Chapter25 Potential

8/12/2019 Chapter25 Potential

1/36

Quiz on the math needed today

What is the result of this integral: B

A

dxx 21

)?(iswhat,)()(andknownis)(If xzdxxzxyxy

)?(iswhat,)()(andknownis)(If integralline rzsrzrr

dyy

BAxdx

x

B

A

B

A

11112

dx

xdyxz

)()(

( ) ( ),

In a Cartesian coordinate systemx

y

y z

z r r

i j k


2/36

Chapter 25

Electric Potential


3/36

A review of gravitational

potential

B

When object of mass mis on ground level B, we define thatit has zero gravitational potential energy. When we let go of

this object, if will stay in place.

When this object is moved to elevation A, we say that it has

gravitational potential energy mgh. his the distance from B

to A. When we let go of this object, if will fall back to level B.

A

When this object is at elevation A, it has gravitational

potential energy UA-UB= mgh. UA is the potential energy at

point A with reference to point B. When the object falls from

level A to level B, the potential energy change:U=UB-UA

The gravitational force does work and causes the potentialenergy change: W = mgh = UA-UB= -U

h

mg

We also know that the gravitational force is conservative:

the work it does to the object only depends on the two

levels A and B, not the path the object moves.


4/36

Introduction of the electric potential, a

special case: the electric field is a constant.

EF

0q

When a charge q0is placed inside an electric field, itexperiences a force from the field:

When the charge is released, the field moves it from A

to B, doing work:

Edqq 00W dEdF

If we define the electric potential energy of the

charge at point A UAand B UB, then:

UUUEdq BA 0W

If we define UB=0, then UA= q0Edis the electricpotential energy the charge has at point A. We can

also say that the electric field has an electric potential

at point A. When a charge is placed there, the charge

acquires an electric potential energy that is the charge

times this potential.


5/36

Electric Potential Energy, the

general caseWhen a charge is moved from point A to point B in anelectric field, the charges electric potential energy inside

this field is changed from UAto UB: AB UUU

EF

0qThe force on the charge is:

B

A

BA dqWUUU sE

0force)fieldthe(of

So we have this final formula for electric potential energy

and the work the field force does to the charge:

When the motion is caused by the electric field force onthe charge, the work this force does to the charge cause

the change of its electric potential energy, so: UW


6/36

Electric Potential Energy, final

discussion

Electric force is conservative. The line

integral does not depend on the path from A

to B; it only depends on the locations of A

and B.

B

A

BA dqUUU sE

0

A BLine integral paths


7/36

The electric potential energy of a

charge q0in the field of a charge Q?

Q

q0

R

Reference point:

We usually define the electric potential of a

point charge to be zero (reference) at a point

that is infinitely far away from the point charge.

B

A

BA dqUUU sE

0

Applying this formula:

Where point A is where the charge q0 is, point B is

infinitely far away.

R

Qkdr

r

Qk

drr

Qkdd

ddrQk

e

R

e

e

e

2

2

2

and

so

,

rEsE

rsrE

And the result is a scalar!

So the final answer is

R

QqkRU e

0)(


8/36

Electric Potential, the

definition The potential energy per unit charge, U/qo, is the

electric potential The potential is a characteristic of the field only

The potential energyis a characteristic of the charge-fieldsystem

The potential is independent of the value of qo The potential has a value at every point in an electric field

The electric potential is

As in the potential energy case, electric potential alsoneeds a reference. So it is the potential differenceVthat matters, not the potential itself, unless areference is specified (then it is againV).

o

UV

q


9/36

Electric Potential,

the formula

The potential is a scalar quantity

Since energy is a scalar

As a charged particle moves in an electric

field, it will experience a change in potential

B

A

BA dVVV sE

reference)(often the


10/36

Potential Difference in a

Uniform Field

dEsEsE

B

A

B

A

BA ddVVV

The equations for electric potential can besimplified if the electric field is uniform:

BABA VV,VVV

or0

direction,sametheandi.e.,

dE0,dEWhen:

This is to say that electric field lines alwayspoint in the direction of decreasing electricpotential


11/36

Electric Potential, final

discussion

The differencein potential is themeaningful quantity

We often take the value of the potential to

be zero at some convenient point in thefield

Electric potential is a scalar characteristic

of an electric field, independent of anycharges that may be placed in the field


12/36

Electric Potential, electric

potential energy and Work

When there is electric field, there is electric potential V.

When a charge q0is in an electric field, this charge

has an electric potential energy Uin this electric field:

U= q0 V.

When this charge q0is moved by the electric field force,

the work this field force does to this charge equals the

electric potential energy change -U:W= -U= -q0V.


13/36

Units The unit for electric potential energy is the unit for energy joule

(J).

The unit for electric potential is volt (V):

1 V = 1 J/C This unit comes from U= q0 V (hereU is electric potential energy,

V is electric potential, not the unit volt) It takes one joule of work to move a 1-coulomb charge through a

potential difference of 1 volt

But from

We also have the unit for electric potential as 1 V = 1 (N/C)m

So we have that 1 N/C (the unit of ) = 1 V/m

This indicates that we can interpret the electric field as ameasure of the rate of change with position of the electricpotential

B

A

dV sE

E


14/36

Direction of Electric Field,

energy conservation

As pointed out before, electric fieldlines always point in the directionof decreasing electric potential

So when the electric field is

directed downward, point Bis at alower potential than pointA

When a positive test charge movesfromAto B, the charge-fieldsystem loses potential energythrough doing work to this charge

Where does this energy go?

2502

It turns into the kinetic energy of the object

(with a mass) that carries the charge q0.


15/36

Equipotentials = equal potentials

Points B andCare at a

lower potential than pointA

Points Band Care at the

same potential

All points in a planeperpendicular to a uniform

electric field are at the same

electric potential

The name equipotential

surfaceis given to anysurface consisting of a

continuous distribution of

points having the same

electric potential


16/36

Charged Particle in a Uniform

Field, ExampleQuestion: a positive charge (massm) is released from rest and moves

in the direction of the electric field.

Find its speed at point B.

Solution: The system loses potential

energy: -U=UA-UB=qEd

The force and acceleration are in

the direction of the field

Use energy conservation to find its

speed:

m

qEdv

qEdmv

2

2

1 2


17/36

Potential and Point Charges

A positive point charge

produces a field

directed radially

outward The potential difference

between pointsAand B

will be

1 1B A e

B A

V V k qr r

No line integral needed!


18/36

Potential and Point Charges,

cont.

The electric potential is independent of the

path between pointsAand B

It is customary to choose a reference

potential of V= 0 at rA=

Then the potential at some point ris

e

qV k r


19/36

Electric Potential of a Point

Charge

The electric potential in

the plane around a

single point charge is

shown The red line shows the

1/rnature of the

potential


20/36

Electric Potential with Multiple

Charges

The electric potential due to several pointcharges is the sum of the potentials due toeach individual charge

This is another example of the superpositionprinciple

The sum is the algebraic sum

V= 0 at r=

i

e i i

qV k

r


21/36

Immediate application:

Electric Potential of a Dipole

The graph shows thepotential (y-axis) of anelectric dipole

The steep slope betweenthe charges represents thestrong electric field in thisregion

)11

(

r-rr-r

qkVVV e


22/36

Potential Energy of Multiple

Charges

Consider two charged

particles

The potential energy of

the system is

1 2

12

e

q qU k

r

2509


23/36

More About Uof Multiple

Charges

If the two charges are the same sign, Uis

positive and external work (not the one from

the field force) must be done to bring the

charges together

If the two charges have opposite signs, Uis

negative and external work is done to keep

the charges apart


24/36

Uwith Multiple Charges, take 3 as an

example

If there are more thantwo charges, then findUfor each pair ofcharges and add them

For three charges:

The result is independentof the order of thecharges

1 3 2 31 2

12 13 23

e

q q q qq qU k

r r r


25/36

Find Vfor an Infinite Sheet of

Charge

We know that , a constant

From

We have

The equipotential lines are thedashed blue lines

The electric field lines are the

brown lines

The equipotential lines are

everywhere perpendicular to thefield lines

02

E

sE

dV

EdV d


26/36

Eand Vfor a Point Charge

The equipotential lines

are the dashed blue

lines

The electric field linesare the brown lines


are everywhere

perpendicular to thefield lines


27/36

Eand Vfor a Dipole


are the dashed blue

lines

The electric field linesare the brown lines


are everywhere

perpendicular to thefield lines


28/36

When you use a computer (program) to calculate

electric Potential for a Continuous Charge

Distribution:

Consider a small

charge element dq

Treat it as a point charge

The potential at somepoint due to this charge

element is

e

dqdV k

r


29/36

Vfor a Continuous Charge

Distribution, cont.

To find the total potential, you need to

integrate to include the contributions from all

the elements

This value for Vuses the reference of V= 0 when

Pis infinitely far away from the chargedistributions

e

dqV k

r


30/36

Vfor a Uniformly Charged

Ring

Pis located on the

perpendicular central

axis of the uniformly

charged ring The ring has a radius a

and a total charge Q

2 2

e

e

k Qdq

V k r a x


31/36

Vfor a Uniformly Charged Disk

The ring has a radius R

and surface charge

density of

P is along theperpendicular central

axis of the disk

1

2 2 22 eV k R x x


32/36

Vfor a Finite Line of Charge

A rod of line has a

total charge of Qand a

linear charge density of

2 2

lnek Q aV

a


33/36

Prove thatVis everywhere the same on a

charged conductor in equilibrium

Inside the conductor, becauseis 0, , soV=0

On the surface, consider twopoints on the surface of the

charged conductor as shown is always perpendicular to

the displacement

Therefore,

Therefore, the potential

difference betweenAand Bisalso zero

E

ds

0d E s

E 0d E s


34/36

Summarize on potential Vof a

charged conductor in equilibrium

Vis constant everywhere on the surface of acharged conductor in equilibrium V= 0 between any two points on the surface

The surface of any charged conductor inelectrostatic equilibrium is an equipotential surface

Because the electric field is zero inside theconductor, we conclude that the electric potential isconstant everywhere inside the conductor and equal

to the value at the surface


35/36

ECompared 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


36/36

Van de Graaff

Generator

Charge is delivered continuously to ahigh-potential electrode by means of amoving belt of insulating material

The high-voltage electrode is a hollow

metal dome mounted on an insulatedcolumn

Large potentials can be developed byrepeated trips of the belt

Protons accelerated through such large

potentials receive enough energy toinitiate nuclear reactions

Documents

Chapter25 Potential