Lecture 7-1 Gravitational vs Electrostatic Potential Energy a b GravityCoulomb b a.

Lecture 7-Lecture 7-11

Gravitational vs Electrostatic Potential Energy

( ) ( )b

a

U U b U a

dF l

a

b

qE��

mg��

Gravity Coulomb

b

aldF

b

a


E from V

x

VE

x

y

VE

y

z

VE

z

VE

r

zyx

r

dzEdyEdxEldErV )()(

Potential are the same everywhere on a conductor

Lecture 7-Lecture 7-33Electric Potential Energy and Electric Potential

negative charge

High U

Low U

positive charge

High U (potential energy)

Low U

High V (potential)

Low V

Electric field direction

High V

Low V

Electric field direction

r

qkrV )(

Lecture 7-Lecture 7-44Reference Point for Potential of Uniformly Charged Infinite Sheet

Take a reference point at O.

0

0

( )

| | 2 | |2

xV x E dl

x k x

��

Or take it at some other point so that V(0)=V0:

0( ) 2 | |V x k x V

Px

O

equipotential

since V=V(x)

Lecture 7-Lecture 7-55Potential from uniformly charged spherical shell

• Potential

• r > R:

• r < R:

( )r

l

QV r E dl k

r

• Electric field (Gauss’s Law)

• r < R: E = 0

• r > R: E = kQ/r2

( )r R r

r r

l R

QV r E dl E dr E dr k

R

0

(or a charged solid spherical conductor)

++

++

+

+

Lecture 7-Lecture 7-66Potential of a Uniformly Charged (solid) Sphere

(2) r < R

( )Q

V r kr

(1) r > R(same as shell or conducting sphere)

2

3 3

2

2

( )

2

32

r R r

r r

l R

rr

R R

V r E dl E dr E dr

kQ kQ kQ kQ rr dr

R R R R

kQ r

R R

(very different!)

V

rR

+++

+

+

+

+

insulator

Lecture 7-Lecture 7-77Physics 241 –warm-up quiz

A infinite plane with uniform charge density +σ. What is the potential difference VB-VA?

A B

1 m2 m

+σ. a) (3/2)/0

b) /20

c) 3/0

d) /20

) 3/0


Charged Concentric Spherical Conductors

(a) r > c

a

b

c

Qin

Qout

r

V

( ) in outQ QV r k

r

(b) b < r < c

( ) .V r const

(c) a < r < b

( ) .inQV r k const

r

(d) r < a

( ) .V r const


Potential from continuous charge distribution: ring

2 2

( )

dqdV k

ra d

kx a

At point P on axis of ring Sum scalar contributions dV

2

2 20

2 2

k a dV dV

x akQ

x a

vector

Lecture 7-Lecture 7-1010Potential and Field of a Ring

At point P on axis of ring2 2 2

1

| | 1 ( / )

kQ kQV

xx a a x

vector

x

V(x)

0

x

y z

VE

xE E

x

Ex

Lecture 7-Lecture 7-1111High Electric Field at Sharp Tips

Two conducting spheres are connected by a long conducting wire. The total charge on them is Q = Q1+Q2.

Potential is the same: 1 2

1 2

kQ kQ

R R 1 1

2 2

Q R

Q R

1 2

2 1

E R

E R

The smaller the radius of curvature, the larger the electric field.

21

11 R

kQE

22

22 R

kQE

With same potential, sphere with smaller radius carry smaller amount of charge

Lecture 7-Lecture 7-1212 Lightning rod

Air “Break down” before too much charge accumulated, i.e. much weaker lightning which is much less destructive.

Golf court

Lecture 7-Lecture 7-1313 Equipotential Surfaces• An equipotential surface is a surface on which the potential is the same everywhere.

V E r E r ��

•Equipotential surfaces are drawn at constant intervals of V

•Potential difference between nearby equipotentials is approximately equal to E times the separation distance.

E an equipotential surface everywhere.

|| rE

Equipotential surfaces


Potential of a Uniformly Charged Sheet

02E

02

E

• Electric field is uniform on each side of the sheet as shown.

• Equipotential surfaces are to the electric fields.

• Separation between equipotential surfaces are equal to the potential differences divided by the magnitude of electric field.

V E r E r �� || rE

Lecture 7-Lecture 7-1515Physics 241 –Quiz 5a

A spherical shell of radius 50 cm is charged uniformly with a total charge of +Q Coulombs. What is the potential difference VB-VA?

a) kQ/2b) -2kQc) -(3/2)kQd) kQe) -kQ/2

QA B

1 m2 m

Lecture 7-Lecture 7-1616Physics 241 –Quiz 5b

Two parallel planes, 2 m apart, are charged with uniform charge densities 2 and (in C/m2). Respectively as shown. What is the potential difference VB-VA (in volts)?

a) (3/2)/0

b) 2/0

c) 3/0

d) /0

) 3/0

A B

2

2 m

Lecture 7-Lecture 7-1717Physics 241 –Quiz 5c

A spherical shell of radius 20 cm is charged uniformly with a total charge of Q Coulombs. What is the potential difference VB-VA?

a) kQ/2b) -2kQc) -(3/2)kQd) kQe) -kQ/2

QA B

1 m2 m

Lecture 7-1 Gravitational vs Electrostatic Potential Energy a b GravityCoulomb b a.

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