Physics 1202: Lecture 4Today’s Agenda

• Announcements:– Lectures posted on:

www.phys.uconn.edu/~rcote/

– HW assignments, solutions etc.

• Homework #1:Homework #1:– On Masterphysics: due this coming FridayOn Masterphysics: due this coming Friday

– Go to masteringphysics.com and register– Course ID: MPCOTE62465

• Labs: Begin next week

Today’s Topic :

• Chapter 16: Electric energy & potential– Definition– How to compute them– Point charges– Capacitors

Electric potential EnergyRecall 1201

•Total mechanical energy– Constant for conservative forces

•Potential energy U– Depends only on position (ex: U = mgy)– Change in U is independent of path

kinetic potential

U2 , y2

U1 , y1

Electric potential Energy•Gravitational force (magnitude)

•Gravitational Potential energy U

•By analogy:

Electric force Electric potential energy

Electric potential Energy•Meaning: recall

•Total energy is conserved– Variation of U with r variation of kinetic energy

•For multiple charges– Simple sum – Ex: 3 charges

q1

q3

q2

r13

r12

r23

Lightning

Energy UnitsMKS: U = QV 1 coulomb-volt = 1 joule

for particles (e, p, ...) 1 eV = 1.6x10-19 joules

Accelerators• Electrostatic: VandeGraaff

electrons 100 keV ( 105 eV)

• Electromagnetic: Fermilab

protons 1TeV ( 1012 eV)

Electric Potential

qA

C

B

rA

Br

path independence equipotentials

R

R

R r

VQ

4 rQ

4 R

Electric Potential• By analogy with the electric field

• Defined using a test charge q0

• We define a potential V due to a charge q

– Using potential energy by charge q and a test charge q0

Electric Potential• Suppose charge q0 is moved from pt

A to pt B through a region of space described by electric field E.

• Since there will be a force on the charge due to E, a certain amount of work WAB will have to be done to accomplish this task. We define the electric potential difference as:

• The potential difference is meaningful

• Because only potential energy difference is meaningful

• U=V=0 can be chosen arbitrarily (like for gravity)

A B

q0E

Lecture 4, ACT 1• A single charge ( Q = -1C) is fixed at the origin.

Define point A at x = + 5m and point B at x = +2m.

– What is the sign of the potential difference between A and B? (VAB VB - VA )

(a) VAB < (b) VAB = (c) VAB >

x-1C AB

• The simplest way to get the sign of the potential difference is to imagine placing a positive charge at point A and determining whether positive or negative work would be done in moving the charge to point B.

• A positive charge at A would be attracted to the -1C charge; therefore NEGATIVE work would be done to move the charge from A to B.

E

Electric Potential• Define the electric potential of a point in space as the potential

difference between that point and a reference point.

• a good reference point is infinity ... we typically set V = 0

• the electric potential is then defined as:

• for a point charge, the formula is:

Potential from N charges

The potential from a collection of N charges is just the algebraic sum of the potential due to each charge separately.

xr1

r2 r3

q1

q3

q2

Electric Dipole z

a

a

+q

-q

r

r1r2

The potential is much easier to calculate than the field since it is an algebraic sum of 2 scalar terms.

• Rewrite this for special case r>>a:

Can we use this potential somehow to calculate the E field of a dipole?

(remember how messy the direct calculation was?)

r2-r1

E from V?• We can obtain the electric field E from the potential V by

inverting our previous relation between E and V:

• Consider 2 plates and a charge q

• force on q

• Work done on q

++

++

++

++

++

++

++

++

++

++

++

++

++

- -

- -

- -

- -

- -

- -

- -

- -

- -

- -

- -

- -

- -

+

F

• But work-energy theorem

• Conservative force

E from V?• We can obtain the electric field E from the potential V by

inverting our previous relation between E and V:

• We have

++

++

++

++

++

++

++

++

++

++

++

++

++

- -

- -

- -

- -

- -

- -

- -

- -

- -

- -

- -

- -

- -

+

F

• So that

• Therefore

Equipotentials

• GENERAL PROPERTY: – The Electric Field is always perpendicular to an

Equipotential Surface.

• Why??

Dipole Equipotentials

Defined as: The locus of points with the same potential.

• Example: for a point charge, the equipotentials are spheres centered on the charge.

The gradient ( ) says E is in the direction of max rate of change.

Along the surface, there is NO change in V (it’s an equipotential!)

So, there is NO E component along the surface either… E must therefore be normal to surface

• ClaimThe surface of a conductor is always an equipotential surface(in fact, the entire conductor is an equipotential)

• Why??

If surface were not equipotential, there would be an Electric Field component parallel to the surface and the charges would move!!

• NotePositive charges move from regions of higher potential to lower potential (move from high potential energy to lower PE).

Equilibrium means charges rearrange so potentials equal.

Conductors

+ +

+ +

+ +

+ + +

+ + +

+ +

Charge on Conductors?• How is charge distributed on the surface of a

conductor? – KEY: Must produce E=0 inside the conductor and E normal to

the surface .

Spherical example (with little off-center charge):

E=0 inside conducting shell.

- ---

- --

-

-

-

-

--

-

- charge density induced on inner surface non-uniform.

+

+

+

++

+

+

++

+ +

+ +

+

+

+

charge density induced on outer surface uniform

E outside has spherical symmetry centered on spherical conducting shell.

+q

A Point Charge Near Conducting Plane

+

a

q

- - -- -- - - -- - - - -- --- ---------------- --- -- ---- --------V=0

A Point Charge Near Conducting Plane

+

-

a

q

The magnitude of the force is

The test charge is attracted to a conducting plane

Image Charge

Equipotential Example• Field lines more closely

spaced near end with most curvature .

• Field lines to surface near the surface (since surface is equipotential).

• Equipotentials have similar shape as surface near the surface.

• Equipotentials will look more circular (spherical) at large r.