Weds., Jan. 29, 2014PHYS 1442-004, Dr. Andrew Brandt 1 PHYS 1442 – Section 004 Lecture #5...

Weds., Jan. 29, 2014 PHYS 1442-004, Dr. Andrew Brandt 1

PHYS 1442 – Section 004 Lecture #5

Wednesday January 29, 2014Dr. Andrew Brandt

CH 17

• Electric Potential due to Point Charges• Shape of the Electric Potential• Equi-potential Lines and Surfaces• Electron-volt • Capacitance

Announcements

a) Avg was 81 on HW. Good job, you should dominate that material on the test. b) New HW assigned on ch 17

Weds., Jan. 29, 2014 2PHYS 1442-004, Dr. Andrew Brandt


Electric Potential and Potential Energy • The electric potential difference gives potential energy (or

the possibility to do work) based on the charge of the object.• So what is happening in batteries or generators?

– They maintain a potential difference.– The actual amount of energy used or transformed depends on how

much charge flows.– How much is the potential difference maintained by a car’s battery?

• 12Volts– If for a given period, 5C charge flows through the headlight lamp,

what is the total energy transformed?• Etot=5C*12V=60 What is the unit?

– If it is left on twice as long? Etot=10C*12V=120J.C*J/C=J (Joules)


Example 17 – 2 Electrons in a TV tube: Suppose an electron in the picture tube of a television set is accelerated from rest through a potential difference Vba=+5000V. (a) What is the change in potential energy of the electron? (b) What is the speed of the electron (m=9.1x10-31kg) as a result of this acceleration? (c) Repeat for a proton (m=1.67x10-27kg) that accelerates through a potential difference of Vba=-5000V.

• (a) What is the charge of an electron?–

• So what is the change of its potential energy?

U baqV 19 161.6 10 5000 8.0 10baeV C V J

191.6 10e C


Example 17 – 2 • (b) Speed of the electron?

– The entire potential energy of the electron is transformed into kinetic energy. Thus the equation is

ev

K

• (c) Speed of a proton that accelerates through V=-5000V?

pv

K

210

2 e em v W U ( )baU eV

16 168.0 10 8.0 10J J

2

e

K

m

167

31

2 8.0 104.2 10 /

9.1 10m s

210

2 p pm v 168.0 10ba baW U e V eV J

2 ba

p

eV

m

165

27

2 8.0 109.8 10 /

1.67 10m s


Electric Potential and Electric Field• The effect of a charge distribution can be

described in terms of electric field or electric potential.– What kind of quantities are the electric field and the

electric potential?• Electric Field:• Electric Potential:

– Since electric potential is a scalar quantity, it often can make problem solving easier.

VectorScalar


50V

5cm

Example 17 – 3 Uniform electric field obtained from voltage: Two parallel plates are charged to a voltage of 50 V. If the separation between the plates is 5.0 cm, calculate the magnitude of the electric field between them, ignoring any fringe effects.

E V

d

50

5.0

V

cm

What is the relationship between electric field and the potential for a uniform field? ( )

/ /

W Fd Eq d

V U q W q Ed

V Ed

Solving for E 2

50

5 10

V

m

1000 /V m


Electric Potential due to Point Charges• Since only the differences in potential have physical

meaning, we can choose at .• The electrical potential V at a distance r from a

single point charge is

• So the absolute potential from a single point charge depends only on the magnitude of the point charge and the distance from it

V 0

1

4

Q

r

0bV br


• What are the differences between the electric potential and the electric field?– Electric potential

• Electric potential energy per unit charge• Inversely proportional to the distance• Simply add the potential from each of the charges to obtain the total potential

from multiple charges, since potential is a scalar quantity

– Electric field• Electric force per unit charge• Inversely proportional to the square of the distance• Need vector sums to obtain the total field from multiple charges

• Potential for a positive charge is large near the charge and decreases to 0 at large distances.

• Potential for the negative charge is small (large magnitude but negative) near the charge and increases with distance to 0

Properties of the Electric Potential

20

1

4

QE

r

r

0

1

4

QV

r


Shape of the Electric Potential• So, what does the electric potential look like as a function of

distance?– What is the formula for the potential by a single charge?

V 0

1

4

Q

rPositive Charge Negative Charge

A uniformly charged sphere would have the same potential as a single point charge.

What does this mean? Uniformly charged sphere behaves like all the charge is on the single point in the center.


Since we obtain

Example 17.5Work to bring two positive charges close together: What minimum work is required by an external force to bring a charge q=3.00 μC from a great distance away ( ) to a point 0.500 m from a charge Q=20.0 μC?What is the work done by the electric field in terms of potential energy and potential?

W

0.500 ,b ar m r

W

In other words, the external force must input work of +1.08J to bring the charge 3.00C from infinity to 0.500m from the 20.0C charge.

baqV 04 b a

q Q Q

r r

0

04 b

q Q

r

04 b

q Q

r 9 2 2 6 68.99 10 3.00 10 20.00 10

1.080.500

N m C C CJ

m

r

12Weds., Jan. 29, 2014 PHYS 1442-004, Dr. Andrew Brandt

Since we obtain

More on Example 17-5 Work to bring two positive charges close together: What minimum work is required by an external force to bring a charge q=3.00 μC from a great distance away ( ) to a point 0.500 m from a charge Q=20.0 μC?What is the work done by the electric field in terms of potential energy and potential?

W

0.500 ,b ar m r

W

In other words, the external force must input work of +1.08J to bring the charge 3.00C from infinity to 0.500m from the 20.0C charge.

baqV 04 b a

q Q Q

r r

0

04 b

q Q

r

04 b

q Q

r 9 2 2 6 68.99 10 3.00 10 20.00 10

1.080.500

N m C C CJ

m

r


Electrostatic Potential Energy: Two charges• What is the electrostatic potential energy of a configuration of

charges? (Choose V=0 at r=)– If there are no other charges around, a single point charge Q1 in

isolation has no potential energy and feels no electric force

• If a second point charge Q2 is to a distance r12 from Q1 ,the potential at the position of Q2 is

• The potential energy of the two charges relative to V=0 at r= is

-- This is the work that needs to be done by an external force to bring Q2 from infinity to a distance r12 from Q1.

– It is also a negative of the work needed to separate them to infinity.

V 1

0 12

1

4

Q

r

2U Q V 1 2

0 12

1

4

Q Q

r


Electrostatic Potential Energy: Three Charges• So what do we do for three charges? • Work is needed to bring all three charges together

– There is no work needed to bring Q1 to a certain place without the presence of any other charge

– The work needed to bring Q2 to a distance to Q1 is

– The work need to bring Q3 to a distance to Q1 and Q2 is

• So the total electrostatic potential of the three charge system is

12U

3U

U

1 2

0 12

1

4

Q Q

r

13U 1 3

0 13

1

4

Q Q

r 2 3

0 23

1

4

Q Q

r23U

12 13 23U U U 1 3 2 31 2

0 12 13 23

1 0 at

4

Q Q Q QQ QV r

r r r


Equi-potential Surfaces• Electric potential can be visualized using equipotential lines in

2-D or equipotential surfaces in 3-D• Any two points on equipotential surfaces (lines) have the same

potential• What does this mean in terms of the potential difference?

– The potential difference between the two points on an equipotential surface is 0.

• How about the potential energy difference?– Also 0.

• What does this mean in terms of the work to move a charge along the surface between these two points?– No work is necessary to move a charge between these two points.


Equi-potential Surfaces• An equipotential surface (line) must be perpendicular to the electric field. Why?

– If there are any parallel components to the electric field, it would require work to move a charge along the surface.

• Since the equipotential surface (line) is perpendicular to the electric field, we can draw these surfaces or lines easily.

• There can be no electric field inside a conductor in static case, thus the entire volume of a conductor must be at the same potential.

• So the electric field must be perpendicular to the conductor surface.

Point charges

Parallel Plate Just like a topographic map


Electrostatic Potential Energy: electron Volt• What is the unit of electrostatic potential energy?

– Joules• Joules is a very large unit in dealing with electrons, atoms or

molecules• For convenience a new unit, electron volt (eV), is defined

– 1 eV is defined as the energy acquired by a particle carrying the charge equal to that of an electron (q=e) when it moves across a potential difference of 1V.

– How many Joules is 1 eV then?• eV however is not a standard SI unit. You must convert the

energy to Joules for computations.

1eV 191.6 10 1C V 191.6 10 J


Capacitors (or Condensers)• What is a capacitor?

– A device that can store electric charge without letting the charge flow• What does it consist of?

– Usually consists of two oppositely charged conducting objects (plates or sheets) placed near each other without touching

– Why can’t they touch each other?• The charges will neutralize each other

• Can you give some examples?– Camera flash, surge protectors, computer keyboard, binary circuits…

• How is a capacitor different than a battery?– Battery provides potential difference by storing energy (usually chemical

energy) while the capacitor stores charge but very little energy.

19

Capacitors• A simple capacitor consists of a pair of parallel plates

of area A separated by a distance d.– A cylindrical capacitors are essentially parallel plates

wrapped around as a cylinder.

• Symbols for a capacitor and a battery:– Capacitor -||-– Battery (+) -|l- (-) Circuit

Diagram

Weds., Jan. 29, 2014 PHYS 1442-004, Dr. Andrew Brandt


• What do you think will happen if a battery is connected (voltage is applied) to a capacitor?– The capacitor gets charged quickly, one plate positive and the other

negative with an equal amount. of charge• Each battery terminal, the wires and the plates are

conductors. What does this mean?– All conductors are at the same potential. – the full battery voltage is applied across the capacitor plates.

• So for a given capacitor, the amount of charge stored in the capacitor is proportional to the potential difference Vba between the plates. How would you write this formula?

– C is a proportionality constant, called capacitance of the device.– What is the unit?

Capacitors

baQ CV

C/V or Farad (F)

C is a property of a capacitor so does not depend on Q or V.

Normally use mF or pF.


Determination of Capacitance• C can be determined analytically for capacitors w/ simple

geometry and air in between.• Let’s consider a parallel plate capacitor.

– Plates have area A each and separated by d.• d is smaller than the length, so E is uniform.

– For parallel plates E= /s e0, where s is the surface charge density.

• E and V are related• So from the formula:

– What do you notice?

baV Ed0

0ba

AQ QC

V Qd A d

C only depends on the area (A) and the separation (d) of the plates and the permittivity of the medium between them.

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