electric potential with integration potential difference in a variable e-field if e varies, we...

Electric Potential with Integration

Potential Difference in a Variable E-field

• If E varies, we integrate the potential difference for a small displacement dl over the path from a to b

b

a

ab dVVV

ddV

lE

lE

©2008 by W.H. Freeman and Company

Potential Difference in a Variable E-field

• Like gravity, the electric field is conservative.

• This means that the potential does not depend on the path taken from a to b.


Potential vs. Potential Difference

• Like gravitational potential energy, only differences in electrical potential energy have physical meaning.

• We can choose a convenient reference level to be the zero of potential.

• The potential V of a point is the potential difference between that point and the reference level.

Example: Potential of a point charge

• Calculate the electric potential at a distance r away from a point charge q.

• We choose the reference level to be at infinity.

• We integrate along a radius, so dl = dr.

•

2r

kqE

PP rr

EdrdV lE

drr

kqV

Pr

2

r

kq

r

kqV

P

r

kqV

Quick Review: Potential

• The potential difference between two points is the change in potential energy per unit charge if a charge were moved from one point to the other.

• The potential difference can be found using the electric field.

• The potential of a point change is

r

kqV

0q

UV

b

a

dV lE

Potential and Potential Energy

• We take the zero of potential energy the same as the zero of potential.

• Under those conditions, UE = qV, where q is the charged placed at the position with potential V.

• The formula chart gives the potential energy of charge q2 located a distance r away from q1 as

r

qqU E

21

04

1

Superposition of Potentials

• If more than one charge is present, the potential at a point is the sum of the potentials of each charge.

• The formula on the formula chart is

i i

i

r

qV

04

1

Example- Potential of two charges

•

• In this arrangement, r1 = |x| and r2 = |x-a|

•

2

2

1

1

r

kq

r

kqV

ax

kq

x

kqV

21


Calculating V for Continuous Charge Distributions

Integrating over a charge distribution

• If instead of point charges, we have a distribution of charge, we treat each small element of charge as a point charge, and integrating over all the charge elements.

r

dqkV

r

kqV

i i

i

V on the Axis of a Ring

elements. charge allfor constant a is

22

r

azr

22

22

az

kQV

dqaz

kV

22

az

dqk

r

dqkV


Charge density

• Linear ()– = Q/L = dQ/dL

• Surface ()– = Q/A = dQ/dA

• Volume ()– = Q/V = dQ/dV

Electric Field

• Spherical symmetry E = 1 q

4o r2

• Continuous charge distributiondE = 1 dq

4o r2

For linear distribution

dE = 1 dq

4o r2

dE = 1 dx

4o r2

E = E = dE

For linear distribution

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

P

cylindrical symmetry in electric field

V near a plane of Charge

• We cannot do as we did for electric field, that is, calculate V for a disk, and then let the size of the disk grow to infinity.– This would yield an

infinite potential.

• We can’t put the zero of potential at infinity, for the same reason.


V near a plane of Charge

• Instead, we start with the electric field of a plane of charge, and integrate along a path from the plane to a distance x away.

x

We say the potential at the surface of the sheet is V0.


Potential near a plane of charge

.on only depends and

constant, is field electric The

2 0

E

0

V

dxdVx

0 02

lE

x

dxV002

002

V

xV

x


Potential near a plane of charge

00 2

x

VV 0

V

00 2

VV


Potential near a charged shell

• We consider first the potential at a point outside the shell. (r>R)


Potential near a charged shell: Outside

2

2

r

drkQdrEdV

r

kQE

rr

r

drkQ

r

drkQV

22

Rrr

kQV


Potential near a charged shell: Outside

Rrr

kQV

• Outside a charged shell, the potential is the same as for a point charge.


Potential near a charged shell: Inside

RrR

kQV

• “Inside” the shell, there is no charge.

• The field is zero inside.

• If the field is zero, the potential cannot change, so V is what it is at the surface.


Energy Storage

Potential Energy of a System of Charges

• Consider a system of two equal charges, q1 and q2.

• Putting the first charge in place requires no energy.

• Putting the second charge requires q2V, where V is the potential of the charges.

• q2 is ½ the total charge Q, so the energy can also be written

• QVU 21

Potential Energy of a System of Charges

• The potential energy of a system of charges qi is given by

i

iiVqU 21

Computing the Electric Field from the Potential

• The electric field points in the direction of greatest change in potential.

• In the one dimensional case,

dxEV x

dx

dVEx


• In general,

dr

dVEr


• In vector calculus notation

z

VE

y

VE

x

VE zyx

kjiE ˆˆˆz

V

y

V

x

VV

electric potential with integration potential difference in a variable e-field if e varies, we...

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