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W03D2Work, Potential Energy and Electric Potential

Announcements

Math Review Tuesday Tues Feb 28 from 9-11 pm in 32-082

PS 3 due Tuesday Tues Feb 28 at 9 pm in boxes outside 32-082 or 26-152

W03D3 Reading Assignment Course Notes: Sections 3.5, 3.7-3.8, 4.8.4

Exam One Thursday March 1 7:30-9:30 pm Room Assignments (See Stellar webpage announcements)

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Outline

Electrical Work

Electric Potential Energy

Electric Potential Difference

Calculating Electric Potential Difference

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Electrical Work

Work done by electrical force moving object 1 from A to B:

1 12 12

B

AW d

F s

Electrical force on object 1 due to interaction between charged objects 1 and 2:

1 212 122

12

ˆe

q qk

r

F r

PATH

INTEGRAL

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Concept Question: Sign of WSuppose a fixed positively charged object (charge qs > 0) is at the origin and we move a negatively charged object (charge q1 < 0) from A to B with rA < rB , where r is the distance from the origin. The work done on the negatively charged object is

1. is positive and we do a positive amount of work

2. is positive and the electrical force does a positive amount of work

3. is negative and we do a positive amount of work

4. is negative and electrical force does a positive amount of work

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Concept Question Ans.: Sign of W

W is the work done by the electrical force. This is the opposite of the work that we must do in order to move a charged object in an electric field due to source. The electrical force is attractive and we are moving the positively charged object away from the source (opposite the direction of the electric field).

Answer 3: W is negative and we do a positive amount

of work

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Group Problem: Work Done by Electrical Force

A point-like charged source object (charge qs) is held fixed. A second point-like charged object (charge q1)is initially at a distance rA from the fixed source and moves to a final distance rB from the fixed source. What is the work done by the electrical force on the moving object? Hint: What coordinate system is best suited for this problem?

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Sign of W: Negative Work

Suppose a fixed positively charged source (charge qs > 0) is at the origin and a positively charged object (charge q1 > 0) moves from A to B with rA > rB , where r is the distance from the origin, then W < 0.

rA r

B

1

rB

1

rA

0 and qsq

1 0 W k

eq

sq

1

1

rB

1

rA

0

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Work and Change in Kinetic Energy

W K

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Group Problem: Work-Kinetic Energy In a Uniform Electric field

Consider two thin oppositely uniform charged thin plates separated by a distance d. The surface charge densities on the plates are uniform and equal in magnitude. An electron with charge –e and mass m is released from rest at the negative plate and moves to the positive plate. What is the speed of the electron when it reaches the positive plate?

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Potential Energy Difference

12 1

1 1B

B A e sAB A

U U U d W k q qr r

F s

Suppose charged object 1 is fixed and located at the origin and charge object 2 moves from an initial position A a distance rA from the origin to a final position B, a distance rB from the origin.

The potential energy difference due to the interaction is defined to be the negative of the work done object 2 in moving from A to B:

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Potential Energy: Zero Point

U (r) U () U (r)

keq

sq

1

r

Choose the zero point for the potential energy at infinity.

Then set rA = ∞ and rB = r .

The potential energy difference between ∞ and any point on a circle of radius r is

U () 0

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Concept Question: Motion of Charged Objects

Two oppositely charged are released from rest in an electric field.

1. Both charged objects will move from lower to higher potential energy.

2. Both charged objects will move from higher to lower potential energy.

3. The positively charged object will move from higher to lower potential energy; the negatively charged object will move from lower to higher potential energy.

4. The positively charged object will move from higher to lower potential energy; the negatively charged object will move from lower to higher potential energy.

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Concept Q. Ans.: Motion of Charged Objects

2. Both charged objects will move from higher to lower potential energy so that

U 0

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Configuration EnergyWhat is the potential energy stored in a configuration of charged objects? Start with all the charged objects at infinity. Choose

(1)Bring in the first charged object.

(2)Bring in the second charged object

(3)Bring in the third charged object

(4)Configuration energy

U2U

12k

eq

1q

2/ r

12

U () 0

U3U

23U

13k

eq

2q

3/ r

23 k

eq

1q

3/ r

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U10

U U12

U23

U13

keq

1q

2/ r

13 k

eq

2q

3/ r

23 k

eq

1q

3/ r

13

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Group Problem: Build It

How much energy does it take you to assemble the charges into the configuration at left, assuming they all started out an infinite distance apart?

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Electric Potential Difference

B B

t tA A

UV d d

q q

F

s E s

Units: Joules/Coulomb = Volts

Change in potential energy per test object in moving the test object (charge qt) from A to B:

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DemonstrationVan de Graaf

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Breakdown of dry air 33 kV/cm

Video of Tesla Coil

http://www.youtube.com/watch?v=FY-AS13fl30

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How Big is a Volt?AA Batteries 1.5 V High Voltage

Transmission Lines100 kV-700 kV

Car Batteries 12 V Van der Graaf 300 kV

US Outlet (AC) 120 V Tesla Coil 500 kV

Distribution Power Lines

120 V- 70 kV

Lightning 10-1000 MV

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E Field and Potential: Effects

qF E

If you put a charged particle, (charge q), in a field:

U qV

To move a charged particle, (charge q), in a field

and the particle does not change its kinetic energy

then:

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Concept Question: Motion of Charged Objects

Two oppositely charged are released from rest in an electric field.

1. Both charged objects will move from lower to higher electric potential.

2. Both charged objects will move from higher to lower electric potential.

3. The positively charged object will move from higher to lower electric potential; the negatively charged object will move from lower to higher electric potential.

4. The positively charged object will move from higher to lower electric potential; the negatively charged object will move from lower to higher electric potential.

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Concept Q. Ans.: Motion of Charged Objects

Two oppositely charged are released from rest in an electric field.

3. The positively charged object will move from higher to lower electric potential; the negatively charged object will move from lower to higher electric potential.

For the positively charged object:

For the negatively charged object: V 0 U qV 0

V 0 U qV 0

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Potential & External WorkChange in potential energy in moving the charged object (charge q) from A to B:

U qV

Conservation of Energy Law:

WextK U

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Demonstration: Kelvin Water Drop (32-082)

orWimshurst Machine (26-152)

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Potential Created by Pt Charge

ˆˆdr r d ds r

B

B A AV V V d

E s

2

ˆ

rkQ

rE

2 2

ˆB B

A A

drkQ d kQ

r r

rs

kQ

1

rB

1

rA

Take V = 0 at r = ∞:

r

kQrV )(ChargePoint

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Concept Question: Two Point Charges

The work done in moving a positively charged object that starts from rest at infinity and ends at rest at the point P midway between two charges of magnitude +Q and –Q

1. is positive.

2. is negative.

3. is zero.

4. can not be determined – not enough info is given.

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Concept Question Answer: Two Point Charges

The potential at ∞ is zero.The potential at P is zero because equal and opposite potentials are superimposed from the two point charges (remember: V is a scalar, not a vector)

3. Work from ∞ to P is zero.

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Potential Landscape

Negative Charge

Positive Charge

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Continuous Charge Distributions

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Continuous Charge DistributionsBreak distribution into

infinitesimal charged elements of charge dq. Electric Potential difference between infinity and P due to dq.

Superposition Principle:

Reference Point:

dV V

dq(P) V

dq() k

e

dq

r 2

V (P) V () k

e

dq

r 2source

V () 0 V (P) k

e

dq

r 2source

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Worked Example

Consider a uniformly charged ring with total charge Q. Find the electric potential difference between infinity and a point P along the symmetric axis a distance z from the center of the ring.

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Worked Example: Charged Rod

d q d s Rd

dV

1

40

d q

r

1

40

Rd

R2 z2

V (z) ke

Rd

R2 z20

2

k

eR

R2 z2R d

a

b

V (z) k

e2 R

R2 z2

keQ

R2 z2

V () 0Choose

Q / 2 R

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Group Problem

A thin rod extends along the x-axis from x = -l /2 to x = l/2 . The rod carries a uniformly distributed positive charge +Q. Calculate the electric potential difference between infinity and at a point P along the x-axis.