Electric Potential, Energy, and Capacitance

Objective: TSW understand, transfer and apply energy concepts to electric fields and charges by solving problems involving electric fields and forces.

Let’s revisit energy concepts for a gravitational field. A 250 gram baseball is thrown upward with an initial velocity of 25m/s. What is the maximum height reached by the ball?

EARTH

v = 25m/s

v = 0m/s fEWE 0gUK

mghmv 22

1

g

vh

2

2

)10(2

)25( 2

h

mh 3.31

Let’s make a parallel comparison of Gravitational Potential Energy (Ug) to Electric Potential Energy (UE).

mghU g Units: JNmmkg

Nkg

qEdUE Units: JNmmC

NC

Now we will do an example:

An electron is released from rest in an electric field of 2000N/C. How fast will the electron be moving after traveling 30cm?

_v = 0m/s

_v = ?

30cm

fEWE 0KUE

2

2

1mvqEd

m

qEdv

2

31

19

1011.9

)3)(.2000)(106.1(2

v

s

mv 71045.1

A new quantity is defined called the Electric Potential Difference. It is the Electric Potential Energy per unit Charge.

The Electric Potential Difference can be considered to be the electric “push” motivating a charge to move from one point to another. It is independent of the size of the charge.

eCh

EnergyPotentialElectricDifferencePotentialElectric

arg

q

UV E

q

qEdV

EdV Units: Vvolt

C

J

The Electric Potential Difference can also called the Electric Potential, the Potential, or the voltage.

Remember Energy and voltage are scalars, so you don’t have to deal with vectors (direction)

The electric potential energy can now be written in terms of the electric potential.

qEdUE EdV

qVUE

Let’s do an example using this new concept:

The potential difference between to charge plates is 500V. Find the velocity of a proton if it is accelerated from rest from one plate to the other.

500V

+

+

+

+

+

-

-

-

-

-

+

s

mv

v

m

qVv

mvqV

KU

EWE

E

f

5

27

19

2

0

101.3

1067.1

)500)(106.1(2

2

2

1

High

Potential

Low

Potential

Positive charges move from high to low potential

Negative charges move from low to high potential

Let’s revisit a gravitational field again:

The gravitational potential energy if the field is constant is given by:

mghU g

But, what if the field is not constant. In other words the gravitational potential energy of a mass located far away from the earth.

m

r

The gravitational potential energy of mass m is given by:

r

mMGU E

g Notice r is not squared

The gravitational potential energy at infinity is zero

Let’s us this new equation to do an example:

With what velocity must a rocket be fired in order to escape the earth’s gravitational field? (Neglect air resistance)

r

GMv

r

mMGmv

r

mMGmv

UK

EWE

E

E

E

g

f

2

2

1

02

1

0

2

2

0

s

mv

v

11000

1038.6

)1098.5)(1067.6(26

2411

Let’s make a parallel comparison of Gravitational Potential Energy (Ug) to Electric Potential Energy (UE) when dealing with individual masses or charges.

r

mMGU E

g

The gravitational potential energy between two masses is given by:

The electric potential energy between two masses is given by:

r

qqkU E

21

Example:

Two 40 gram masses each with a charge of -6µC are 20cm apart. If the two charges are released, how fast will they be moving when they are a very, very long way apart. (infinity)

s

m

rm

kqv

rm

kqv

mvr

qk

mvmvr

qqk

KKU

EWE

E

f

0.2)4)(.2(.

)106)(109(

2

1

2

1

2692

22

22

2221

21

0

Let’s use the electric potential energy between two charges to derive an equation for the electric potential (voltage) due to a single point charge.

q1

q2

r

The electric potential energy is given by:

r

qqkUE

21Let’s remove charge q2 and consider the electric potential (voltage) at point P.

P

r

qk

qrqkq

q

UV E 1

2

21

To find the potential due to more than one point charge simply add up all the individual potentials:

i

i

r

qkV

Example 1: The electron in the Bohr model of the atom can exist at only certain orbits. The smallest has a radius of .0529nm, and the next level has a radius of .212m.

a) What is the potential difference between the two levels?

b) Which level has a higher potential?

+er1

r2

r

qkV

11 r

ekV

VV 2.27100529.

106.1)109(

9

199

1

VV 79.6100212.

106.1)109(

9

199

2

VVdiffpotential 4.2079.62.27

r1 is at a higher potential.

Example 2

What is the electric potential at the center of the square?

45º45º

r

r r

r

mr

r

rr

071.

01.2

10.02

222

C

JV

V

r

qkV

6

69

1027.1

071.

1010109

C

JV

V

r

qkV

5

69

1034.6

071.

105109

C

JV

V

r

qkV

total

total

i

i

6

5566

1027.1

1034.61034.6)1027.1(1027.1

Example 3: A proton is moved from the negative plate to the positive plate of a parallel-plate arrangement. The plates are 1.5cm apart, and the electric field is uniform with a magnitude of 1500N/C.

a) How much work would be required to move a proton from the negative to the positive plate?

b) What is the potential difference between the plates?

c) If the proton is released from rest at the positive plate, what speed will it have just before it hits the negative plate?

cosxFW qEFE

JW

mC

NCW

xqEW

18

19

106.3

)015)(.1500)(106.1(

1

Example 3: A proton is moved from the negative plate to the positive plate of a parallel-plate arrangement. The plates are 1.5cm apart, and the electric field is uniform with a magnitude of 1500N/C.

b) What is the potential difference between the plates?

C

JV

mC

NV

EdV

5.22

)015)(.1500(

Example 3: A proton is moved from the negative plate to the positive plate of a parallel-plate arrangement. The plates are 1.5cm apart, and the electric field is uniform with a magnitude of 1500N/C.

c) If the proton is released from rest at the positive plate, what speed will it have just before it hits the negative plate?

Use conservation of energy

fEWE 0

s

mv

v

m

qVv

mvqV

KUE

4

27

19

2

1057.6

1067.1

)5.22)(106.1(2

2

2

1

Example 4:

Compute the energy necessary to bring together the charges in the configuration shown below:

Calculate the electric potential energy between each pair of charges and add them together.

JU

U

r

qqkU

72.02.

)104)(104()109(

12

669

12

2112

JU

U

r

qqkU

72.02.

)104)(104()109(

13

669

13

3113

JU

U

r

qqkU

72.02.

)104)(104()109(

23

669

23

3223

JU

JJJU

total

total

72.

)72.()72.(72.

Capacitors

Two parallel conducting plates used to store charge, which in turn

stores energy

Charge flows through the circuit until the

capacitor is charge at which point no more

charge flows.

Capacitance

The amount of charge stored per volt of potential difference.

V

QC

C = Capacitance (C/V = Farad (F).

Q = The stored charge (C).

V = The potential difference (voltage) between the plates (V).

Capacitance based on the physical characteristics of the capacitor

d

AC 0

Є0 = Permittivity of free space = 8.85 x 10-12

A = Surface area of one plate.

d = distance between plates.

Example 1: Find the Charge stored in the capacitor.

12V 20μF

CQ

VV

CQ

CVQV

QC

4

6

104.2

)12)(1020(

Capacitors in Series

........1111

321

CCCCs

i is CC

11

Example 2: Find the total charge stored in each the capacitor.

FC

C

CC

s

s

i is

45.530

1

10

1

20

11

11

12V 10μF

20μF

30μF

First calculate the total capacitance.Now find the total charge

CQ

VFQ

CVQV

QC

5

6

1054.6

)12)(1045.5(

In a series circuit the charge stored on each capacitor is the same and the voltage is split.

Capacitors in Parallel

......321 CCCC p

i

ip CC

Example 3: Find the total charge stored in all the capacitors.

FC

C

CC

p

p

iip

60

301020

12V 10μF20μF 30μF

First find the total capacitance.

Now calculate the total charge.

CQ

VFQ

CVQV

QC

4

6

102.7

)12)(1060(

In a parallel circuit the charge stored on each capacitor can be different and the voltage must be the same.

Example 4: Find the total capacitance of all the capacitors.

12V

12μF

8μF 4μF

First find the total capacitance of the parallel capacitors

FC p 1248

Now, combine this value in series with 12μF capacitor.

FC

C

total

total

612

1

12

11

Energy Stored in a Capacitor

22

2

1

22

1CV

C

QQVUC

Units: Joules (J)