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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. 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
MIT Class: Electric Potential

2


Potential energy and potential

Potential Energyand Potential

Start with Gravity


Gravity force and work
Gravity: Force and Work

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

Work done by gravity moving m from A to B:

Work Done by Earth’s Gravity


Prs question sign of w g

PRS Question:Sign of Wg


Prs sign of w g
PRS: 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


Prs answer sign of w g
PRS Answer: Sign of Wg

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

Work done by gravity moving m from A to B:

Work Near Earth’s Surface

G roughly constant:

Wg depends only on endpoints

– not on path taken –

Conservative Force


Potential energy joules
Potential Energy (Joules)

  • U0: constant depending on reference point

  • Only potential difference DU has physical significance


Gravitational potential joules kilogram

Define gravitational potential difference:

Gravitational Potential(Joules/kilogram)

That is, two particle interaction  single particle effect


Prs question masses in potentials

PRS Question:Masses in Potentials


Prs masses in potentials
PRS: 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


Prs answer masses in potentials
PRS Answer: Masses in Potentials

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



Gravity electrostatics
Gravity - Electrostatics

Mass M Charge q (±)

Both forces are conservative, so…


Potential potential energy
Potential & Potential Energy

Units: Joules/Coulomb = Volts

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

Joules


Potential external work
Potential & External Work

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

18


How big is a volt
How Big is a Volt?

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


Potential summary thus far
Potential: Summary Thus Far

Charges CREATE Potential Landscapes


Potential landscape
Potential Landscape

Positive Charge

Negative Charge


Potential summary thus far1
Potential: Summary Thus Far

Charges CREATE Potential Landscapes

Charges FEEL Potential Landscapes

We work with DU (DV) because only changes matter


2 prs questions potential potential energy

2 PRS Questions:Potential & Potential Energy


Prs positive charge

11

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


Prs answer positive charge
PRS Answer: Positive Charge

Answer:

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


Prs negative charge
PRS: Negative Charge

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

18


Prs answer negative charge
PRS Answer: Negative Charge

Answer:

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:


Potential landscape1
Potential Landscape

Positive Charge

Negative Charge


Creating potentials calculating from e two examples

Creating Potentials:Calculating from E,Two Examples


Potential in a uniform field
Potential in a Uniform Field

Just like gravity, moving in field direction reduces potential


Potential created by pt charge
Potential Created by Pt Charge

Take V = 0 at r = ∞:


Prs question point charge potential

PRS Question:Point Charge Potential


Prs two point charges

-q

+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


Prs answer two point charges

-q

+q

P

PRS Answer: Two Point Charges

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)


Potential landscape2
Potential Landscape

Positive Charge

Negative Charge


Group problem superposition
Group Problem: Superposition

Consider the 3 point charges at left.

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



Deriving e from v1
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


Deriving e from v2
Deriving E from V

If we do all coordinates:

Gradient (del) operator:



Prs e from v
PRS: 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

15


Prs answer e from v
PRS Answer: E from V

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!


Prs e from v1
PRS: E from V

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

43


Prs answer e from v1
PRS Answer: E from V

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

44


Prs e from v2
PRS: E from V

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


Prs answer e from v2
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

46


Group problem e from v
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?


Demonstration making measuring potential lab preview

Demonstration:Making & Measuring Potential(Lab Preview)



Configuration energy1
Configuration Energy

How much energy to put two charges as pictured?

  • First charge is free

  • Second charge sees first:


Configuration energy2
Configuration Energy

How much energy to put three charges as pictured?

  • Know how to do first two

  • Bring in third:

Total configuration energy:


Group problem build it
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?




Equipotential curves
Equipotential Curves

All points on equipotential curve are at same potential.

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


Direction of electric field e
Direction of Electric Field E

E is perpendicular to all equipotentials

Constant E field

Point Charge

Electric dipole


Properties of equipotentials
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


Summary e field and potential creating
Summary: E Field and Potential: Creating

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

Use superposition for systems of charges

They are related:

58


E field and potential effects
E Field and Potential: Effects

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:

59


Experiment 1 equipotentials
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

60


Prs questions midpoint check

PRS Questions:Midpoint Check

61


Prs lab midpoint equipotential
PRS: Lab Midpoint: Equipotential

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


Prs answer equipotential
PRS Answer: Equipotential

5

Answer:

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.

63


Prs lab midpoint field lines
PRS: Lab Midpoint: Field Lines

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

64


Prs answer field lines
PRS Answer: Field Lines

2

Answer:

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

65


Experiment 1 equipotentials1
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)

66


Prs questions lab summary

PRS Questions:Lab Summary

67


Prs lab summary potentials
PRS: Lab Summary: Potentials

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:

20

68


Prs answer potentials
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.

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)

69


Prs lab summary e field
PRS: Lab Summary: E Field

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:

20

70


Prs answer e fields
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.

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)

71


Prs lab summary charge
PRS: Lab Summary: Charge

|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:

20

72


Prs answer charge
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.

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)|

73


Prs kelvin water dropper
PRS: Kelvin Water Dropper

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

20

Water Drop

74


Prs answer kelvin water dropper
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

75


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