1/23/07 184 Lecture 9 1

PHY 184PHY 184PHY 184PHY 184

Spring 2007Lecture 9

Title: The Electric Potential

1/23/07 184 Lecture 9 2

AnnouncementsAnnouncementsAnnouncementsAnnouncements

Homework Set 2 done, Set 3 ongoing and Set 4 will open on Thursday

Helproom hours of the TAs are listed on the syllabus in LON-CAPA• Honors Option students will provide help in the in the

SLC starting this week.

Remember Clicker’s Law…

Up to 5% (but not more!)

1/23/07 184 Lecture 9 3

Review - Potential Energy Review - Potential Energy Review - Potential Energy Review - Potential Energy

When an electrostatic force acts between charged particles, assign an electric potential energy, U.

The difference in U of the system in two different states, initial i and final f, is

Reference point: Choose U=0 at infinity. If the system is changed from initial state i to the

final state f, the electrostatic force does work, W

Potential energy is a scalar.

if UUU

WUUU if

1/23/07 184 Lecture 9 4

Review - Work Review - Work Review - Work Review - Work

Work done by an electric field

cos so

and

qEddEqW

EqFdFW

Q is the angle between electric field and displacement

(1) Positive W U decreases(2) Negative W U increases

1/23/07 184 Lecture 9 5

Clicker Question Clicker Question Clicker Question Clicker Question

In the figure, a proton moves from point i to point f in a uniform electric field directed as shown. Does the electric field do positive, negative or no work on the proton?

A: positive B: negative C: no work is done on the

proton

1/23/07 184 Lecture 9 6

Clicker Question Clicker Question Clicker Question Clicker Question

In the figure, a proton moves from point i to point f in a uniform electric field directed as shown. Does the electric field do positive, negative or no work on the proton?

B: negative

1/23/07 184 Lecture 9 7

Electric Potential Electric Potential VVElectric Potential Electric Potential VV

The electric potential, V, is defined as the electric potential energy, U, per unit charge

The electric potential is a characteristic of the electric field, regardless of whether a charged object has been placed in that field. (because U q)

The electric potential energy is an energy of a charged object in an external electric field (or more precisely, an energy of the system consisting of the charged object and the external field).

V U

q

1/23/07 184 Lecture 9 8

Electric Potential Difference Electric Potential Difference VVElectric Potential Difference Electric Potential Difference VV

The electric potential difference between an initial point i and final point f can be expressed in terms of the electric potential energy of q at each point

Hence we can relate the change in electric potential to the work done by the electric field on the charge

V Vf Vi U f

qUiq

Uq

V We

q

1/23/07 184 Lecture 9 9

Electric Potential Difference (2)Electric Potential Difference (2)Electric Potential Difference (2)Electric Potential Difference (2)

Taking the electric potential energy to be zero at infinity we have

where We, is the work done by the electric field on the charge as it is brought in from infinity.

The electric potential can be positive, negative, or zero, but it has no direction. (i.e., scalar not vector)

The SI unit for electric potential is joules/coulomb, i.e., volt.

V We,

q

Explain: i = , f = x, so that V = V(x) 0

1/23/07 184 Lecture 9 10

The VoltThe VoltThe VoltThe Volt

The commonly encountered unit joules/coulomb is called the volt, abbreviated V, after the Italian physicist Alessandro Volta (1745 - 1827)

With this definition of the volt, we can express the units of the electric field as

For the remainder of our studies, we will use the unit V/m for the electric field.

1 V = 1 J

1 C

m

V

C

J/m

C

N

][

][][

qF

E

1/23/07 184 Lecture 9 11

Example - Energy Gain of a ProtonExample - Energy Gain of a ProtonExample - Energy Gain of a ProtonExample - Energy Gain of a Proton

A proton is placed between two parallel conducting plates in a vacuum as shown.The potential difference between the two plates is 450 V. The proton is released from rest close to the positive plate.

What is the kinetic energy of the proton when it reaches the negative plate?

+ -

= V(+)-V()The potential difference between the two plates is 450 V.

The change in potential energy of the proton is U, and V = U / q (by definition of V), soU = q V = e[V()V(+)] = 450 eV

1/23/07 184 Lecture 9 12

Example - Energy Gain of a Proton (2)Example - Energy Gain of a Proton (2)Example - Energy Gain of a Proton (2)Example - Energy Gain of a Proton (2)

Because the acceleration of a charged particle across a potential difference is often used in nuclear and high energy physics, the energy unit electron-volt (eV) is common.

An eV is the energy gained by a charge e that accelerates across an electric potential of 1 volt

The proton in this example would gain kinetic energy of 450 eV = 0.450 keV.

1 eV 1.6022 10 19 J

initial finalConservation of energyK = U = + 450 eV

Because the proton started at rest,K = 1.6x10-19 C x 450 V = 7.2x10-17 J

1/23/07 184 Lecture 9 13

The Van de Graaff GeneratorThe Van de Graaff GeneratorThe Van de Graaff GeneratorThe Van de Graaff Generator

A Van de Graaff generator is a device that creates high electric potential.

The Van de Graaff generator was invented by Robert J. Van de Graaff, an American physicist (1901 - 1967).

Van de Graaff generators can produce electric potentials up to many 10s of millions of volts.

Van de Graaff generators can be used to produce particle accelerators.

We have been using a Van de Graaff generator in lecture demonstrations and we will continue to use it.

1/23/07 184 Lecture 9 14

The Van de Graaff Generator (2)The Van de Graaff Generator (2)The Van de Graaff Generator (2)The Van de Graaff Generator (2) The Van de Graaff generator

works by applying a positive charge to a non-conducting moving belt using a corona discharge.

The moving belt driven by an electric motor carries the charge up into a hollow metal sphere where the charge is taken from the belt by a pointed contact connected to the metal sphere.

The charge that builds up on the metal sphere distributes itself uniformly around the outside of the sphere.

For this particular Van de Graaff generator, a voltage limiter is used to keep the Van de Graaff generator from producing sparks larger than desired.

1/23/07 184 Lecture 9 15

The Tandem Van de Graaff The Tandem Van de Graaff AcceleratorAccelerator

The Tandem Van de Graaff The Tandem Van de Graaff AcceleratorAccelerator

One use of a Van de Graaff generator is to accelerate particles for condensed matter and nuclear physics studies.

Clever design is the tandem Van de Graaff accelerator.

A large positive electric potential is created by a huge Van de Graaff generator.

Negatively charged C ions get accelerated towards the +10 MV terminal (they gain kinetic energy).

Terminal at +10MV

C-1 C+6

Stripper foilstrips electrons from C

Electrons are stripped from the C and the now positively charged C ions are repelled by the positively charged terminal and gain more kinetic energy.

1/23/07 184 Lecture 9 16

Example - Energy of Tandem AcceleratorExample - Energy of Tandem AcceleratorExample - Energy of Tandem AcceleratorExample - Energy of Tandem Accelerator

Suppose we have a tandem Van de Graaff accelerator that has a terminal voltage of 10 MV (10 million volts). We want to accelerate 12C nuclei using this accelerator.

What is the highest energy we can attain for carbon nuclei?

What is the highest speed we can attain for carbon nuclei?

There are two stages to the acceleration• The carbon ion with a -1e charge gains energy

accelerating toward the terminal• The stripped carbon ion with a +6e charge gains

energy accelerating away from the terminal15 MV Tandem Van de Graaff at Brookhaven

1/23/07 184 Lecture 9 17

Example - Energy of Tandem Accelerator Example - Energy of Tandem Accelerator (2)(2)

Example - Energy of Tandem Accelerator Example - Energy of Tandem Accelerator (2)(2)

The mass of a 12C nucleus is 1.99 10-26 kg

K 1

2mv2

v 2K

m

2 1.12 10 11 J

1.99 10-26 kg3.36 107 m/s

v 11% of the speed of light

J 1012.1eV 1

J 101.602 MeV70

MeV70 MV107

6 and 1

1119-

21

21

K

eK

eqeq

VqVqUK

1/23/07 184 Lecture 9 18

Equipotential Surfaces and LinesEquipotential Surfaces and LinesEquipotential Surfaces and LinesEquipotential Surfaces and Lines

When an electric field is present, the electric potential has a given value everywhere in space. V(x) = potential function

Points close together that have the same electric potential form an equipotential surface. i.e, V(x) = constant value

If a charged particle moves on an equipotential surface, no work is done.

Equipotential surfaces exist in threedimensions.

We will often take advantageof symmetries in the electric potentialand represent the equipotential surfacesas equipotential lines in a plane.

Equipotential surface from eight point chargesfixed at the corners of a cube

1/23/07 184 Lecture 9 19

If a charged particle moves perpendicular to electric field lines, no work is done.

If the work done by the electric field is zero, then the electric potential must be constant

Thus equipotential surfaces and lines must always be perpendicular to the electric field lines.

if d E

General ConsiderationsGeneral ConsiderationsGeneral ConsiderationsGeneral Considerations

V We

q0 V is constant

1/23/07 184 Lecture 9 20

Electric field lines and equipotential surfaces

1/23/07 184 Lecture 9 21

Constant Electric FieldConstant Electric FieldConstant Electric FieldConstant Electric Field

Electric field lines: straight lines parallel to E Equipotential surfaces (3D): planes perp to E Equipotential lines (2D): straight lines perp to E

1/23/07 184 Lecture 9 22

Electric Field from a Single Point ChargeElectric Field from a Single Point ChargeElectric Field from a Single Point ChargeElectric Field from a Single Point Charge

Electric field lines: radial lines emanating from the point charge.

Equipotential surfaces (3D): concentric spheres Equipotential lines (2D): concentric circles

1/23/07 184 Lecture 9 23

Electric Field from Two Oppositely Charged Point ChargesElectric Field from Two Oppositely Charged Point ChargesElectric Field from Two Oppositely Charged Point ChargesElectric Field from Two Oppositely Charged Point Charges

The electric field lines from two oppositely charge point charges are a little more complicated.

The electric field lines originate on the positive charge and terminate on the negative charge.

The equipotential lines are always perpendicular to the electric field lines.

The red lines represent positiveelectric potential.

The blue lines represent negativeelectric potential.

Close to each charge, the equipotentiallines resemble those from a pointcharge.

1/23/07 184 Lecture 9 24

ELECTRIC DIPOLE

1/23/07 184 Lecture 9 25

Electric Field from Two Identical Point ChargesElectric Field from Two Identical Point ChargesElectric Field from Two Identical Point ChargesElectric Field from Two Identical Point Charges

The electric field lines from two identical point charges are also complicated.

The electric field lines originate on the positive charge and terminate at infinity.

Again, the equipotential linesare always perpendicular tothe electric field lines.

There are only positivepotentials.

Close to each charge, theequipotential lines resemblethose from a point charge.

1/23/07 184 Lecture 9 26

TWO POSITIVE CHARGES

1/23/07 184 Lecture 9 27

Calculating the Potential from the Calculating the Potential from the FieldField

Calculating the Potential from the Calculating the Potential from the FieldField

To calculate the electric potential from the electric field we start with the definition of the work dW done on a particle with charge q by a force F over a displacement ds

In this case the force is provided by the electric fieldF = qE

Integrating the work done by the electric force on the particle as it moves in the electric field from some initial point i to some final point f we obtain

sdFdW

sdEqdW

f

isdEqW

1/23/07 184 Lecture 9 28

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

Remembering the relation between the change in electric potential and the work done …

…we find

Taking the convention that the electric potential is zero at infinity we can express the electric potential in terms of the electric field as

qW

V e

f

i

eif sdE

qW

VVV

i

sdExV

)( ( i = , f = x)

1/23/07 184 Lecture 9 29

Example - Charge moves in E fieldExample - Charge moves in E fieldExample - Charge moves in E fieldExample - Charge moves in E field

Given the uniform electric field E, find the potential difference Vf-Vi

by moving a test charge q0 along the path icf.

Idea: Integrate Eds along the path connecting ic then cf. (Imagine that we move a test charge q0 from i to c and then from c to f.)

1/23/07 184 Lecture 9 30

Example - Charge moves in E fieldExample - Charge moves in E fieldExample - Charge moves in E fieldExample - Charge moves in E field

EdVV

EdsEsdE

sdE

sdEsdEVV

if

f

c

f

c

c

i

f

c

c

iif

21distance)45cos(

E)lar toperpendicu (ds 0

distance = sqrt(2) d by Pythagoras

1/23/07 184 Lecture 9 31

Clicker QuestionClicker QuestionClicker QuestionClicker Question

We just derived Vf-Vi for the path i -> c -> f. What is Vf-Vi when going directly from i to f ?

A: 0B: -EdC: +EdD: -1/2 Ed

Quick: V is independent of path.Explicit: V = - E . ds = E ds = - Ed