Physics 1202: Lecture 5Today’s Agenda

• Announcements:– Lectures posted on:

www.phys.uconn.edu/~rcote/

– HW assignments, solutions etc.

• Homework #2:Homework #2:– On Masterphysics today: due Friday next weekOn Masterphysics today: due Friday next week

– Go to masteringphysics.com

• Labs: Begin next week

Today’s Topic :• End of Chapter 16: Electric potential

– Equipotentials and Conductors

– Electric Potential Energy

» of Charge in External Electric Field

• Capacitors:Electrostatic energy– Definition and concept

– Capacitors in parallel and in series

– Energy stored

– Dielectrics

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 of a charge q and a test charge q0

Electric Potential Energy• The Coulomb force is a CONSERVATIVE force (i.e. the work

done by it on a particle which moves around a closed path returning to its initial position is ZERO.)

• Therefore, a particle moving under the influence of the Coulomb force is said to have an electric potential energy defined by:

• The total energy (kinetic + electric potential) is then conserved for a charged particle moving under the influence of the Coulomb force.

this “q” is the ‘test charge” in other examples...

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

inverting our previous relation between E and V:

• We found

++

++

++

++

++

++

++

++

++

++

++

++

++

- -

- -

- -

- -

- -

- -

- -

- -

- -

- -

- -

- -

- -

+

F

• In general true for all direction

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.

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

+ +

+ +

+ +

+ + +

+ + +

+ +

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.

Definitions & Examples

d

A

- - - - -

+ + + +a

b L

a

b

a b

Capacitance• A capacitor is a device whose purpose is to store electrical

energy which can then be released in a controlled manner during a short period of time.

• A capacitor consists of 2 spatially separated conductors which can be charged to +Q and -Q respectively.

• The capacitance is defined as the ratio of the charge on one conductor of the capacitor to the potential difference between the conductors.

• Is this a "good" definition? Does the capacitance belong only to the capacitor, independent of the charge and voltage ?

+ -

• Calculate the capacitance. We assume +, - charge densities on each plate with potential difference V:

Example:Parallel Plate Capacitor

d

A

- - - - -

+ + + +

• Need Q:

• Need V: recall

where x = dor

so

Recall:Two infinite planes• Same charge but opposite

• Fields of both planes cancel out outside

• They add up inside

++++++++++++++++++++++++++

- - - - - - - - - - - - - - - - - - - - - - - - - -

Perfect to store energy !

Example:Parallel Plate Capacitor

d

A

- - - - -

+ + + +

• Calculate the capacitance:

Assume +Q,-Q on plates with potential difference V.

• As hoped for, the capacitance of this capacitor depends only on its geometry (A,d).

Dimensions of capacitance

• C = Q/V => [C] = F(arad) = C/V = [Q/V]

• Example: Two plates, A = 10cm x 10cm d = 1cm apart

=> C = A0/d = = 0.01m2/0.01m * 8.852e-12 C2/Jm = 8.852e-12 F

Lecture 5 – ACT 1

d

A

- - - - -

+ + + +• Suppose the capacitor shown here is charged to Q and then the battery disconnected.

• Now suppose I pull the plates further apart so that the final separation is d1. d1 > d d1

A

- - - - -

+ + + +

If the initial capacitance is C0 and the final capacitance is C1, is …

A) C1 > C0 B) C1 = C0 C) C1 < C0

• Can we define the capacitance of a single isolated sphere ?

• The sphere has the ability to store a certain amount of charge at a given voltage (versus V=0 at infinity)

Example :Isolated Sphere

• Need V: V = 0

VR = keQ/R

• So, C = R/ke

+

+

+

+

Capacitors in Parallel

• Find “equivalent” capacitance C in the sense that no measurement at a,b could distinguish the above two situations.

V

a

b

Q2Q1 V

a

b

Q

Parallel Combination:

Equivalent Capacitor:

Total charge: Q = Q1 + Q2

C = C1 + C2

Capacitors in Series

• Find “equivalent” capacitance C in the sense that no measurement at a,b could distinguish the above two situations.

• The charge on C1 must be the same as the charge on C2 since applying a potential difference across ab cannot produce a net charge on the inner plates of C1 and C2 .

a b+Q -Q

a b+Q -Q

RHS:

LHS:

Examples:Combinations of Capacitors

a

b

a b

• How do we start??

• Recognize C3 is in series with the parallel combination on C1 and C2. i.e.

Energy of a Capacitor• How much energy is stored in a charged capacitor?

– Calculate the work provided (usually by a battery) to charge a capacitor to +/- Q:

Calculate incremental work W needed to add charge q to capacitor at voltage V: - +

• But W is also the change in potential energy U

Qq

q

Vq

Vq=q/CV

• The total U to charge to Q is shaded triangle:

• In terms of the voltage V:

Lecture 5 – ACT 2

d

A

- - - - -

+ + + +

d1

A

- - - - -

+ + + +

The same capacitor as last time.

The capacitor is charged to Q and then the battery disconnected.

Then I pull the plates further apart so that the final separation is d1. d1 > d

If the initial energy is U0 and the final

capacitance is U1, is …

A) U1 > U0 B) U1 = U0 C) U1 < U0

Summary

d

A

- - - - -

+ + + +• Suppose the capacitor shown here is charged to Q and then the battery disconnected.

• Now suppose I pull the plates further apart so that the final separation is d1.

• How do the quantities Q, W, C, V, E change?

• How much do these quantities change?.. exercise for student!!

• Q:• W:• C:• V:• E:

remains the same.. no way for charge to leave.

increases.. add energy to system by separating

decreases.. since energy , but Q remains sameincreases.. since C , but Q remains sameremains the same.. depends only on chg density

answers:

Where is the Energy Stored?• Claim: energy is stored in the Electric field itself. Think of the

energy needed to charge the capacitor as being the energy needed to create the field.

• The Electric field is given by:

• The energy density u in the field is given by:

• To calculate the energy density in the field, first consider the constant field generated by a parallel plate capacitor:

Units: J/m3

Dielectrics• Empirical observation:

Inserting a non-conducting material between the plates of a capacitor changes the VALUE of the capacitance.

• Definition:

The dielectric constant of a material is the ratio of the capacitance when filled with the dielectric to that without it. i.e.

– values are always > 1 (e.g., glass = 5.6; water = 78)

– They INCREASE the capacitance of a capacitor (generally good, since it is hard to make “big” capacitors

– They permit more energy to be stored on a given capacitor than otherwise with vacuum (i.e., air)

Parallel Plate Example +++++++++++++

- - - - - - - - - - - - -

• Charge a parallel plate capacitor filled with vacuum (air) to potential difference V0.

• An amount of charge Q = C V0 is deposited on each plate.

+++++++++++++

- - - - - - - - - - - - -

• Now insert material with dielectric constant .

– Charge Q remains constant

+

- +

-

+

- +

-

+

-

+

-+

-

– So…, C = C0

Voltage decreases from V0 to

Electric field decreases also: