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# MIT Class: Electric Potential - PowerPoint PPT Presentation

Workshop: Using Visualization in Teaching Introductory E&M AAPT National Summer Meeting, Edmonton, Alberta, Canada. Organizers: John Belcher, Peter Dourmashkin, Carolann Koleci, Sahana Murthy. MIT Class: Electric Potential. 2. Potential Energy and Potential. Start with Gravity.

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Workshop: Using Visualization in Teaching Introductory E&MAAPT National Summer Meeting, Edmonton, Alberta, Canada.Organizers: John Belcher, Peter Dourmashkin, Carolann Koleci, Sahana Murthy

MIT Class: Electric Potential

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### Potential Energyand Potential

Gravitational force on m due to M:

Work done by gravity moving m from A to B:

PATH

INTEGRAL

Work Done by Earth’s Gravity

### PRS Question:Sign of Wg

Thinking about the sign and meaning of this…

Moving from rA to rB:

• Wg is positive – we do work

• Wg is positive – gravity does work

• Wg is negative – we do work

• Wg is negative – gravity does work

• I don’t know

:19

Answer: 3. Wg is negative – we do work

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

Work Near Earth’s Surface

G roughly constant:

Wg depends only on endpoints

– not on path taken –

Conservative Force

• U0: constant depending on reference point

• Only potential difference DU has physical significance

Gravitational Potential(Joules/kilogram)

That is, two particle interaction  single particle effect

### PRS Question:Masses in Potentials

Consider 3 equal masses sitting in different gravitational potentials:

A) Constant, zero potential

B) Constant, non-zero potential

C) Linear potential (V  x) but sitting at V = 0

Which statement is true?

• None of the masses accelerate

• Only B accelerates

• Only C accelerates

• All masses accelerate, B has largest acceleration

• All masses accelerate, C has largest acceleration

• I don’t know

:19

Answer: 3. Only C (linear potential) accelerates

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

### Move to Electrostatics

Mass M Charge q (±)

Both forces are conservative, so…

Units: Joules/Coulomb = Volts

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

Joules

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

Joules

The external work is

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

then the external work equals the change in potential energy

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Know These!

• 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

Charges CREATE Potential Landscapes

Positive Charge

Negative Charge

Charges CREATE Potential Landscapes

Charges FEEL Potential Landscapes

We work with DU (DV) because only changes matter

### 2 PRS Questions:Potential & Potential Energy

PRS: Positive Charge

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

• higher to lower electric potential; lower to higher potential energy

• higher to lower electric potential; higher to lower potential energy

• lower to higher electric potential; lower to higher potential energy

• lower to higher electric potential; higher to lower potential energy

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

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

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

• higher to lower electric potential; lower to higher potential energy

• higher to lower electric potential; higher to lower potential energy

• lower to higher electric potential; lower to higher potential energy

• lower to higher electric potential; higher to lower potential energy

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4. Neg. acc. from lower to higher electric potential higher to lower potential energy

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

Positive Charge

Negative Charge

### Creating Potentials:Calculating from E,Two Examples

Just like gravity, moving in field direction reduces potential

Take V = 0 at r = ∞:

### PRS Question:Point Charge Potential

+q

P

PRS: Two Point Charges

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

• is positive.

• is negative.

• is zero.

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

• I don’t know

:16

+q

P

3. Work from  to P is zero

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)

Positive Charge

Negative Charge

Consider the 3 point charges at left.

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

### Deriving E from V

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

Ex = Rate of change in V with y and z held constant

If we do all coordinates:

### PRS Questions:E from V

Consider the point charges you looked at earlier:

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

• Yes, its kQ/a2 (up)

• Yes, its kQ/a2 (down)

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

• No, you can’t get E(P) from V(P)

• I don’t know

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4. No, you can’t get E(P) from V(P)

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!

larger than that for x < 0

smaller than that for x < 0

equal to that for x < 0

I don’t know

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

:20

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The slope is smaller for x > 0 than x < 0

Translation: 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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Ex > 0 is > 0 and Ex < 0 is > 0

Ex > 0 is > 0 and Ex < 0 is < 0

Ex > 0 is < 0 and Ex < 0 is < 0

Ex > 0 is < 0 and Ex < 0 is > 0

I don’t know

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

20

45

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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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?

### Configuration Energy

How much energy to put two charges as pictured?

• First charge is free

• Second charge sees first:

How much energy to put three charges as pictured?

• Know how to do first two

• Bring in third:

Total configuration energy:

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?

### Equipotentials

All points on equipotential curve are at same potential.

Each curve represented by V(x,y) = constant

E is perpendicular to all equipotentials

Constant E field

Point Charge

Electric dipole

• 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

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

Use superposition for systems of charges

They are related:

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

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

and the particle does not change its kinetic energy

then:

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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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1

2

3

4

5

6

5

6

4

1

3

2

The circle is at +5 V relative to the plate. Which of the below is the most accurate equipotential map?

:20

62

5

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.

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1

2

3

4

5

6

6

5

4

3

2

1

The circle is at +5 V relative to the plate. Which of the below is the most accurate electric field line map?

20

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2

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

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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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V(A) > V(B) > V(C) > V(D)

V(A) > V(B) ~ V(C) > V(D)

V(A) ~ V(B) > V(C) ~ V(D)

V(D) > V(C) ~ V(B) > V(A)

V(B) > V(C) > V(D) ~ V(A)

V(A) > V(D) ~ V(C) > V(B)

A

C

B

D

Holding 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:

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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.

A

C

B

D

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)

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E(A) > E(B) > E(C) > E(D)

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

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

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

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

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

A

C

B

D

Holding 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:

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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.

A

C

B

D

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)

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|Q(A)| ~ |Q(C)| > |Q(B)| ~ |Q(D)|

|Q(A)| > |Q(B)| ~ |Q(C)| > |Q(D)|

|Q(A)| ~ |Q(B)| > |Q(C)| ~ |Q(D)|

|Q(D)| ~ |Q(C)| > |Q(B)| ~ |Q(A)|

|Q(B)| ~ |Q(D)| > |Q(A)| ~ |Q(C)|

|Q(A)| > |Q(D)| ~ |Q(C)| > |Q(B)|

A

C

D

B

Holding 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:

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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.

A

C

D

B

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)|

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no net charge

a positive charge

a negative charge

I don’t know

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

Can

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Water Drop

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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

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