Electric Potential Energy &

Voltage

Tesla Envy http://www.youtube.com/watch?v

=jlzEQZ4EfqA&feature=related

Ch 23 & 24:

Electric Force and Field

1 212 2

ˆq q

kr

F r

qEF

11 2

Electric Field E due to q :kq

Er

Ch 25: ENERGY & VOLTAGE

E is the FORCE FIELD: VECTOR

V is the ENERGY FIELD: SCALAR

The force field lines are everywhere

perpendicular to the potential energy field lines

Gravitational Force Field and

Potential Energy & Potential

Review: Work

• If F is constant then

• If F varies with r then along any radial segment

• The total work is

( )dW d F r dr F r

( ) f

i

r

rW F r dr

A force moving a mass through a displacement does WORK.

cosW Fr F r

Review: Conservative Forces

and Potential Energy

• A conservative force is defined as a vector field

that is equal to the negative of the

• gradient of a scaler field –

the ‘Poetential Energy’ function U:

• Leads to Work Energy Theorem and Conservation of

Energy:

f

i

x

xx

W F dx U K

x

dUF

dx

The gradient of a scalar field at a point is a vector pointing in the direction of

the steepest slope or grade at that point. The steepness of the slope at that

point is THE FORCE!!!

0

U K

E K U

Electric Potential Energy

• The work done by F along any radial segment is

• The work done by a force that is perpendicular to the displacement is 0. ds = dr

• The total work is

Since the Force is Conservative, the Work is independent of path.

W U K Recall that the work done by a

conservative force on an object is:

(As a rock falls, it loses PE but gains KE!!!)

0( ) ( )dW r ds q r ds F E

0 ( )B

AW q r d E r

Potential Difference in a Uniform Field

• Electric field lines always point in

the direction of decreasing electric

potential

• When the electric field is directed

downward, point B is at a lower

potential than point A

• When a positive test charge moves

from A to B, the charge-field system

loses potential energy

B B

B AA A

V V V d E d Ed E s s

The Electric Field does

work on the charge to

move it a distance d from

A to B, converting

Potential Energy into

Kinetic Energy, same as

with the gravitational field.

Conservation of Energy

and the Work-Energy

Theorem apply:

U K W F r

E

g

U qEd

U mgh

r d

(Uniform Field Only)

Electric Potential

B AV V V

The Electric Potential Energy per

unit charge between A and B is

called the Electric Potential

Difference between A and B.

For a uniform field:

0 0/q Ed q

Ed

V Ed

0/U q

ABV Ed

Electric Potential

•Unit for Electric Potential is the VOLT:

•1 V = J/C = N m /C

• Electric Potential is a SCALAR!

• Only DIFFERENCES in potential between

two points can be measured. There is no

potential for a single point unless the other

point is zero – the ground.

• Electric Potential is NOT Potential Energy!

• + charges move from hi to low potential

(think gravity – water fall)

• - charges move from low to hi potential

(think antigravity – water pump)

High Potential

Low Potential

B AV V V E s

Potential Difference: VOLTAGE

As a + particle moves from A to B, its

electric potential energy changes by

0 ( )B

B AA

U U U W q r d E r

Define the potential difference between points

A and B 0/V U q

B

B AA

V V V E ds

• The electric potential is independent of the path

• It is customary to choose a reference potential of V = 0 at rA = ∞

• E and V are perpendicular!!!

First: The Simplest Case Constant Force -Uniform Electric Fields

B AV V V E s

Electron Volts

What is the kinetic energy gained by

an electron when it moves across a

potential difference of 1 volt?

K U Vq

19( 1 )( 1.6 10 )V x C

191 / (1.6 10 )J C x C

191.6 10x J

191 1.6 10eV x J

Note: 5eV electron breaks organic bonds in molecules.

Nuclear decay energies ~ 1MeV

High Potential Low Potential

Example What is the final speed of a free

electron accelerated from rest through

a potential difference of 1V?

2qVv

m

K U

21

2mv q V

195

31

2( 1.6 10 )15.93 10 /

9.11 10

x C Vv x m s

x kg

Potential

Worksheet

High Potential

Low Potential

B AV V V E s

Potential Difference: Signs

Example A 12V motorcycle battery moves 5000C of charge. How much

energy does it deliver to that charge?

12V V

0/V U q

0U Vq

12 (5000 )V C

12 / (5000 )J C C

46.00 10x J

Potential (Voltage) is NOT Energy!

Problem

What is the magnitude of the maximum voltage that can be

sustained between 2 parallel plates separated by 2.5 cm of dry

air? Dry air supports max field strength (dielectric strength CH

26) of 3x 106 V/m .

V E s 6(3 10 )(.025 )x V m

47.5 10x V

75kV

More than this and the air breaks down and

becomes a conductor. LIGHTENING!

B AV V V E s

The dielectric breakdown strength of dry air, at

Standard Temperature and Pressure (STP), between

spherical electrodes is approximately 33 kV/cm.

Jacob’s Ladder

WARNING!

Exposure to an arc-producing device can pose health hazards. In a closed space such as a classroom or home, the continuous arc formation of an open-air Jacob's Ladder will ionize oxygen and nitrogen, which then reforms into reactive molecules such as ozone and nitric oxide. These free radicals can be damaging to the mucous membranes of people near the spark gap.

Time exposure

Sources of Electric Potential:

Batteries 1.5 Volt Battery is the Voltage across the terminals.

A

B

Case 2: Potential Due to Point

Charges & Varying Force

Potential and Point Charges • A positive point charge produces a

field directed radially outward

• The potential difference between

points A and B will be

1 1B A e

B A

V V k qr r

2

1 1B B B

B AA A A

B A

kqV V V E ds E dr dr kq

r r r

• The electric potential is independent of the

path between points A and B

• It is customary to choose a reference

potential of V = 0 at rA = ∞

• Then the potential at some point r is:

e

qV k

r

PROBLEM What is the electric potential 1.2m from a

point charge Q = + 4x10-8C. How does the potential

change if the charge is positive or negative?

r

Point Charge: ( )Q

V r kr

2 89

2

4 10( ) 9.00 10

1.2

Nm x CV r x

C m

( ) 300Nm

V rC

( ) 300V r V (Note: If Q is negative, V is negative!)

r

dVE

dr

The Electric Field is the Negative of the

Slope (Gradient) of the Electric Potential!

e

qV k

r

2e

qE k

r

The gradient of a scalar field at a point is a vector pointing in the direction of

the steepest slope or grade at that point. The steepness of the slope at that

point is given by the magnitude of the gradient vector.

B AV V V E sV

Es

B

B AA

V V V E ds

VE

s

VE

s

VE

s

In Sum: Potential due to a Point Charge

r

The potential a distance r from a point charge is given by:

( )Q

V r kr

V is the potential difference between r

and infinity where the potential is taken

to be zero (the ground is at infinity.)

V gives the amount of energy per unit

charge that it takes to bring a test charge

from infinity to r.

The total potential due to a distribution of charge is equal to the

algebraic sum of all the potentials (not vector sum!) The sign of

the charge matters!!!

Multiple Charges • The electric potential due to several

point charges is the sum of the potentials due to each individual charge, taking the potential at infinity equal to zero and r is the distance from the charge to the point P:

• The potential energy of the system is

1 2

12

q qU k

r

i

i i

qV k

r

0U q V

Problem Determine the electric potential at point A & B due to a dipole

A proton is placed at point A and released.

What is the change in potential energy of the proton between A and B?

What is the velocity of the proton at B?

Electric Potential of a Dipole

U for System of Static Charges

• The potential energy of a system of charges is the energy needed to assemble them from infinity where U = 0. The total U includes a term for every pair of charges!!!

• For two charges it is:

• For three charges:

1 3 2 31 2

12 13 23

q q q qq qU k

r r r

1 2

12

q qU k

r

i j

net

ij ij

q qU k

r

Potential Energy Problem Determine the electric potential energy of the triangular group. That

is, find the work done to bring them from infinity to the triangular

arrangement. Is there a place interior to the group where the

potential is zero? The side of the triangle is 0.50m.

1 3 2 31 2

12 13 23

e

q q q qq qU k

r r r

i j

net

ij ij

q qU k

r

i

i i

qV k

r

Sample Problem –Old Exam

Problem

Case 4: Electric Potential for ANY

Continuous Charge Distribution In general, the potential at

some point due to a small

charge element dq is

P e

dqdV k

r

V( )=0P e

dqV k

r

Nonuniform Continuous Charge

Distribution, total charge Q

V( )=0P e

dqV k

r

0x

As shown in the text…

V for a Uniformly Charged Ring

Easier to calculate V

because it is a scalar! Find

E from the gradient.

2 2

dq kQV k

r a x

3/2

2 2

ˆ( )kxQ

E x ix a

x

dVE

dx

V for a Uniformly Charged Disk

• The ring has a radius R

and surface charge

density of σ

• P is along the

perpendicular central

axis of the disk

12 2 22V πkσ R x x

2 22 1Disk

xE k

x R

x

dVE

dx

Cavity in a Conductor

• Assume an irregularly

shaped cavity is inside

a conductor

• Assume no charges

are inside the cavity

• The electric field

inside the conductor

must be zero

Cavity in a Conductor, cont

• The electric field inside does not depend on the charge distribution on the outside surface of the conductor

• For all paths between A and B,

• A cavity surrounded by conducting walls is a field-free region as long as no charges are inside the cavity

0B

B AA

V V d E s

V Due to a Charged Conductor

• The potential difference between A and B is also zero

• The charge density is high where the radius of curvature is small

– And low where the radius of curvature is large

• The electric field is large near the convex points having small radii of curvature and reaches very high values at sharp points

Corona Discharge

• If the electric field near a conductor is sufficiently strong, electrons resulting from random ionizations of air molecules near the conductor accelerate away from their parent molecules

• These electrons can ionize additional molecules near the conductor

• This creates more free electrons

• The corona discharge is the glow that results from the recombination of these free electrons with the ionized air molecules

• The ionization and corona discharge are most likely to occur near very sharp points

Low energy corona discharge

surrounding a circular conductor.

Corona Discharge

Corona Discharge Supersonic Free-Jet

(CDSFJ) to generate excited-state molecular

nitrogen for growing III-N semiconductors. .

Corona Discharge

Corona discharge into free air

Jacob’s Ladder

WARNING!

Exposure to an arc-producing device can pose health hazards. In a closed space such as a classroom or home, the continuous arc formation of an open-air Jacob's Ladder will ionize oxygen and nitrogen, which then reforms into reactive molecules such as ozone and nitric oxide. These free radicals can be damaging to the mucous membranes of people near the spark gap.

Time exposure

Van de Graaff

Generator

• Charge is delivered continuously to a high-potential electrode by means of a moving belt of insulating material

• The high-voltage electrode is a hollow metal dome mounted on an insulated column

• Large potentials can be developed by repeated trips of the belt

• Protons accelerated through such large potentials receive enough energy to initiate nuclear reactions

Electrostatic Precipitator • An application of electrical discharge in gases

is the electrostatic precipitator

• It removes particulate matter from

combustible gases

• The air to be cleaned enters the duct and

moves near the wire

• As the electrons and negative ions created by

the discharge are accelerated toward the outer

wall by the electric field, the dirt particles

become charged

• Most of the dirt particles are negatively

charged and are drawn to the walls by the

electric field

Application – Xerographic

Copiers

• The process of xerography is used for

making photocopies

• Uses photoconductive materials

– A photoconductive material is a poor conductor

of electricity in the dark but becomes a good

electric conductor when exposed to light

The Xerographic Process

Application – Laser Printer

• The steps for producing a document on a laser printer is similar to the steps in the xerographic process

• A computer-directed laser beam is used to illuminate the photoconductor instead of a lens

Millikan Oil-Drop Experiment –

Experimental Set-Up

PLAY

ACTIVE FIGURE

Millikan Oil-Drop Experiment

• Robert Millikan measured e, the magnitude

of the elementary charge on the electron

• He also demonstrated the quantized nature

of this charge

• Oil droplets pass through a small hole and

are illuminated by a light

Oil-Drop Experiment, 2

• With no electric field

between the plates, the

gravitational force and

the drag force

(viscous) act on the

electron

• The drop reaches

terminal velocity with D mF g

Oil-Drop Experiment, 3

• When an electric field is set up between the plates

– The upper plate has a higher potential

• The drop reaches a new terminal velocity when the electrical force equals the sum of the drag force and gravity

Oil-Drop Experiment, final

• The drop can be raised and allowed to fall

numerous times by turning the electric field on

and off

• After many experiments, Millikan determined:

– q = ne where n = 0, -1, -2, -3, …

– e = 1.60 x 10-19 C

• This yields conclusive evidence that charge is

quantized

• Use the active figure to conduct a version of the

experiment

Case 3: Electric Potential Given the

Electric Field

If there is sufficient symmetry

then you can use Gauss’s Law

to find the E field and find the

potential difference from

B

B AA

V V V dE s

Choose V = 0 at some convenient

point, usually infinity.

E Compared to V • The electric potential is a

function of r

• The electric field is a

function of r2

• The effect of a charge on

the space surrounding it:

– The charge sets up a

vector electric field

which is related to the

force

– The charge sets up a

scalar potential which

is related to the energy

Bohr Orbital Binding Energy for

Single Electron Atoms

2

213.6n

ZE eV

n

Ground State: n = 1

First Excited: n = 2

2nd Excited: n = 3

1. -13.6eV is the energy of the H ground state.

2. Negative because it is the Binding Energy and work must be

done on the atom (by a photon) to ionize it.

3. A 13.6eV photon must be absorbed to ionize the ground state

Binding Energy in Molecules