Physics 2112Unit 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:

b

a

b

aba ldEqldFU

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

qUV

qFE

Unit 6, Slide 2

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?

Example 6.2.b (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.

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?

Unit 6, Slide 4

Example 6.2.c (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.

e) What is the difference in electrical potential between points 1 and 2?

f) What is the electrical potential at point 2?

Unit 6, Slide 5

Electric Potential from E fieldConsider 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

ACA ldEldEV

00 < xEldEV

C

DCA

Unit 6, Slide 6

CheckPoint: Zero Electric Field

Electricity & Magnetism Lecture 6, Slide 7

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

ABA ldEV

Remember the definition

Example 6.2 (V from point charges)

Unit 6, Slide 8

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 9

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

aba ldEV

VE

In Cartesian coordinates:

dxVEx

dyVEy

dzVEz

Electricity & Magnetism Lecture 6, Slide 10

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

Unit 2, Slide 11

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

Electricity & Magnetism Lecture 6, Slide 12

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

Electricity & Magnetism Lecture 6, Slide 13

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

Example 6.5 (V near line of charge)

Unit 2, Slide 14

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).

Example 6.5 (V for charges)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: Spherical symmetry: Use Gauss’ Law to calculate E everywhere Integrate E to get V

metal

metal

+Q

a1

a2

a3

a4

+q

cross-section

Main Idea: Charges q and Q will create an E field throughout space

dErVr

r

0

)(

Electricity & Magnetism Lecture 6, Slide 15

Unit 2, Slide 16

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.

Equipotentials

Equipotentials produced by a point

charge

SPACING of the equipotentials indicates STRENGTH of the E field.

Electricity & Magnetism Lecture 6, Slide 18

0V 0elecW 0 ldE

Equipotentials always perpendicular to field lines.

CheckPoint: Electric Field Lines 1

Electricity & Magnetism Lecture 6, Slide 19

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 2

Electricity & Magnetism Lecture 6, Slide 20

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

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

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

D. Cannot determine without performing the calculation

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