Energy of a System of Charges

Start by putting first charge in position

Next bring second charge into place

No work is necessary to do this

Now work is done by the electric field of the first

charge. This work goes into the potential energy

between these two charges.Now the third charge is put into place

Work is done by the electric fields of the two previous

charges. There are two potential energy terms for this

step.

We continue in this manner until all the charges are in place

ji ji

jisystem

r

qqPE

04

1

The total potential is then

given by

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Example 1

Two test charges are brought separately to the

vicinity of a positive charge Q

A

qrQ

BQ2q

2r

Charge +q is brought to pt A, adistance r from Q

Charge +2q is brought to pt B,

a distance 2r from Q

(a) UA < UB (b) UA = UB (c) UA > UB

I) Compare the potential energy ofq (UA) to that of 2q (UB)

(a) (b) (c)

II) Suppose charge 2qhas mass mand is released from restfrom the above position (a distance 2r from Q). What is itsvelocity vfas it approaches r = ?

mr

Qqvf

04

1

mr

Qqvf

02

1

0fv

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Therefore, the potential energies UA and UB are EQUAL!!!

Example 2

Two test charges are brought separately to the

vicinity of a positive charge Q

A

qrQ

BQ2q

2r

Charge +q is brought to pt A, adistance r from Q

Charge +2q is brought to pt B,

a distance 2r from Q

(a) UA < UB (b) UA = UB (c) UA > UB

I) Compare the potential energy ofq (UA) to that of 2q (UB)

The potential energy ofq is proportional to Qq/r

The potential energy of 2q is proportional to Q(2q)/(2r) = Qq/r

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The principle at work here is CONSERVATION OF ENERGY.Initially:The charge has no kinetic energy since it is at rest.The charge does have potential energy (electric) = UB.

Finally:

The charge has no potential energy (U

1/R)The charge does have kinetic energy = KE

(a) (b) (c)

II) Suppose charge 2qhas mass mand is released from restfrom the above position (a distance 2r from Q). What is itsvelocity vfas it approaches r = ?

mr

Qqvf

04

1

mr

Qqvf

02

1

0fv

Example 3

KEUB 2

0 2

1

2

)2(

4

1fmv

r

qQ

mrQq

vf0

2

2

1

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i i

i

q r

qqPE 0

04

1

0

Recall Case 1 from before

The potential energy of the

test charge, q0, was given by

Notice that there is a part of this equation that would

remain the same regardless of the test charge, q0,

placed at point a

i i

iq

r

qqPE

00

4

10

The value of the test charge can

be pulled out from the

summation

Electric Potential

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Electric Potential

We define the term to the right of the summation as

the electric potential at point a

i i

ia

r

qPotential

04

1

Like energy, potential is a scalarWe define the potential of a given point charge as

being

r

q

VPotential 04

1

This equation has the convention that the potential

is zero at infinite distance

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coulomb

joules

charge

Energy

Volts

The potential at a given point

Represents the potential energy that a positiveunit charge would have, if it were placed at that

point

It has units of

Electric Potential

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General Points for either positive or negative charges

The Potential increases if you move in thedirection oppositeto the electric field

andThe Potential decreases if you move in the samedirection as the electric field

Electric Potential

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What is the potential difference between points A and B?

VAB = VB - VAa)VAB >0 b)VAB =0 c)VAB

VBC

b)VAC

=

VBC

c)VAC

0 (c) Ex< 0

To obtain Exeverywhere, use

Example 10

The electric potential in a region of space is given by

The x-component of the electric field Ex at x = 2 is323)( xxx

We know V(x)everywhere

VE

dx

dVE x

2

36 xxE x

0)2(3)2(6)2( 2 xE