1/24/07184 Lecture 101 PHY 184 Spring 2007 Lecture 10 Title: The Electric Potential, V(x)

1/24/07 184 Lecture 10 1

PHY 184PHY 184PHY 184PHY 184

Spring 2007Lecture 10

Title: The Electric Potential, V(x)

1/24/07 184 Lecture 10 2

Review - Electric Potential V(x)Review - Electric Potential V(x)Review - Electric Potential V(x)Review - Electric Potential V(x)

The change in electric potential energy U of a charge q that moves in an electric field is related to the change in electric potential V

The unit of electric potential is the volt, V. The unit of electric field is V/m. For reference state at infinity,

qΔΔΔUqU

V

or

…a scalar function of position

x

sdExV

)(

1/24/07 184 Lecture 10 3

Electric Potential for a Point ChargeElectric Potential for a Point ChargeElectric Potential for a Point ChargeElectric Potential for a Point Charge

We’ll derive the electric potential for a point source q, as a function of distance R from the source. That is, V(R).

Remember that the electric field from a point charge q at a distance r is given by

The direction of the electric field from a point charge is always radial.

We integrate from distance R (distance from the point charge) along a radial to infinity:

1/24/07 184 Lecture 10 4

Electric Potential of a Point Charge (2)Electric Potential of a Point Charge (2)Electric Potential of a Point Charge (2)Electric Potential of a Point Charge (2) The electric potential V from a point charge q at a distance r is then

Positive point charge

Negative point charge

rkq

rV )(

1/24/07 184 Lecture 10 5

Electric Potential from a System of ChargesElectric Potential from a System of ChargesElectric Potential from a System of ChargesElectric Potential from a System of Charges

We calculate the electric potential from a system of n point charges by adding the potential functions from each charge

This summation produces an electric potential at all points in space – a scalar function

Calculating the electric potential from a group of point charges is usually much simpler than calculating the electric field. (because it’s a scalar)

V Vii1

n

kqirii1

n

1/24/07 184 Lecture 10 6

Example - Superposition of Electric Example - Superposition of Electric PotentialPotential

Example - Superposition of Electric Example - Superposition of Electric PotentialPotential

Assume we have a system of three point charges:q1 = +1.50 Cq2 = +2.50 Cq3 = -3.50 C.

q1 is located at (0,a)q2 is located at (0,0)q3 is located at (b,0)a = 8.00 m and b = 6.00 m.

What is the electric potential at point P located at (b,a)?

1/24/07 184 Lecture 10 7

Example - Superposition of Electric Potential (2)Example - Superposition of Electric Potential (2)Example - Superposition of Electric Potential (2)Example - Superposition of Electric Potential (2)

The electric potential at point P is given by the sum of the electric potential from the three charges

V kqirii1

3

kq1

r1q2

r2q3

r3

k

q1

b

q2

a2 b2q3

a

V 8.99 109 N/C 1.50 10 6 C

6.00 m

2.50 10 6 C

8.00 m 2 6.00 m 2

3.50 10 6 C

8.00 m

V 562 V

r1

r2

r3

1/24/07 184 Lecture 10 8

Clicker Question - Electric PotentialClicker Question - Electric PotentialClicker Question - Electric PotentialClicker Question - Electric Potential

Rank (a), (b) and (c) according to the net electric potential V produced at point P by two protons. Greatest first!

A: (b), (c), (a)B: all equalC: (c), (b), (a)D: (a) and (c) tie, then (b)

1/24/07 184 Lecture 10 9

Clicker Question - Electric PotentialClicker Question - Electric PotentialClicker Question - Electric PotentialClicker Question - Electric Potential

Rank (a), (b) and (c) according to the net electric potential V produced at point P by two protons. Greatest first!

B: all equal

aqd

V2

1/24/07 184 Lecture 10 10

Calculating the Field from the Calculating the Field from the PotentialPotential

Calculating the Field from the Calculating the Field from the PotentialPotential

We can calculate the electric field from the electric potential starting with

Which allows us to write

If we look at the component of the electric field along the direction of ds, we can write the magnitude of the electric field as the partial derivative along the direction s

V We,

q

ES Vs

1/24/07 184 Lecture 10 11

Math Reminder - Partial DerivativesMath Reminder - Partial DerivativesMath Reminder - Partial DerivativesMath Reminder - Partial Derivatives

Given a function V(x,y,z), the partial derivatives are

Example: V(x,y,z)=2xy2+z3

act on x, y and z independently

Meaning: partial derivatives give the slope along the respective direction

1/24/07 184 Lecture 10 12

Calculating the Field from the Potential (2)Calculating the Field from the Potential (2)Calculating the Field from the Potential (2)Calculating the Field from the Potential (2)

We can calculate any component of the electric field by taking the partial derivative of the potential along the direction of that component.

We can write the components of the electric field in terms of partial derivatives of the potential as

In terms of graphical representations of the electric potential, we can get an approximate value for the electric field by measuring the gradient of the potential perpendicular to an equipotential line

Ex Vx

; Ey Vy

; Ez Vz

VE

1/24/07 184 Lecture 10 13

Example - Graphical Extraction of the Field from the PotentialExample - Graphical Extraction of the Field from the PotentialExample - Graphical Extraction of the Field from the PotentialExample - Graphical Extraction of the Field from the Potential

Assume a system of three point chargesq1 6.00 C q2 3.00 C q3 9.00 C

x1, y1 1.5 cm,9.0 cm x2 , y2 6.0 cm,8.0 cm x3, y3 5.3 cm,2.0 cm

1/24/07 184 Lecture 10 14

Example - Graphical Extraction of the Field Example - Graphical Extraction of the Field from the Potential (2)from the Potential (2)

Example - Graphical Extraction of the Field Example - Graphical Extraction of the Field from the Potential (2)from the Potential (2)

We calculate the magnitude of the electric field at point P.

To perform this task, we draw a line through point P perpendicular to the equipotential line reaching from the equipotential line of +1000 V to the line of –1000V.

The length of this line is 1.5 cm. So the magnitude of the electric field can be approximated as

The direction of the electric field points from the positive equipotential line to the negative potential line.

ES Vs

2000 V 0 V

1.5 cm1.3105 V/m

1/24/07 184 Lecture 10 15

Clicker Question - E Field from PotentialClicker Question - E Field from PotentialClicker Question - E Field from PotentialClicker Question - E Field from Potential

Pairs of parallel plates with the same separation and a given V of each plate. The E field is uniform between plates and perpendicular to them. Rank the magnitude of the electric field E between them. Greatest first!

A: (1), (2), (3)B: (3) and (2) tie, then (1)C: all equal

D: (2), then (1) and (3) tie

1/24/07 184 Lecture 10 16

Clicker Question - E Field from PotentialClicker Question - E Field from PotentialClicker Question - E Field from PotentialClicker Question - E Field from Potential

Pairs of parallel plates with the same separation d and a given V of each plate. The E field is uniform between plates and perpendicular to them. Rank the magnitude of the electric field E between them. Greatest first!

D: (2), then (1) and (3) tie

Use and take the magnitude only

1/24/07 184 Lecture 10 17

Electric Potential Energy for aElectric Potential Energy for aSystem of ParticlesSystem of Particles

Electric Potential Energy for aElectric Potential Energy for aSystem of ParticlesSystem of Particles

So far, we have discussed the electric potential energy of a point charge in a fixed electric field.

Now we introduce the concept of the electric potential energy of a system of point charges.

In the case of a fixed electric field, the point charge itself did not affect the electric field that did work on the charge.

Now we consider a system of point charges that produce the electric potential themselves.

We begin with a system of charges that are infinitely far apart. Reference state, U = 0.

To bring these charges into proximity with each other, we must do work on the charges, which changes the electric potential energy of the system.

1/24/07 184 Lecture 10 18

Electric Potential Energy for aElectric Potential Energy for aSystem of Particles (2)System of Particles (2)


To illustrate the concept of the electric potential energy of a system of particles we calculate the electric potential energy of a system of two point charges, q1 and q2 .

We start our calculation with the two charges at infinity. We then bring in point charge q1; ; that requires no work. Because there is no electric field and no corresponding electric force, this action requires no work to be done on the charge. Keeping this charge (q1) stationary, we bring the second point charge (q2) in from infinity to a distance r from q1; that

requires work q2 V1(r).

q1 q2r

1/24/07 184 Lecture 10 19



So, the electric potential energy of this two charge system is

where

Hence the electric potential of the two charge system is

If the two point charges have the same sign, then we must do positive work on the particles to bring them together.

If the two charges have opposite signs, we must do negative work on the system to bring them together from infinity.

)(12 rVqU

q1 q2

r

rkq

rV 11 )(

rqkq

U 21

1/24/07 184 Lecture 10 20

Example - Electric Potential Energy (1)Example - Electric Potential Energy (1)Example - Electric Potential Energy (1)Example - Electric Potential Energy (1)

Consider three point charges at fixed positions. What is the electric potential energy U of the assembly of charges?

Key Idea: The potential energy is equal to the work we must do to assemble the system, bringing in each charge from an infinite distance.

Strategy: Let’s build the system by starting with one charge in place and bringing in the others from infinity.

q1=+q, q2=-4q, q3=+2q

1/24/07 184 Lecture 10 21


Strategy: Let’s say q1 is in place and we bring in q2.

We then bring in charge 3. The work we must do to bring 3 to its place relative to q1 and q2 is then:

q1=+q, q2=-4q, q3=+2q

1/24/07 184 Lecture 10 22


Total potential energy: Sum over U’s for all pairs of charges

Answer :

q1=+q, q2=-4q, q3=+2q

The negative potential energy means that negative work would have to be done to assemble the system starting with three charges at infinity.

Picture D

1/24/07 184 Lecture 10 23

Example - Four ChargesExample - Four ChargesExample - Four ChargesExample - Four Charges

Consider a system of four point charges as shown. The four point charges have the values q1 =+1.0 C, q2 = +2.0 C, q3 = -3.0 C, and q4 = +4.0 C. The charges are placed such that a = 6.0 m and b = 4.0 m.

What is the electric potential energy of this system of four point charges?

1/24/07 184 Lecture 10 24

Example - Four Charges (2)Example - Four Charges (2)Example - Four Charges (2)Example - Four Charges (2)

aqq

bqq

Dqq

Dqq

bqq

aqq

kU

434232

413121

Energy of the complete assembly

= sum of pairs

Answer: 1.2 x 10-3 J

1/24/07 184 Lecture 10 25

Example - 12 Electrons on a CircleExample - 12 Electrons on a CircleExample - 12 Electrons on a CircleExample - 12 Electrons on a Circle Consider a system of 12 electrons

arranged on a circle with radius R as indicated in the figure. Relative to V=0 at infinity, what are the electric potential V and the electric field E at point C?

Superposition principle:

Symmetry: E=0

The electric field of any given electron is canceled by the field due to the electron located at the diametrically opposite position.

R

ek

Re

kCVi

12)(

12

1

0)( CE

1/24/07 184 Lecture 10 26

Clicker QuestionClicker QuestionClicker QuestionClicker Question

Consider a rearrangement of the 12 electrons as indicated in the figure. Relative to previous arrangement of the electrons, what are the electric potential V and the electric field E at point C?

A: V is unchanged, E not 0 anymore B: V is bigger, still E=0 C: V is smaller, E not 0 anymore D: V is the same, still E=0

The symmetry is lost, there is no total cancellation of the electric fields anymore

1/24/07184 Lecture 101 PHY 184 Spring 2007 Lecture 10 Title: The Electric Potential, V(x)

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