1 electric potential reading: chapter 21 chapter 21

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Electric Potential

Reading: Chapter 21

Chapter 21

2

If is an infinitesimal displacement of test charge, then the work done by electric force during the motion of the charge is given by

Electric Potential Energy

s

When a test charge is placed in an electric field, it experiences a force

E

q

F qE

s

FW s F

Fs

3

Electric Potential Energy

E

q

F qE

s This is the work done by electric field.

In this case work is positive.

Because the positive work is done, the potential energy of charge-field system should decrease. So the change of potential energy is

F FU s F q s E

minus sign

F mg g

s This is very similar to

gravitational force: the work done by force is

F Fs F mgs

The change of potential energy is

F FU s F mgs

F FW s F q s E

4

Electrical Potential Energy

E

q

F qE

s

A

B

B A ABU U Work

The electric force is conservative

For all paths:

B A ABU U U Work

Work is the same for all paths

5

Electric Potential

Electric potential is the potential energy per unit charge,

The potential is independent of the value of q.The potential has a value at every point in an electric fieldOnly the difference in potential is the meaningful quantity.

E

B A ABB A

U U WorkV V

q q q

UV

q

q

F qE

s

FsA

B

6

Electric Potential

To find the potential at every point 1. we assume that the potential is equal to 0 at some point, for example at point A, 2. we find the potential at any point B from the expression

E

AB ABB A

Work WorkV V

q q

0AV

B FV s E

q

F qE

s

FsA

B

7

Electric Potential: Example

Plane: Uniform electric field

E

0

σh

2εB AV V hE hE

A

σ 0

σ>0

h

0AV

h

s

E

s

0

σ

2εE

B

C

0

σh

2εC AV V hE hE

8



E

0

σh

2εBV

A

σ>0

h

hE

B

C

0

σh

2εCV

V

h0

σ 0

V

h0

σ 0

All points with the same h have the same potential

9



0

σh

2εV

0

σh

2εV

The same potential

equipotential lines

10


Point Charge

E

B e

QV k

r

0Q

0V

r

s

2

| |r e

QE k

rB

0V

11


Point Charge

r e

QV k

r0V

equipotential lines

12

Units

• Units of potential: 1 V = 1 J/C

– V is a volt

– It takes one joule (J) of work to move a 1-coulomb (C) charge through a potential difference of 1 volt (V)

Another unit of energy that is commonly used in atomic and nuclear physics is the electron-volt One electron-volt is defined as the energy a charge-field system gains or loses when a charge of magnitude e (an electron or a proton) is moved through a potential difference of 1 volt

1 eV = 1.60 x 10-19 J

13

Potential and Potential Energy

• If we know potential then the potential energy of point charge q is

U qV

(this is similar to the relation between electric force and electric field)

F qE

14

Potential Energy: Example

What is the potential energy of point charge q in the field of uniformly charged plane?

U QV

Q

h

σ

0

σh

2εV

0

σQh

2εU qV

U

h0

σQ>0repulsion

U

h0

σQ>0attraction

15


What is the potential energy of two point charges q and Q?

q QU qV

QQ e

QV k

rq

The potential energy of point charge q in the field of point charge Q

rq Q e

qQU qV k

r

This can be calculated by two methods:

A

Q qU QV

Qq e

qV k

rq

The potential energy of point charge Q in the field of point charge q

rQ q e

qQU QV k

r

B

In both cases we have the same expression for the energy. This expression gives us the energy of two point charges.

e

qQU k

r

16


Q q

r

Potential energy of two point charges:

e

qQU k

r

U

r0

σQ>0repulsion

U

r0

σQ<0attraction

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1q 2q

12r

Find potential energy of three point charges:

1 212

12e

q qU k

r

3q

23r13r

12 13 23U U U U

1 313

13e

q qU k

r 2 3

2323

e

q qU k

r

1 3 2 31 212 13 23

12 13 23e e e

q q q qq qU U U U k k k

r r r

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Potential Energy: Applications: Energy Conservation

1q 2q12,initialr

For a closed system: Energy Conservation:

The sum of potential energy and kinetic energy is constant

3q

23,initialr13r

12 13 23U U U U

1 3 2 31 212 13 23

12, 13 23,initial e e e

initial initial

q q q qq qE T U T U U U k k k

r r r

Example: Particle 2 is released from the rest. Find the speed of the particle when it will reach point P.2q

12,finalr

23,finalr2v

Initial Energy is the sum of kinetic energy and potential energy (velocity is zero – kinetic energy is zero)

2

2

mvT

- Potential energy

- Kinetic energy

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1q 2q12,initialr

For a closed system: Energy Conservation:


3q

23,initialr13r

21 3 2 32 2 1 2

12 13 2312, 13 23,2final e e e

final final

q q q qm v q qE T U T U U U k k k

r r r

2q12,finalr

23,finalr2v

Final Energy is the sum of kinetic energy and potential energy (velocity of particle 2 is nonzero – kinetic energy)

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1q 2q12,initialrFor a closed system: Energy Conservation:


3q

23,initialr13r

2q12,finalr

23,finalr2v

Final Energy = Initial Energy

21 3 2 3 1 3 2 31 2 2 2 1 2

12, 13 23, 12, 13 23,2e e e e e einitial initial final final

q q q q q q q qq q m v q qk k k k k k

r r r r r r

22 3 2 32 2 1 2 1 2

12, 12, 23, 23,2 e e e einitial final initial final

q q q qm v q q q qk k k k

r r r r

2 1 2 2 32 12, 12, 23, 23,

2 1 1 1 1e e

initial final initial final

v k q q k q qm r r r r

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Electric Potential of Multiple Point Charge

r e

QV k

r

1q2q 3q

4q

P

31 2 41 2 3 4

1 2 3 4r e e e e

qq q qV V V V V k k k k

r r r r

The potential is a scalar sum.

The electric field is a vector sum.

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Spherically Symmetric Charge Distribution

?V

Uniformly distributed charge

Q

Q

23

Spherically Symmetric Charge Distribution

2r e

QE k

r

E

B e

QV k

r

0Q

r

ds

B

0V

0V

r a

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σ>0

Important Example: Capacitor

σ>0

σ<0

σ<0

E

0E

0

σ

εE

0E

?V

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σ 0

Important Example

σ 0

E

0E

0

σ

εE

0E

0

σ

εV Eh h

0

σ

εV x

0V

x

0

σ

εh

V

x

h

x

x

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Electric Potential: Charged Conductor

The potential difference between A and B is zero!!!!

Therefore V is constant everywhere on the surface of a charged conductor in equilibrium– ΔV = 0 between any two points on the

surface The surface of any charged conductor is an

equipotential surface Because the electric field is zero inside the

conductor, the electric potential is constant everywhere inside the conductor and equal to the value at the surface

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Electric Potential: Conducting Sphere: Example

r e

QV k

r

sphere e

QV k

R

for r > R

for r < R

The potential of conducting sphere!!

28

Conducting Sphere: Example

69 10

9 10 900000.1sphere e

Q CV k V

R m

What is the potential of conducting sphere with radius 0.1 m and charge ?1μC

29

Capacitance

Chapter 21

30

Capacitors

Capacitors are devices that store electric charge• A capacitor consists of two conductors

– These conductors are called plates– When the conductor is charged, the plates carry

charges of equal magnitude and opposite directions• A potential difference exists between the plates due to

the charge

Q

Q

Q - the charge of capacitor

A

B

A

B A BV V V - a potential difference of capacitor

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Capacitors

• A capacitor consists of two conductors

A

B

conductors (plates)

Plate A has the SAME potential at all points because this is a conductor .

A BV V V

Plate B has the SAME potential at all points.

So we can define the potential difference between the plates:

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Capacitance of Capacitor

A

B

The SI unit of capacitance is the farad (F) = C/V.

Capacitance is always a positive quantity

The capacitance of a given capacitor is constant

and determined only by geometry of capacitor

The farad is a large unit, typically you will see

microfarads ( ) and picofarads (pF)

QC

V

μF

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Capacitor: Parallel Plates

0ε SQC

V h

E

σ 0

σ 02

1

0

σ

εV x

0V

0

σ

εV h

0

σ

εE

0E

0E

E0Q

0Q

Qσ

S

The potential difference:

0 0

σ Qh= h

ε ε SV

The capacitance:

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• Main assumption - the electric field is uniform

• This is valid in the central region, but not at the ends of the plates

• If the separation between the plates is small compared to the length of the plates, the effect of the non-uniform field can be ignored

Capacitor: Parallel Plates: Assumptions

Dielectric (insulator) inside capacitor. Capacitance = ?

Capacitors with Dielectric (Insulator)

Q

Q

Q C V

?C

dielectric

h

36

• The molecules that make up the dielectric are modeled as dipoles

• An electric dipole consists of two charges of equal magnitude and opposite signs

• The molecules are randomly oriented in the absence of an electric field

Dielectric (insulator)

37

Dielectric in External Electric Field

The molecules partially align with

the electric field

The degree of alignment of the

molecules with the field depends on

temperature and the magnitude of

the field

Dielectric in Electric Field

Polarization

38

Dielectric inside capacitor. Capacitance = ?

Capacitors with Dielectric

Q

Q

Q C V

?C

dielectric

h

The electric field inside dielectric

0 indE E

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Q

Q

Q C V

?C

dielectric

h

Q Q

If we have the same charge as without dielectric then the potential difference is increased, since

0( )indV h E E

without dielectric it was

0 0V hE

0V V then 00

Q QC C

V V

Capacitance is increased.

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Q

Q

Q C V

?C

dielectric

h

Q Q

Capacitance is increased

To characterize this increase the coefficient (dielectric constant of material) is introduced, so

(this is true only if dielectric completely fills the region between the plates)

0C κC

42

Type of Capacitors: Tubular

• Metallic foil may be interlaced

with thin sheets of paper

• The layers are rolled into a

cylinder to form a small

package for the capacitor

43

Type of Capacitors: Oil Filled

• Common for high- voltage

capacitors

• A number of interwoven

metallic plates are immersed

in silicon oil

44

Type of Capacitors: Electrolytes

• Used to store large amounts

of charge at relatively low

voltages

• The electrolyte is a solution

that conducts electricity by

virtue of motion of ions

contained in the solution

45

Type of Capacitors: Variable

• Variable capacitors consist of

two interwoven sets of metallic

plates

• One plate is fixed and the

other is movable

• These capacitors generally

vary between 10 and 500 pF

• Used in radio tuning circuits

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Capacitor: Charging

Battery- produces the fixed voltage – the fixed potential difference

Each plate is connected to a terminal of the battery The battery establishes an electric field in the connecting wires This field applies a force on electrons in the wire just outside of the plates The force causes the electrons to move onto the negative plate This continues until equilibrium is achieved

The plate, the wire and the terminal are all at the same potential

At this point, there is no field present in the

wire and there is no motion of electrons

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Capacitance and Electrical Circuit

Chapter 21

48

Electrical Circuit

A circuit diagram is a simplified representation of an actual circuit Circuit symbols are used to represent the various elements Lines are used to represent wires The battery’s positive terminal is indicated by the longer line

49

Electrical Circuit

Conducting wires. In equilibrium all the points of the wires have the same potential

50

Electrical Circuit

The battery is characterized by the voltage – the potential difference between the contacts of the battery

V

In equilibrium this potential difference is equal to the potential difference between the plates of the capacitor.

V

Then the charge of the capacitor is

Q C V

If we disconnect the capacitor from the battery the capacitor will still have the charge Q and potential differenceV

V

51

Electrical Circuit

V

VQ C V

If we connect the wires the charge will disappear and there will be no potential difference

V

0V

52

One of the main application of capacitor:

capacitors act as energy reservoirs that can be

slowly charged and then discharged quickly to

provide large amounts of energy in a short pulse

Energy Stored in a Capacitor: Application

Q

Q

221 1

( )2 2 2

QU Q V C V

C

Q C V

53

Electric Potential and Electric Field

Can we find electric field if we know electric potential?

E

x

B AV V x E

A B

B AV V V

Ex x

V

Ex

x

54

Electric Potential and Electric Field

V

Es

Equipotential lines are everywhere perpendicular to the electric field.

Equipotential lines

1 electric potential reading: chapter 21 chapter 21

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