Electric Potential - Austin Community College District · 2012. 8. 25. · MFMcGraw-PHY 2426...

MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 1

Chapter 24

Electric Potential

1. Potential Difference

2. Potential Due to a System of Point Charges

3. Computing the Electric Field from the Potential

4. V for Continuous Charge Distributions

5. Equipotential Surfaces

6. Electrostatic Potential Energy

Conservative Force

For conservative forces we can introduce

the concept of a potential energy.

Potential energy is the energy of position

whereas kinetic energy is the energy of

motion.

Gravitational Analogy

The positive charge is used to make the

analogy more transparent.

If you throw a ball straight up in the air its initial kinetic

energy is soon reduced to zero at the top of its motion.

The “loss of kinetic energy” is only apparent. The kinetic

energy is restored to its original value when the ball returns

to its starting position.

The use of the concept of potential energy allows us to

maintain the conservation of total mechanical energy.

Total Mechanical Energy = KE + PE

An Example from Last Semester

Examples of Forces

Conservative Forces

Gravitation

Electric force

Magnetic force

Spring force

Non-Conservative Forces

Friction

Air resistance

The magnetic force has some restrictions on it since in

general a velocity dependent force is not conservative.

The kinetic energy is transformed into heat

and is then outside the mechanical system.

Work and the Conservative Force

The characteristic of a conservative force is that the work

done on an object moving it, in a force field, from point A

to point B is independent of the path taken between A and

If the path forms a closed loop then the work done is zero.

⋅∫� �

W = F dL = 0

Potential Energy - Potential Function

There is a potential energy (U) and a potential function

(V) and they are closely related.

� �i

� �

� �i

dU = -F dL

F = qE

dU = -qE dL

∆∫

� �i

dU = -q E dL

dUdV = -E dL

U∆V = V - V = = - E dL

Potential Energy Potential Function

∆∫� �i

U∆V = V - V = = - E dL

The Change in Potential a b

The integral is equal to

the difference between

the potential function

evaluated at the end

points.

Potentials for Discrete Charges

Potential and the Reference Point

The potential

function needs a

common reference

point so that the

calculated potential

differences will

have physical

meaning.

Potential and the Point Charge

∫ ∫ ∫

� � �i i

P ref 2 2

ref ref ref

P ref P 2

P refref

kqE = r

kq kqV - V = - E dL = - r dL = - dr

dr kq kqV - V V - 0 = -kq =

kqV = ; as r goes to

rFinal form

Potential Due to Two Point Charges

Find V at points P1 and P2 ∑ i 1 2

i i 1 2

kq kq kqV = = +

kq kq kq 9x10 * 5.0x10V = + = 2 = 2 = 2247 V

r r r .040

9 -9 9 -9

1 2P ' '

kq kq 9x10 * 5.0x10 9x10 * 5.0x10V = + = + = 1200 V

r r .060 .010

Potential Calculation for 2 Point Charges

∑ i 1 2

i i 1 2

kq kq kqV = = +

1 2kq kqV = +

x x - a

Potential for 2 Point Charges

In contrast to the electric field component graphs the potential

graphs, being scalar quantities, depend on the sign of the charge,

there is no direction.

Electric Dipole Geometry for Potential

Calculation

The dipole configuration requires equal and opposite charges.

kql lV = ; x >

2lx -4

≈ �2 2

kql kpV = ; x l

p = ql is the dipole moment

3D Image of the Electric Dipole Potential

Electric Field from the Electric Potential

In general

For one dimension

ˆ ˆ�� i i i xdV(x) = -E dl = -E dxi = -(E i)dx = -E dx

dV(x)E = -

ˆˆ ˆ ∂ ∂ ∂

∇ ∂ ∂ ∂

� V V VE = - V = - i + j + k = -gradV

Potentials for Continuous Charge

Distributions

Potential Due to a Ring Charge

= = =∫ ∫Q Q

kdq k kQV dq

2 2 2= +r z a

1 1 12 2 2 22 2 21

2 21 1

= = = + + +

kQ kQ kQV

z a a az z

Electric Field Due to a Ring Charge

312 2 2 22 2

32 2 2

3 3 3 22 2 22 23

ˆ ˆ ˆ ˆ1

− −

∂ ∂ = − = − + = + ∂ ∂

= = = = +

VE kQ z a kQ z a z

kQzE E k k

kQz kQz kQz aE E k k k k

zzz a az

Electric Field Due to a Ring Charge

Far Away from the Ring

3 32 2

3ˆ ˆ ˆ1 12

3 ˆ ˆlim 12

→∞

= = + = −

kQz a kQz aE E k k k

z zz z

kQz a kQk k

This the electric field of a point charge of value Q

Particle Acceleration Due to a Ring

Charge

The particle and the ring both have the

same sign charge. Therefore the particle

experiences an accelerating force.

Do calculation

Potential Due to a Disk Charge

For simplicity we only consider potentials along the z-axis

Potential Due to a Disk Charge

2σ σ π

dq dA ada

12 2 2

( ) 2 1 1

= + −

∫ ∫V R

k adakdqdV

adadV V k

RV z k z

Potential Close to a Disk Charge

2( ) 2 1 1π σ

= + −

RV z k z

Very close to the disk z<<R. The disk

appears to effectively be an infinite plane.

( ) 2 1 1 2 1

( ) 2 2 2

π σ π σ

π σ π σ π σ

= + − ≈ −

= − = −

R RV z k z k z

V z k R k z V k z

The Absolute Value Operation

Potential Due to an Infinite Plane of

Charge

The potential function is required to be continuous but its

derivatives need not be continuous.

The Electric Field Due to an Infinite

Plane of Charge (z>0)

( ) ( )

π σ π σ

σπ σ

πε ε

∂ ∂= − = =

∂ ∂

V z V k z

VE z k z k

Potential Due to a Thin Spherical Shell

No electric field inside, therefore

V is constant inside the sphere.

Do calculation

Potential Due to a Thin Spherical Shell

ˆ For r > R,

= − =

�� i

kQdV E dl E r

kQdV r dl

+∞ +∞

= − = − =∫ ∫p pr r

kQ kQV dr kQ r dr

For r < R

20 −

+∞ +∞

= − = − − =∫ ∫ ∫p pr rR

kQ kQV dr kQ r dr dr

Potential Due to a Uniformly Charged

Solid Sphere

kQFor r R E r

kQFor r R E

Potential For an Infinite Line Charge

Here we are going

to use our

knowledge of the

field to determine

the potential.

Due to the geometry

here we are unable

to place the

reference point at

infinity.

Since it is the “difference in potential” that has a physical meaning, whatever

reference point we pick will drop out upon taking the difference of potential.

⋅∫� �P

∆V = V - V = - E dL

P ref P

RV - V = V - 0 = 2kλln

Equipotential Surfaces

The equipotentials (gray

lines) are represented by

contours along which the

value of the potential

function is constant.

Potential lines and

electric field lines are

everywhere

perpendicular.

Equipotential Surfaces

Close to the surface of

the conductor the

equipotential surfaces

follow the shape of the

conductor.

The surface of a

conductor is an

equipotential surface.

Equipotential Surfaces-Point Charges

Field Lines

Potential Lines

Potentials Due to Spherical Shells

Do calculation

Primitive Method of Storing a Large

Charge or Creating a High Potential

Spherical conductor on the end of an

insulating rod

Spherical conductor is charged positively by

removing a quantity of electrons.

Touching the charged sphere to the inner

surface of the large hollow conducting sphere

causes electrons to transfer to the small

spherical conductor - the large sphere is left

with a net positive charge.

Repeat the process to build up the charge.

Eventually the process saturates.

The Van de Graff Generator

Robert J. Van de Graaff

1901-1967

The energies produced by Van de Graaff

atomic particle accelerators are limited to

about 30 MeV, even with tandem generators

accelerating doubly charged (for example

alpha) particles.

More modern particle accelerators using

different technology produce much higher

energies, thus Van de Graaff particle

accelerators have become largely obsolete.

They are still used to some extent for

graduate student research at colleges and

universities and as ion sources for high

energy bursts.

Ernest O. Lawrence

1901- 1958

Ernest Orlando Lawrence was an American physicist and

Nobel Laureate, known for his invention, utilization, and

improvement of the cyclotron atom-smasher beginning in

1929, based on his studies of the works of Rolf Widerøe, and

his later work in uranium-isotope separation for the

Manhattan Project.

Lawrence had a long career at the University of California,

where he became a Professor of Physics. In 1939, Lawrence

was awarded the Nobel Prize in Physics for his work in

inventing the cyclotron and developing its applications.

Chemical element number 103 is named "lawrencium" in

Lawrence's honor.

Source: http://en.wikipedia.org/wiki/E._O._Lawrence

Lawrence read a journal article about producing very high-energy particles

required for atomic disintegration by means of a succession of very small

"pushes."

In an attempt to make the accelerator more compact, Lawrence decided to

use a circular accelerating chamber situated between the poles of a large

electromagnet. The magnetic field would hold the charged particles in a

planar spiral path as they were accelerated between the two semicircular

electrodes. These electrodes would be sandwiched between the magnetic

poles and connected to an alternating potential. After about a hundred

circular orbits, the particles would impact the target as a beam of high-

energy particles. Lawrence had discovered a method for obtaining particles

of very high energy without the use of any high voltage.

The Beginning of the Cyclotron

Giant calutron plants developed at Lawrence's laboratory were used at Site X during

World War II to purify uranium for use in the first atomic bomb. Alpha Track Calutron at

the Y-12 Plant (Site X) at Oak Ridge, Tennessee from the Manhattan Project, used for

uranium enrichment.

Alpha Track Calutron at the Y-12 Plant (Site X) at

Oak Ridge, Tennessee

Ernest O. Lawrence

1901- 1958

Robert J. Van de Graaff

1901-1967

Linear

Acceleration

Circular

Acceleration

Van de Graaff was jealous of

his more successful rival -

Lawrence. In later years at

HVE his engineers could not

even discuss a design that

involved particles traveling

in a circular motion.

Dielectric Breakdown

High E-Field

Low E-Field

“Air by nature is an Insulator. For it to become a Conductor, it must experience Dielectric

Breakdown. This occurs if a voltage of 30,000V per centimeter is applied to the air. The air is

then Ionized. In a Jacobs Ladder, Dielectric breakdown occurs when a voltage of 30,000 V is

supplied to the bottom of the Rods.”

“An Electric Arc is formed between the Rods and rises along with the hot air. As the arc length

increases, its power consumption increases and there is insufficient Voltage to maintain the

Electric Arc. The arc extinguishes and the cycle repeats with a new arc forming at the bottom

of the rods.”

Source: http://macao.communications.museum/eng/Exhibition/secondfloor/moreinfo/2_1_1_JacobsLadder.html

Jacob’s Ladder serves no

scientific purpose but

every mad scientist in the

movies had to have at least

one in his lab.

Dielectric Strengths

Source: http://en.wikipedia.org/wiki/Dielectric_strength

Two Charged Spherical Conductors

The thin conducting wire connecting the two spheres serves

the same purpose as allowing the two spheres to touch.

This contact allows charge to transfer (remember only the

electrons move) between the spheres and to bring the

conductor surfaces to the same potential.

Do calculation

U - Electrostatic Potential Energy

• The electrostatic energy of a

system of charges is equal to the

Work required to assemble the

charges from infinity.

• The Work is the difference in

potential energy between the

charge at infinity and at its final

position.

• Since U at infinity is zero, the

work is just U evaluated at the

final charge positions.

U - Electrostatic Potential Energy

1U = q V

1U = QV

For point charges

A conductor with a charge Q

at potential V

For a system of conductors

Extra Slides

Conservative Forces and Potentials

from Vector Analysis

Work around a closed loop = 0

Stokes Theorem

Therefore a potential function V exists for a conservative force.

⋅ = ∇ ⋅

∇ ⋅ ⇒ ∇

∇ ∇ ∇

∫ ∫

��

� � � ��

� � � �

W = F dl = 0

F dl × F da

× F da = 0 × F = 0

F = - V since × V = 0

Vector Analysis

ψ��

φ and are scalar functions

F and G are vector functions

∇ ⋅

� � � �

φ= grad φ= gradient of φ

F = div F = divergence of F

× F = curl F = curl of F

Vector Analysis

Gradient

Divergence

ˆˆ ˆφ φ φ∂ ∂ ∂

∇∂ ∂ ∂

�φ= i + j + k

∂∂ ∂∇ ⋅

∂ ∂ ∂

� �yx z

FF FF = + +

ˆˆ ˆ

∂ ∂ ∂∇ ×

∂ ∂ ∂

� �

F =x y z

Electric Potential - Austin Community College District · 2012. 8. 25. · MFMcGraw-PHY 2426...

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