2.5 Conductors

2.5.1 Basic Properties of Conductors

2.5.2 Induced Charges

2.5.3 The Surface Charge on a Conductor; the Force on a Surface Charge

2.5.4 Capacitors

2.5.1 Basic Properties of Conductors

eare free to move in a conductor

E

otherwise, the free charges that produce will move to make 0E

inside a conductor

( ) ( ) 0

( ) ( )

b

aV b V a E dl

V b V a

0E

0 0E

0E

inside a conductor(1)

(2) ρ = 0 inside a conductor

(3) Any net charge resides on the surface

(4) V is constant, throughout a conductor.

(5) is perpendicular to the surface, just outside a conductor. Otherwise, will move the free charge to make

in terms of energy, free charges staying on the surface have a minimum energy.

uniform

0inE 0inW

2

0

1

8

qEnergy

R

uniform

0inE 0Win

2

0

3

20

qEnergy

R

2.5.1 (2)

E

E

0E

0

0in

enc induced

induecd

E

Q q q

q q

Example : A point charge q at the center of a spherical conducting shell. How much induced charge will accumulate there?

charge conservation2 24 4b ab a

21

4bq

b

2

0

4

in

a

E

a q

21

4aq

a

q induced

Solution :

2.5.1 (3)

2.5.2 Induced Charge

Example 2.9 Within the cavity is a charge +q. What is the field outside the sphere?

-q distributes to shield q and to make 0inE i.e., surfaceV constant

from charge conservation and symmetry.+q uniformly distributes at the surface

20

1ˆ( )

4

qE P r

r

a,b are arbitrary chosen

0

0b

a

E dl

E dl

0incavityE

a Faraday cage can shield out strayE

2.5.3 The Surface Charge on a Conductor

0ˆabove belowE E n

0 or 0in belowE E

0ˆaboveV E n

0

V

n

if we know V, we can get .s

or

why the average?

0

ˆ2above otherE E k

0

ˆ2below otherE E k

1( )

2other above below averageE E E E 0

electrostatic pressure2

20

02 2P E

In case of a conductor2

0

1ˆ

2 2f E nabove

2.5.3

1( )

2average above belowf E E E

Force on a surface charge,

2.5.4 Capacitors

Consider 2 conductors (Fig 2.53)

The potential difference

V V V E dl

20

ˆ1

4

rE d

r

Define the ratio between Q and V to be capacitanceQ

CV

a geometrical quantity

in mks 1 farad(F)= 1 Coulomb / volt

inconveniently large ; 6

12

10 :

10 :

F microfarad

F picofarad

Q E V ;;;;;;;;;;;;;;

double double double double

(V is constant.)

Example 2.10Find the capacitance of a “parallel-plate capacitor”?

0 0

QE

A

0

QV E d d

A

0AC

d

2.5.4 (2)

+Q

-QAd

Solution:

Example 2.11Find capacitance of two concentric spherical shellswith radii a and b .

20

1ˆ

4

QE r

r

20 0

1 1 1

4 4

a a

b b

Q QV E dl dr

a br

04Q ab

CV b a

-Q+Q

2.5.4 (3)

Solution:

The work to charge up a capacitor

( )q

dW Vdq dqC

22

0

1 1 1( )

2 2 2

Q QqW dq QV CV

C C

2.5.4 (4)

Today’s agenda:

Energy Storage in Capacitors.You must be able to calculate the energy stored in a capacitor, and apply the energy storage equations to situations where capacitor configurations are altered.

Dielectrics.You must understand why dielectrics are used, and be able include dielectric constants in capacitor calculations.

Energy Storage in Capacitors,Dielectrics and Capacitors With Dielectrics

Energy Storage in Capacitors

Let’s calculate how much work it takes to charge a capacitor.The work required for an external force to move a charge dq through a potential difference V is dW = dq V.

V+ -

+q -q

+dq

From Q=CV ( V = q/C):

qdW dq

C

q is the amount of charge on the capacitor during the time the charge dq is being moved.

We start with zero charge on the capacitor, and end up with Q, so

Q2 2Q Q

0 00

q q QW dW dq .

C 2C 2C

The work required to charge the capacitor is the amount of energy you get back when you discharge the capacitor (because the electric force is conservative).

Thus, the work required to charge the capacitor is equal to the potential energy stored in the capacitor.

2QU .

2C

Because C, Q, and V are related through Q=CV, there are three equivalent ways to write the potential energy.

2 2Q CV QVU .

2C 2 2

2 2Q CV QVU .

2C 2 2

All three equations are valid; use the one most convenient for the problem at hand.

It is no accident that we use the symbol U for the energy stored in a capacitor. It is just another “version” of electrical potential energy. You can use it in your energy conservation equations just like any other form of potential energy!

Example: a camera flash unit stores energy in a 150 F capacitor at 200 V. How much electric energy can be stored?

2CVU

2

6 2150 10 200U

2

U 3 J

If you keep everything in SI (mks) units, the result is “automatically” in SI units.

Energy Stored in Electric Fields

V+ -

+Q -Q

E

21U C V

2

d

area A

20A1U Ed

2 d

20

1U Ad E

2

Energy is stored in the capacitor:

The “volume of the capacitor” is Volume=Ad

Energy stored per unit volume (u):

V+ -

+Q -Q

E

d

area A

20

20

1Ad E 12u EAd 2

The energy is “stored” in the electricfield!

We’ve gone from the concrete (electric charges experience forces)……to the abstract (electric charges create electric fields)…

…to an application of the abstraction (electric field contains energy).

20

1u E

2 V

+ -

+Q -Q

E

f

area A

This is not a new “kind” of energy. It’s the electric potential energy resulting from the coulomb force between charged particles.

Or you can think of it as the electric energy due to the field created by the charges. Same thing.

“The energy in electromagnetic phenomena is the same as mechanical energy. The only question is, ‘Where does it reside?’ In the old theories, it resides in electrified bodies. In our theory, it resides in the electromagnetic field, in the space surrounding the electrified bodies.”—James Maxwell

Today’s agenda:

Energy Storage in Capacitors.You must be able to calculate the energy stored in a capacitor, and apply the energy storage equations to situations where capacitor configurations are altered.

Dielectrics.You must understand why dielectrics are used, and be able include dielectric constants in capacitor calculations.

If an insulating sheet (“dielectric”) is placed between the plates of a capacitor, the capacitance increases by a factor , which depends on the material in the sheet. is the dielectric constant of the material.

dielectric

In general, C = 0A / d. is 1 for a vacuum, and 1 for air. (You can also define = 0 and write C = A / d).

Dielectrics

AC = .

d

The dielectric is the thin insulating sheet in between the plates of a capacitor.

dielectric

Any reasons to use a dielectric in a capacitor?

Lets you apply higher voltages (so more charge).

Lets you place the plates closer together (make d smaller).

Increases the value of C because >1.

Q=CV

AC =

d

Makes your life as a physics student more complicated.

Gives you a bigger kick when you discharge the capacitor through your tongue!

Gives you a bigger kick when you discharge the capacitor through your tongue!

Gives you a bigger kick when you discharge the capacitor through your tongue!

2.3 Electric potential2.3.1 Introduction to Potential

2.3.2 Comments on Potential

2.3.3 Poisson’s Equation and Laplace’s Equation

2.3.4 The Potential of a Localized Charge Distribution

2.3.5 Electrostatic Boundary Conditions

2.3.1 introduction to potentialAny vector whose curl is zero is equal to the gradient of some

scalar. We define a function:

Where is some standard reference point ; V depends only on the point P. V is called the electric potential.

The fundamental theorem for gradients

( ) ( ) ( )b

aV b V a V dl

( )b b

a aV dl E dl

so

( )

pV p E dl

( ) ( )b a b

aV b V a E dl E dl E dl

E V

2.3.2 Comments on potential

(1)The name Potential is not potential energy

F qE q V

U F X

V : Joule/coulomb U : Joule

(2)Advantage of the potential formulation V is a scalar function, but E is a vector quantity

( )V r ˆ ˆ ˆx y zE E x E y E z

If you know V, you can easily get E: .E V

0E

so

2.3.2 (2)

, ,x y zE E Eare not independent functions

, ,y yx xz zE EE EE E

y x y z z x

2

y z x z y x z x y x z y x y z y x z z x x zE E E E E E E E

1

3

2

1

3

Adding a constant to V will not affect the potential difference between two point:

(3)The reference point Changing the reference point amounts to adds a constant to

the potential

( ) ( )p p

V p E dl E dl E dl K V p

(Where K is a constant)

2.3.2 (3)

( ) ( ) ( ) ( )V b V a V b V a

Since the derivative of a constant is zero:For the different V, the field E remains the same.

Ordinarily we set

V V

( ) 0V

2.3.2 (4)

(4)Potential obeys the superposition principle

1 2F F F F QE

1 2E E E

Dividing through by Q

,

Integrating from the common reference point to p ,

1 2V V V (5)Unit of potential

Volt=Joule/Coulomb

:

: :qV

XF X

Vq

F newton F x Joule

F qE q V

Joule/Coulomb

solution:

0 0

1 14 4

rr q q

V r E dr r

for r>R:

for r<R:

0inE

201

4ˆq

out rE r

0

14

qV r V R

R 0

14

qV R

R

0inE

2.3.2 (5)

Example 2.6 Find the potential inside and outside a spherical shell of radius R, which carries a uniform surface

charge (the total charge is q).

_

2.3.3 Poisson’s Eq. & Laplace’s Eq.

0

2E V V

E V

Laplace’s eq.0

Poisson’s Eq.0

2V

2 0V

pR r r

i i pR r r 0 1

1( )4

ni

ii

qV P

R

2.3.4 The Potential of a Localized Charge Distribution

• Potential for a point charge

• Potential for a collection of charge

E V r

V V Edr 0V

20 0 0

1 1 1( )4 4 4

rr q q q

V r drr rr

R

Ri

0

1( )4

qV P

R

• Potential of a continuous distribution

for volume charge for a line charge for a surface charge

0

1( )4

V P dR

0

1( )4

V P dR

0

1( )4

V P daR

q d q d q da

20

1 ˆ( )4

rE P dR

20

ˆ14

( ) r dR

E P

20

ˆ14

( ) r daR

E P

• Corresponding electric field 2

ˆ1 RR R

[ ]

2.3.4 (2)

2.3.4 (3)

Example 2.7 Find the potential of a uniformly charged spherical shell of radius R.

0

1( ) ,

4V r da

r

r2 2 2 2 cosr R z Rz

2

2 2

2

2 20

2 2 2

0

2 2 2 2

2 2

sin4 ( )

2 cos

sin2

2 cos

12 2 cos

22 2

2( ) ( )

( )

( )

[ ]

R d dV z

R z Rz

R dR z Rz

R R z RzRz

RR z Rz R z Rz

zR

R z R zz

Solution:

2

0 0

0 0

( ) [( ) ( )] ,2

( ) [( ) ( )] ,2

R RV z R z z R outsidez z

R RV z R z R z insidez

2.3.4 (4)

2.3.5 Electrostatic Boundary Condition

Electrostatic problem

The above equations are differential or integral. For a unique solution, we need boundary conditions.

(e.q. , V()=0 )(boundary value problem. Dynamics: initial value

problem.)

• Superposition

• Coulomb law