#### Transcript of F16 PH 333 2.2.1-2.2.4

Divergence and Curl of Electrostatic Fields

2.2.1-2.2.4

2.2.1 Field lines, Flux, Gauss's Law

2.2.2 The divergence of E

2.2.3 Using Gauss's Law

2.2.4 The Curl of E

**2.2.1**

Some integrals are hard. Here we start simplifications. ( but these have issues of only working in nice symmetries)

Field lines:

Never cross

start at + end at - (infinity is ok for either too)

Density matters

Flux is:

Gauss's Law

Using the Divergence theorem, how can we express the LHS of Gauss's law?

D

C

B

A

By definition how can we find the Q enclosed inside the same volume?

D

C

B

A

Together:

or in differential form

Back to divergence of E 2.2.2

Recall

Take the divergence of both sides.

with

Gives

Gauss's law

differential form

2.2.3 Using Gauss's Law

Always true. But not always easy (useful)

E is constant on surface

E is perpendicular to surface

E is parallel to surface

Nice symmetries exist

All the above

Which of these conditions must be met to make it easy (useful)

A

B

C

D

E

**Symmetries**

Spherical Symmetry

Cylindrical Symmetry

Planar Symmetry

Choose Gaussian surface to match the symmetry

L

R

A Gaussian cylinder sits in a constant E field as shown, what is the total E Flux through the Gaussian surface?

D

C

B

A

**Practice**

**Practice**

Curl of E

Choose a point charge.

by inspection

more rigorously, take any path by this point charge.

If the path closed, then = zero

ANY E FIELD

Q

2Q

5

4

3

2

1

d

c

b

a

Two large conductors have charges Q and 2Q and top surfaces A. What is the E field in each region 1-5 and surface charge density a-d

"Could you explain the Gaussian pillbox a little better? I am having a hard time picturing it. "

"Can we go through the derivation of the Divergence of E? I want to see and better understand how the separation vector applies in the integral."

"Could we go over how to to draw the electric field lines when there are multiple charges?"

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