My office change was not reflected on the syllabus. It is now ESCN 2.206.

Our first exam is a week from next Tuesday - Sep 27. It will cover everything I have covered in class including material covered next Thursday.

There will be two review sessions Monday, Sep 26 - at 12:30 PM and at 3:00 PM in the same room as the problem solving session: FN 2.212.

I have put several (37) review questions/problems on Mastering Physics. These are not for credit but for practice. I will review them at the review session Monday.

Example: Potential between oppositely charged parallel plates

From our previous examples

0( )

( )

U y q Ey

V y Ey

abVE

d

Easy way to calculate surface charge density

0 abV

d

Remember! Zero potential doesn’t mean the conducting object has no charge! We can assign zero potential to any place, only difference in potential makes physical sense

Example: Charged wireWe already know E-field around the wireonly has a radial component

0

1;

2rEr

0

ln2

bb

aa

rE dr

r

Vb = 0 – not a good choice as it follows

Why so?

aV

We would want to set Vb = 0 at some distance r0 from the wire

0

0

ln2

rV

r

r - some distance from the wire

Example: Sphere, uniformly charged inside through volume

3' r

q QR Q - total charge

Q

V - volume density of charge

( )r R r

R rR

E dr

eR

k Q

R

2

23

2e

rk Q r

R R

This is given that at infinity

rE03

R

R

2

3|

2re

R Rk Q r

R

Potential Gradientb

a ba

E d l

We can calculate potential difference directlya

a bb

d

x y zE i E j E k E x y zd E dx E dy E dz

: :x y zE E Ex y z

Components of E in terms of

E operator "del"

Frequently, potentials (scalars!) are easier to calculate:

So people would calculate potential and then the field

Superposition for potentials: V = V1 + V2 + …

Example: A positively charged (+q) metal sphere of radius ra is inside of another metal sphere (-q) of radius rb. Find potential at different pointsinside and outside of the sphere.

) : ) : )a a b ba r r b r r r c r r

+q

-q1

2

a) 2

0

10

( )4

( )4

b

a

qV r

r

qV r

r

Total V=V1+V2

0

1 1( )

4 a b

qV r

r r

b)

0

1 1( )

4 b

qV r

r r

c) 0V

Electric field between spheres Er

Equipotential Surfaces

• Equipotential surface—A surface consisting of a continuous distribution of points having the same electric potential

• Equipotential surfaces and the E field lines are always perpendicular to each other

• No work is done moving charges along an equipotential surface

– For a uniform E field the equipotential surfaces are planes

– For a point charge the equipotential surfaces are spheres

Equipotential Surfaces

Potentials at different points are visualized by equipotential surfaces (just like E-field lines).

Just like topographic lines (lines of equal elevations).

E-field lines and equipotential surfaces are mutually perpendicular

Definitions cont

• Electric circuit—a path through which charge can flow

• Battery—device maintaining a potential difference V between its terminals by means of an internal electrochemical reaction.

• Terminals—points at which charge can enter or leave a battery

Definitions

• Voltage—potential difference between two points in space (or a circuit)

• Capacitor—device to store energy as potential energy in an E field

• Capacitance—the charge on the plates of a capacitor divided by the potential difference of the plates C = q/V

• Farad—unit of capacitance, 1F = 1 C/V. This is a very large unit of capacitance, in practice we use F (10-6) or pF (10-12)

Capacitors

• A capacitor consists of two conductors called plates which get equal but opposite charges on them

• The capacitance of a capacitor C = q/V is a constant of proportionality between q and V and is totally independent of q and V

• The capacitance just depends on the geometry of the capacitor, not q and V

• To charge a capacitor, it is placed in an electric circuit with a source of potential difference or a battery

CAPACITANCE AND CAPACITORS

Capacitor: two conductors separated by insulator and charged by opposite and equal charges (one of the conductors can be at infinity)

Used to store charge and electrostatic energy

Superposition / Linearity: Fields, potentials and potential differences, or voltages (V), are proportional to charge

magnitudes (Q)

(all taken positive, V-voltage between plates)

Capacitance C (1 Farad = 1 Coulomb / 1 Volt) is determined by pure geometry (and insulator properties)

1 Farad IS very BIG: Earth’s C < 1 mF

QC

V

Calculating Capacitance

1. Put a charge q on the plates

2. Find E by Gauss’s law, use a surface such that

3. Find V by (use a line such that V = Es)

4. Find C by

0encq

EAAdE

EssdEV

Vq

C

Energy stored in a capacitor is related to the E-field between the plates Electric energy can be regarded as stored in the field itself.

This further suggests that E-field is the separate entity that may exist alongside charges.

Parallel plate capacitor

€

density σ = charge Q /area S

E =σ

ε0

=Q

ε0A; V = Ed =

Qd

ε0A

C =ε0A

d

Generally, we find the potential differenceVab between conductors for a certain charge Q

Point charge potential difference ~ Q

This is generally true for all capacitances

Capacitance configurations

€

V = keQdr

r2a

b

∫ = keQ(1

ra

−1

rb

)

C =rarb

ke (rb − ra )

With rb →∞, C = ra /ke -

capacitance of an individual sphere

Cylindrical capacitor

)ln(2

)ln(22

ab

k

lC

a

b

l

Qk

r

drkV

e

e

b

a

e

Spherical Capacitance