Physics 2112 Unit 6: Electric Potential...Example 6.7: ( V for charges) A solid metal conducting sphere of radius a = 2.5 cm has a net charge Q in = - 3 nC. The sphere is surrounded

Physics 2112

Unit 6: Electric Potential

Today’s Concept:

Electric Potential

(Defined in terms of Path Integral of Electric Field)

Unit 6, Slide 1

Big Idea

Last time we defined the electric potential energy of charge q in an electric field:

∆𝑈𝑎→𝑏 = −න𝑎

𝑏

Ԧ𝐹 ∙ 𝑑Ԧ𝑙 = −න𝑎

𝑏

𝑞𝐸 ∙ 𝑑Ԧ𝑙

The only mention of the particle was through its charge q.

We can obtain a new quantity, the electric potential, which is a PROPERTY OF THE SPACE, as the potential energy per unit charge.

Note the similarity to the definition of another quantity which is also a PROPERTY OF THE SPACE, the electric field.

b

a

baba ldE

q

UV

q

FE

Unit 6, Slide 2

Electricity & Magnetism Lecture 2, Slide 3

We never talk about the “gravitational potential” in 2111. We just talked about a “gravitational force”.

Let’s say we had. What is the magnitude of the gravitational potential close to the surface of the earth?

A) g

B) mgh

C) gh

D) mg

Example 6.2a (Potential from Field)

= 6 N/C|| E

q=+8uC

A +8uC charge is placed in a

downwards pointing electric field

with a magnitude of 6N/C. An

outside force moves the charge

up a distance of 0.4m from point

1 to point 2.

a) What is the force on this charge?

b) How much work was done by the outside force during this

move?

c) How much work was done by the electric field during this

move?

d) What is the change in the electrical potential energy of the

particle?

e) What is the difference in electrical potential between points 1

and 2?

1

2

In the example just worked with a positive

charge q and the field pointing down, we found

that

a) the force on the charge was -4.8 X 10-5Nb) the work done moving the charge up was –1.92 X 10-5Jc) the change in electrical potential energy (EPE) was +1.92 X 10-5Jd) the change in the electrical potential was +2.4V

q

E2

+

+

A. (a)

B. (b)

C. (c)

D. ( d)

E. All of them would change

Which of these values would not change if I redid the

problem, switching q to a negative (-8mC) charge?

1

Question

when we moved the charge upwards 0.4m.



that


q

E2

+

+

A. (a)

B. (b)

C. (c)

D. ( d)

E. All of them would change

Which of these values would not change if we went

back to the original +8mC charge, but now the field

pointed up?

1

Question




that


q

E2

+

+

What it the electrical potential of the original point 1?

1

Question


A. -2.4V

B. 0V

C. 4.8V

D. -4.8V

E. We can’t tell.

Electric Potential from E field

Consider the three points A, B, and C located in a region of constant electric field as shown.

What is the sign of VAC VC VA ?

A) VAC < 0 B) VAC 0 C) VAC > 0

Choose a path (any will do!)

D

Dx

C

D

D

A

CA ldEldEV

00 < xEldEV

C

D

CA

Unit 6, Slide 8

CheckPoint: Zero Electric Field


Suppose the electric field is zero in a certain region of space. Which of the following statements best describes the electric potential in this region?

A. The electric potential is zero everywhere in this region.B. The electric potential is zero at least one point in this

region.C. The electric potential is constant everywhere in this

region.D. There is not enough information given to distinguish

which of the above answers is correct.

B

A

BA ldEV

Remember the definition

Example 6.2 (V from point charges)

Unit 6, Slide 10

A B x+Q Q

10cm 10cm 10cm

+5nc -5nc

What is the electrical potential at

points A and B?

Define V = 0 at r =(standard)

Example 6.3 (V from point charges)

Unit 6, Slide 11

C

x+Q Q

10cm 20cm

10cm+5nc -5nc

What is the electrical potential at

point C?

Define V = 0 at r =(standard)

E from V

If we can get the potential by integrating the electric field:

We should be able to get the electric field by differentiating the potential?

b

a

ba ldEV

VE

In Cartesian coordinates:

dx

VEx

dy

VEy

dz

VEz


Example 6.4 (E-field above a ring of charge)

Unit 2, Slide 13

a

y

P What is the electrical

potential, V, a distance y

above the center of ring

of uniform charge Q and

radius a? (Assume V = 0

at y = )

y

x

What is the electrical

field, E, at that point?

CheckPoint: Spatial Dependence of Potential 1


The electric potential in a certain region is plotted in the following graph

At which point is the magnitude of the E-FIELD greatest?

A. AB. B C. CD. D

CheckPoint: Spatial Dependence of Potential 2


The electric potential in a certain region is plotted in the following graph

At which point is the direction of the E-field along the negative x-axis?

A. AB. B C. CD. D

WARNING!!!!

Unit 6, Slide 16

Good news: They’re going to be

really, really simple. Sometimes

so simple we can do them

conceptually.

We’re about to do some

integrals.

Example 6.5: Charged Plates

sR= +6nC/m2sL= -2nC/m2

10cm

30cm

15cm

P W

Two infinite plates are placed

30cm apart and are charged

with the surface charge

densities shown.

What is the electrical potential

difference between point P

which is 10cm to the left of the

negative plate and point W

which is 15cm to the right of

the positive plate?

Example 6.6: (V near line of charge)

Unit 2, Slide 18

An infinitely long solid insulating cylinder of radius a = 4.1 cm

is positioned with its symmetry axis along the z-axis as

shown. The cylinder is uniformly charged with a charge

density ρ = 27.0 μC/m3. Concentric with the cylinder is a

cylindrical conducting shell of inner radius b = 19.9 cm, and

outer radius c = 21.9 cm. The conducting shell has a linear

charge density λ = -0.36μC/m.

What is V(P) – V(R), the potential

difference between points P and R?

Point P is located at

(x,y) = (46.0 cm, 46.0 cm).

Point R is located at

(x,y) = (0 cm, 46.0 cm).

C

Example 6.7: (V for charges)

A solid metal conducting sphere of radius a = 2.5 cm has a net charge Qin = - 3 nC. The sphere is surrounded by a concentric conducting spherical shell of inner radius b = 6 cm and outer radius c = 9 cm. The shell has a net charge Qout = + 2 nC.

What is V0, the electric potential at the center of the metal sphere, given the potential at infinity is zero?

Plan:

Two Step process:

- Step #1: Use Gauss’ Law to calculate E everywhere

- Step #2: Integrate E to get V

cross-section


B

A

BA ldEV

o

enc

surface

QAdE

Example 6.6 (V for charges)

A solid metal conducting sphere of radius a = 2.5 cm has a net charge Qin = - 3 nC. The sphere is surrounded by a concentric conducting spherical shell of inner radius b = 6 cm and outer radius c = 9 cm. The shell has a net charge Qout = + 2 nC.

What is V0, the electric potential at the center of the metal sphere, given the potential at infinity is zero?

cross-section


E V

Example 6.8: (V for spherical shells)

Point charge q at center of concentric conducting spherical shells of radii a1, a2, a3, and a4. The inner shell is uncharged, but the outer shell carries charge Q.

What is V as a function of r?

Plan:

Two Step process:



metal

metal

+Q

a1

a2

a3

a4

+q

cross-section


B

A

BA ldEV

o

enc

surface

QAdE

Question

A) 0 B) C)


metal

metal

+Q

a1

a2

a3

a4

D) E)

Why?Gauss’ law:

r

0enclosedQ

AdE

+q

cross-section

2

40 )(4

1

ar

Q

2

40 )(2

1

ar

qQ

+

2

04

1

r

qQ

2

04

1

r

qQ +

0

24

qQ

rE+

r > a4 : What is E(r)?

metal

metal

+Q

a1

a2

a3

a4

+q

cross-section

Calculation: Quantitative Analysis


a2 < r < a3 :

r

r < a1 : 2

04

1

r

qE

a4 < r :

a3 < r < a4 : 0E

a1 < r < a2 : 0E

2

04

1

r

qQ +

metal

metal

+Q

a1

a2

a3

a4

+q

cross-section

Question


r

a3 < r < a4 ?

A)

B)

C)

404

1

a

qQV

+

304

1

a

qQV

+

0V

What is V for

( ) 04 + aV

r

qQV

+

04

1

D)

Example 6.9: (V for Insulator)

A spherical insulator has a total charge +Q and a radius of a.

What is V as a function of r?

insulator

+Q

a


Plan:

Two Step process:




B

A

BA ldEV

o

enc

surface

QAdE

insulator

+Q

a

E

r

Unit 2, Slide 27

Equipotentials

In previous example,

all these points had

same V, same

electrical potential

Line is called “equipotenial”

or “a line of equipotential”.

Topographic Map

Lines on topo map

are lines of “equal

gravitational

potential”

Closer the lines are,

the steeper the hill

Gravity does no

work when you walk

along a brown line.

http://en.wikipedia.org/wiki/Image:Topographic_map_example.png

Equipotential for Point Charge

Equipotentials produced by a point charge

SPACING of the equipotentials indicates STRENGTH of the E field.


0V 0elecW 0 ldE

Equipotentials always perpendicular to field lines.

Question

Unit 6, Slide 30

Which of the equipotential

diagrams shown to the left best

describes the spherical insulator

in the described below?

(The colored circle in the center

represents the insulator.)

CheckPoint: Electric Field Lines 1


The field-line representation of the E-field in a certain region in space is shown below. The dashed lines represent equipotential lines.

At which point in space is the E-field the weakest?

A. AB. BC. CD. D

CheckPoint: Electric Field Lines 3


Compare the work done moving a negative charge from A to B and from A to D. Which one requires more work?

The field-line representation of the E-field in a certain region in space is shown below. The dashed lines represent equipotentiallines.

A. More work is required to move a negative charge from A to B than from A to D

B. More work is required to move a negative charge from A to D than from A to B

C. The same amount of work is required to move a negative charge from A to B as to move it from A to D

D. Cannot determine without performing the calculation

The diagram to the right shows

equally space equipotential lines 1V

apart. Where on the above diagram

is the electric field the strongest?

A. On the left hand side

B. On the right hand side

C. In the middle

D. It’s constant throughout

E. We can’t tell about the electric field from the equipotential lines.

12V 14V10V

Question

An electric field is represented

by the equipotential surface

lines shown to the right. If a

positive charge is place in this

field, how will it move?

A. Constant velocity to the left

B. Constant velocity to the right

C. Constant acceleration to the left

D. Constant acceleration to the right

E. None of the above

0V-100V

(Each line represent a change of 20V.)

Question

Which of the below drawings of

possible equipotential surfaces

could represent the electrical field

lines shown to the right?

A. (a)

B. (b)

C. (c)

D. (a) or (c) depending on the sign of the test charge

E

(a)(b) (c)

Question

Which of the below drawings of

possible equipotential surfaces

could represent the electrical field

lines shown to the right?

A. (a)

B. (b)

C. (c)

D. (a) or (c)

E. (a) or (b)

E

(a) (b) (c)

10V

50V

50V

10V -10V

-50V

Question

Documents

Physics 2112 Unit 6: Electric Potential...Example 6.7: ( V for charges) A solid metal conducting sphere of radius a = 2.5 cm has a net charge Q in = - 3 nC. The sphere is surrounded