Box versus open surface

Seem to be able to tellif there are charges inside

…no clue…

Gauss’s law: If we know the field distribution on closed surface we can tell what is inside.

Patterns of Fields in Space

Define Electric Flux Gauss’s Law Apply it

Symmetry

Need a way to quantify pattern of electric field on surface: electric flux

1. Direction

flux>0 : electric field comes outflux<0 : electric field goes in

+1 -10Relate flux to the angle between outward-going normal and E:

flux ~ cos()

Electric Flux: Direction of E

2. Magnitude

flux ~ E

flux ~ Ecos()

Electric Flux: Magnitude of E

𝑓𝑙𝑢𝑥 𝐸 ∙ ��

3. Surface area

flux through small area:

AnEflux ˆ~

Definition of electric flux on a surface:

surface

AnE ˆ

Electric Flux: Surface Area

Perpendicular field

cosˆ AEAnE

AEAnE ˆ

Perpendicular area

coscosˆ yxEAEAnE

x y

AEAnE ˆ

Electric Flux: Perpendicular Field or Area

q

surface

AnE ˆ

dAnE ˆ

Ad

AdE

AdE

surface closed a on flux electric

Adding up the Flux

0

ˆ

inside

surface

qAnE

0

ˆ

insideqdAnE

Features:1. Proportionality constant2. Size and shape independence3. Dependence on sum of charges inside4. Charges outside contribute zero

Gauss’s Law

0

ˆ

inside

surface

qAnE

204

1

r

QE

surface

Anrr

Qˆˆ

4

12

0

surface

Ar

Q2

04

1

0

22

0

44

1

Q

rr

Q

What if charge is negative?

Works at least for one charge and spherical surface

1. Gauss’s Law: Proportionality Constant

0

ˆ

inside

surface

qAnE

204

1

r

QE

2

1~

rE

2~ rA

2

1~

rE universe would be

much different ifexponent was not exactly 2!

2. Gauss’s Law: The Size of the Surface

0

ˆ

inside

surface

qAnE

E nA

surface EA

surface

The flux through the inner sphere is the same as the flux through the outer.

3. Gauss’s Law: The Shape of the Surface

A2 / A1 r22 / r1

2

E2A2 / E1A1 1

A2 R2 (r2 tan)2 r22

∆ 𝐴1⊥∝𝑟12

0

ˆ

inside

surface

qAnE

surfacesurface

AEAnE ˆ

2~ rA

2

1~

rE 2211 EAEA –

Outside charges contribute 0 to total flux

4. Gauss’s Law: Outside Charges

0

11 ˆ

Q

AnEsurface

0

22 ˆ

Q

AnEsurface

0ˆ3 surface

AnE

0

ˆ

inside

surface

qAnE

5. Gauss’s Law: Superposition

0

ˆ

inside

surface

qAnE

0

ˆ

insideqdAnE

Features:1. Proportionality constant2. Size and shape independence3. Independence on number of charges inside4. Charges outside contribute zero

Gauss’s Law and Coulomb’s Law?

204

1

r

QE

Can derive one from another

Gauss’s law is more universal:works at relativistic speeds

Gauss’s Law

0

ˆ

inside

surface

qAnE

1. Knowing E can conclude what is inside2. Knowing charges inside can conclude what is E

Applications of Gauss’s Law

Symmetry: Field must be perpendicular to surfaceEleft=Eright

0

ˆ

inside

surface

qAnE

2EAbox Q / A Abox

0

E Q / A 20

The Electric Field of a Large Plate

Symmetry: 1. Field should be radial2. The same at every location

on spherical surface

0

ˆ

inside

surface

qAnE

A. Outer Dashed Sphere:

0

24

QrE 2

04

1

r

QE

B. Inner Dashed Sphere:

0

2 04

rE 0E

The Electric Field of a Uniform Spherical Shell of Charge

Finally!

𝑟𝑟

0

ˆ

inside

surface

qAnE

Is Gauss’s law still valid?

Can we find E using Gauss’s law?

The Electric Field of a Uniform Cube

Without symmetry, Gauss’s law loses much of its power.

Yes, it’s always valid.

Gauss’s Law for Electric Dipole

No symmetry

Direction and Magnitude of E varies

NumericalSolution

Clicker Question

What is the net electric flux through the box?

A) 0 VmB) 0.36 VmC) 0.84 VmD) 8.04 VmE) 8.52 Vm

Can we have excess charge inside a metal that is in static equilibrium?

Proof by contradiction:

0

ˆ

inside

surface

qAnE

=0

00

insideq

Gauss’s Law: Properties of Metal

0

ˆ

inside

surface

qAnE

=0

00

insideq

Gauss’s Law: Hole in a Metal

+5nC

0

ˆ

inside

surface

qAnE

=0

00

insideq

0 insidesurface qq

nC 5 surfaceq

Gauss’s Law: Charges Inside a Hole

Next Class is a Review