my office change was not reflected on the syllabus. it is now escn 2.206

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