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Workshop: Using Visualization in Teaching Introductory E&M

AAPT National Summer Meeting, Edmonton, Alberta, Canada.

Organizers: John Belcher, Peter Dourmashkin, Carolann Koleci, Sahana Murthy

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MIT Class: Electric Potential

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Potential Energyand Potential

Start with Gravity

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Gravity: Force and Work

Work done by gravity moving m from A to B:

g

B

g AW d F s

Gravitational force on m due to M:

2ˆg

MmGr

F r

PATH

INTEGRAL

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Work Done by Earth’s GravityWork done by gravity moving m from A to B:

ggW d F s

2ˆ ˆˆ

B

A

GMm

rdr rd

r r θ

1 1

B A

GMmr r

2

B

A

r

r

GMmdr

r

B

A

r

r

GMm

r

d s

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PRS Question:Sign of Wg

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PRS: Sign of Wg

Thinking about the sign and meaning of this…

1 1g

B A

W GMmr r

Moving from rA to rB:

0%

0%

0%

100%

0% 1. Wg is positive – we do work

2. Wg is positive – gravity does work

3. Wg is negative – we do work

4. Wg is negative – gravity does work

5. I don’t know

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PRS Answer: Sign of Wg

Wg is the work that gravity does. This is the opposite of the work that we must do in order to move an object in a gravitational field.

We are pushing against gravity we do positive work

Answer: 3. Wg is negative

– we do work

1 1g

B A

W GMmr r

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Work Near Earth’s Surface

2ˆ ˆ

E

GMg

r g y y

G roughly constant:

ˆB

Amg d y s

Work done by gravity moving m from A to B:

By

yAmgdy ( )B Amg y y

g gW d F s

Wg depends only on endpoints

– not on path taken –

Conservative Force

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Potential Energy (Joules)

B

g B A g gAU U U d W F s

02

ˆ(1) g g

GMm GMmU U

r r F r

• U0: constant depending on reference point

• Only potential difference U has physical significance

0ˆ(2) g gmg U mgy U F y

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Gravitational Potential(Joules/kilogram)

Define gravitational potential difference:

g( / )B Bg

g A A

UV m d d

m

F s g s

g

FieldForce Energy Potential

Just as , g gU V F g

That is, two particle interaction single particle effect

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PRS Question:Masses in Potentials

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PRS: Masses in PotentialsConsider 3 equal masses sitting in different gravitational potentials:A) Constant, zero potentialB) Constant, non-zero potentialC) Linear potential (V x) but sitting at V = 0

Which statement is true?

0%

20%

20%

20%

20%

20% 1. None of the masses accelerate2. Only B accelerates3. Only C accelerates4. All masses accelerate, B has largest acceleration5. All masses accelerate, C has largest acceleration6. I don’t know

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PRS Answer: Masses in Potentials

When you think about potential, think “height.” For example, near the Earth:

U = mgh so V = gh

Constant potential (think constant height) does not cause acceleration!

The value of the potential (height) is irrelevant.Only the slope matters

Answer: 3. Only C (linear potential) accelerates

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Move to Electrostatics

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Gravity - Electrostatics

2ˆ

MGr

g rMass M Charge q (±)

2ˆ

e

qkr

E r

g mF g

E qF E

Both forces are conservative, so…

B

g gAU d F s

B

EAU d F s

B

g AV d g s

B

AV d E s

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Potential & Potential Energy

Change in potential energy in moving the charged object (charge q) from A to B:

JoulesB AU U U q V

B

A

V d E s Units:

Joules/Coulomb = Volts

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

JoulesB AU U U q V

If the kinetic energy of the charged object does not change,

The external work is

extW K U

0K

then the external work equals the change in potential energy

extW U q V

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How Big is a Volt?

• AA, C, D Batteries 1.5 V

• Car Battery 12 V

• US Outlet 120 V (AC)

• Residential Power Line

• Our Van de Graaf

• Big Tesla Coil

Know These!

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Potential: Summary Thus Far

Charges CREATE Potential Landscapes

0 "0"

"0"

( )V V V V d r

r E s

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Potential Landscape

Negative Charge

Positive Charge

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Potential: Summary Thus Far

Charges CREATE Potential Landscapes

Charges FEEL Potential Landscapes

0 "0"

"0"

( )V V V V d r

r E s

U qVr r

We work with U (V) because only changes matter

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2 PRS Questions:Potential & Potential Energy

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PRS: Positive Charge

Place a positive charge in an electric field. It will accelerate from

25%

25%

25%

25% 1. higher to lower electric potential; lower to higher potential energy

2. higher to lower electric potential; higher to lower potential energy

3. lower to higher electric potential; lower to higher potential energy

4. lower to higher electric potential; higher to lower potential energy

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PRS Answer: Positive Charge

Objects always “move” (accelerate) to reduce their potential energy. Positive charges do this by accelerating towards a lower potential

Answer: 2. + acc. from higher to lower electric potential; higher to lower potential energy

U q V

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PRS: Negative Charge

Place a negative charge in an electric field. It will accelerate from

40%

20%

20%

20% 1. higher to lower electric potential; lower to higher potential energy

2. higher to lower electric potential; higher to lower potential energy

3. lower to higher electric potential; lower to higher potential energy

4. lower to higher electric potential; higher to lower potential energy

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PRS Answer: Negative Charge

Objects always “move” (accelerate) to reduce their potential energy. Negative charges do this by accelerating towards a higher potential:

Answer: 4. Neg. acc. from lower to higher electric potential

higher to lower potential energy

U q V

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Potential Landscape

Negative Charge

Positive Charge

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Creating Potentials:Calculating from E,

Two Examples

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Potential in a Uniform Field

ˆdyds j

ˆEE j

ˆB B

A AE d E dy j s

Ed

Just like gravity, moving in field direction reduces potential

B

B A ATO FROM

V V V d E s

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Potential Created by Pt Charge

θrsd ˆˆ drdr

B

B A AV V V d E s

2

ˆ

rkQ

rE

2 2

ˆB B

A A

drkQ d kQr r

r

s

1 1

B A

kQr r

Take V = 0 at r = ∞:

r

kQrV )(ChargePoint

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PRS Question:Point Charge Potential

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PRS: Two Point ChargesThe work done in moving a positive test charge from infinity to the point P midway between two charges of magnitude +q and –q:

+q -qP

20%

20%

20%

20%

20%1. is positive.

2. is negative.

3. is zero.

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

5. I don’t know

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PRS 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

+q -qP

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Potential Landscape

Negative Charge

Positive Charge

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Group Problem: Superposition

Consider the 3 point charges at left.

What total electric potential do they create at point P (assuming V = 0)

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Deriving E from V

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Deriving E from V

ˆx s i

A = (x,y,z), B=(x+x,y,z)

x

V VE

x x

Ex = Rate of change in V

with y and z held constant

B

A

V d E s

( , , )

( , , )

x x y z

x y z

V d

E s E s

ˆ( ) xx E x E i

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Gradient (del) operator:

ˆ ˆ ˆx y z

i j+ k

If we do all coordinates:

V E

Deriving E from V

ˆ ˆ ˆV V V

x y z

E i + j k

ˆ ˆ ˆ Vx y z

i + j k

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PRS Questions:E from V

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PRS: E from VConsider the point charges you looked at earlier:

V P kQ a

You calculated V(P). From that can you derive E(P)?

20%

20%

20%

20%

20%1. Yes, its kQ/a2 (up)

2. Yes, its kQ/a2 (down)

3. Yes in theory, but I don’t know how to take a gradient

4. No, you can’t get E(P) from V(P)

5. I don’t know

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PRS Answer: E from V

The electric field is the gradient (spatial derivative) of the potential. Knowing the potential at a single point tells you nothing about its derivative.

People commonly make the mistake of trying to do this. Don’t!

4. No, you can’t get E(P) from V(P)

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PRS: E from V

The graph above shows a potential V as a function of x. The magnitude of the electric field for x > 0 is

0%

0%

0%

0% 1. larger than that for x < 0 2. smaller than that for x < 0 3. equal to that for x < 0 4. I don’t know :20

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PRS Answer: E from V

The slope is smaller for x > 0 than x < 0Translation: The hill is steeper on the left than on the right.

Answer: 2. The magnitude of the electric field for x > 0 is smaller than that for x < 0

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PRS: E from V

The above shows potential V(x). Which is true?

0%

0%

0%

0%

0% 1. Ex > 0 is > 0 and Ex < 0 is > 0

2. Ex > 0 is > 0 and Ex < 0 is < 0

3. Ex > 0 is < 0 and Ex < 0 is < 0

4. Ex > 0 is < 0 and Ex < 0 is > 0

5. I don’t know 20

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PRS Answer: E from V

E is the negative slope of the potential, negative on the left, positive on the right

Translation: “Downhill” is to the left on the left and to the right on the right.

Answer: 2. Ex > 0 is > 0 and Ex < 0 is < 0

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Group Problem: E from V

A potential V(x,y,z) is plotted above. It does not depend on x or y.

What is the electric field everywhere?

Are there charges anywhere? What sign?

-5 0 50

5

10

Pot

entia

l (V

)

Z Position (mm)

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Demonstration:Making & Measuring

Potential(Lab Preview)

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Configuration Energy

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Configuration EnergyHow much energy to put two charges as pictured?

1) First charge is free

2) Second charge sees first:

1 2

12

12 2 2 1

1

4 o

q qU W q V

r

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Configuration EnergyHow much energy to put three charges as pictured?

1) Know how to do first two

2) Bring in third:

3 3 1 2W q V V

1 3 2 31 22 3 12 13 23

0 12 13 23

1

4

q q q qq qU W W U U U

r r r

Total configuration energy:

3 1 2

0 13 234

q q q

r r

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Group Problem: Build It

1) How much energy did it take to assemble the charges at left?

2) How much energy would it take to add a 4th charge +3Q at P?

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Equipotentials

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Topographic Maps

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Equipotential Curves

All points on equipotential curve are at same potential.Each curve represented by V(x,y) = constant

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Direction of Electric Field E

E is perpendicular to all equipotentials

Constant E field Point Charge Electric dipole

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Properties of Equipotentials

• E field lines point from high to low potential

• E field lines perpendicular to equipotentials• Have no component along equipotential

• No work to move along equipotential

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Summary: E Field and Potential: Creating

;B

B A AV V V V d E E s

They are related:

A point charge q creates a field and potential around it:

2ˆ;e e

q qk V kr r

E r

Use superposition for systems of charges

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E Field and Potential: Effects

qF E

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

extW U q V

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

and the particle does not change its kinetic energy

then:

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Experiment 1: Equipotentials

Download LabView file (save to desktop) and run it

Log in to server and add each student to your group (enter your MIT ID)

Each group will do two of the four figures (your choice). We will break about half way through for some PRS

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PRS Questions:Midpoint Check

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PRS: Lab Midpoint: EquipotentialThe circle is at +5 V relative to the plate. Which of the below is the most accurate equipotential map?

1 4

3

52

6

1 2 3 4 5 6

0% 0% 0%0%0%0%

1. 1

2. 2

3. 3

4. 4

5. 5

6. 6

:20

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PRS Answer: Equipotential

The electric field is stronger between the plate and circle than on either outer side, so the equipotential lines must be spaced most closely in between the two conductors.

5Answer:

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PRS: Lab Midpoint: Field LinesThe circle is at +5 V relative to the plate. Which of the below is the most accurate electric field line map?

1 4

3

52

6

1 2 3 4 5 6

0% 0% 0%0%0%0%

1. 1

2. 2

3. 3

4. 4

5. 5

6. 6 20

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PRS Answer: Field Lines

Field lines must be perpendicular to equipotential surfaces, including the conductors themselves.

Answer: 2

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Experiment 1: Equipotentials

Continue with the experiment…

If you finish early make sure that you talk about the extra questions posed at the end of the lab. Labs will be asked about on the exams (see, for example, the final exam from Fall 2005)

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PRS Questions:Lab Summary

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PRS: Lab Summary: PotentialsHolding the red plate at +5 V relative to the ground of the blue plate, what is true about the electric potential at the following locations:

A

BC

D

0%

0%

0%

0%

0%

0% 1. V(A) > V(B) > V(C) > V(D)2. V(A) > V(B) ~ V(C) > V(D)3. V(A) ~ V(B) > V(C) ~ V(D)4. V(D) > V(C) ~ V(B) > V(A)5. V(B) > V(C) > V(D) ~ V(A)6. V(A) > V(D) ~ V(C) > V(B) 20

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PRS Answer: Potentials

The potential at A is nearly +5 V.The potential at B & C ~ 2.5 V (they are both halfway).The potential at D is about 0 V.

Holding the red plate at +5 V relative to the ground of the blue plate… Answer: 2. V(A) > V(B) ~ V(C) > V(D)

A

BC

D

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PRS: Lab Summary: E FieldHolding the red plate at +5 V relative to the ground of the blue plate, what is true about the electric field at the following locations:

A

BC

D

0%

0%

0%

0%

0%

0% 1. E(A) > E(B) > E(C) > E(D)

2. E(A) > E(B) ~ E(C) > E(D)

3. E(A) ~ E(B) > E(C) ~ E(D)

4. E(D) > E(C) ~ E(B) > E(A)

5. E(B) > E(C) > E(D) ~ E(A)

6. E(A) > E(D) ~ E(C) > E(B)

20

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PRS Answer: E Fields

The potential changes most rapidly (and hence E is largest) at B. It also changes at C, but not as fast. The potential is very uniform outside, so the E field out there is nearly zero.

Holding the red plate at +5 V relative to the ground of the blue plate… Answer: 5. E(B) > E(C) > E(D) ~ E(A)

A

BC

D

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PRS: Lab Summary: ChargeHolding the red plate at +5 V relative to the ground of the blue plate, what is true about the amount of charge near the following points: A

BD

C

0%

0%

0%

0%

0%

0% 1. |Q(A)| ~ |Q(C)| > |Q(B)| ~ |Q(D)|2. |Q(A)| > |Q(B)| ~ |Q(C)| > |Q(D)|3. |Q(A)| ~ |Q(B)| > |Q(C)| ~ |Q(D)|4. |Q(D)| ~ |Q(C)| > |Q(B)| ~ |Q(A)|5. |Q(B)| ~ |Q(D)| > |Q(A)| ~ |Q(C)|6. |Q(A)| > |Q(D)| ~ |Q(C)| > |Q(B)|

2020

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PRS Answer: Charge

Charges go where the field is highest (higher field more field lines more charges to source & sink). Field at A & B is the same, so Q is as well. Higher than at C & D.

Holding the red plate at +5 V relative to the ground of the blue plate…Answer: 3. |Q(A)| ~ |Q(B)| > |Q(C)| ~ |Q(D)|

A

BD

C

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PRS: Kelvin Water Dropper

A drop of water falls through the right can. If the can has positive charge on it, the separated water drop will have

Can

Water Drop

0%

0%

0%

0% 1. no net charge

2. a positive charge

3. a negative charge

4. I don’t know

20

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PRS Answer: Kelvin Water Dropper

The positive charge on the can repels positive charge to the top of the drop and attracts negative charge to the bottom of the drop just before it separates. After the drop separates its charge is therefore negative.

Answer: 3. The drop has a negative charge

+

+

+

+

+

+-

+ +

-