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Lesson 7Gausss Law and Electric Fields

Class 18

- Today, we will
- learn the definition of a Gaussian surface
- learn how to count the net number of field lines

passing into a Gaussian surface - learn Gausss Law of Electricity
- learn about volume, surface, and linear charge

density - learn Gausss Law of Magnetism
- show by Gausss law and symmetry that the

electric field inside a hollow sphere is zero

Section 1Visualizing Gausss Law

Gaussian Surface

- A Gaussian surface is
- any closed surface
- surface that encloses a volume
- Gaussian surfaces include
- balloons
- boxes
- tin cans
- Gaussian surfaces do not include
- sheets of paper
- loops

Counting Field Lines

- To count field lines passing through Gaussian

surfaces - Count 1 for every line that passes out of the

surface. - Count -1 for every line that comes into the

surface.

1

-1

Electric Field Lines

We have a 2 charge and a -2 charge.

Electric Field Lines

What is the net number of field lines passing

through the Gaussian surface?

Electric Field Lines

What is the net number of field lines passing

through the Gaussian surface?

8

Electric Field Lines

What is the net number of field lines passing

through the Gaussian surface?

Electric Field Lines

What is the net number of field lines passing

through the Gaussian surface?

8

Electric Field Lines

What is the net number of field lines passing

through the Gaussian surface?

Electric Field Lines

What is the net number of field lines passing

through the Gaussian surface?

-8

Electric Field Lines

What is the net number of field lines passing

through the Gaussian surface?

Electric Field Lines

What is the net number of field lines passing

through the Gaussian surface?

-8

Electric Field Lines

What is the net number of field lines passing

through the Gaussian surface?

Electric Field Lines

What is the net number of field lines passing

through the Gaussian surface?

0

Electric Field Lines

What is the net number of field lines passing

through the Gaussian surface?

Electric Field Lines

What is the net number of field lines passing

through the Gaussian surface?

0

Electric Field Lines

From the field lines coming out of this box, what

can you tell about whats inside?

Electric Field Lines

The net charge inside must be 1 (if we draw 4

lines per unit of charge).

Gausss Law of Electricity

The net number of electric field lines passing

through a Gaussian surface is proportional to the

charge enclosed within the Gaussian surface.

Section 2Charge Density

Charge Density

Charge Volume Charge Area Charge Length

Volume ? Surface s Linear ?

Charge Density

In general, charge density can vary with

position. In this case, we can more carefully

define density in terms of the charge in a very

small volume at each point in space. The density

then looks like a derivative

You need to understand what we mean by this

equation, but we wont usually need to think of

density as a derivative.

Section 3Gausss Law of Magnetism

Gausss Law and Magnetic Field Lines

If magnetic field lines came out from point

sources like electric field lines, then we would

have a law that said The net number of magnetic

field lines passing through a Gaussian surface is

proportional to the magnetic charge inside.

N

Gausss Law and Magnetic Field Lines

But we have never found a magnetic monopole. -

The thread model suggests that there is no reason

we should expect to find a magnetic monopole as

the magnetic field as we know it is only the

result of moving electrical charges. - The field

line model suggests that theres no reason we

shouldnt find a magnetic monopole as the

electric and magnetic fields are both equally

fundamental.

Gausss Law and Magnetic Field Lines

What characteristic would a magnetic monopole

field have?

Gausss Law and Magnetic Field Lines

What characteristic would a magnetic monopole

field have?

Gausss Law and Magnetic Field Lines

All known magnetic fields have field lines that

form closed loops. So what can we conclude

about the number of lines passing through a

Gaussian surface?

Gausss Law of Magnetism

The net number of magnetic field lines passing

through any Gaussian surface is zero.

Section 4Gausss Law and Spherical Symmetry

Spherically Symmetric Charge Distribution

The charge density, ?, can vary with r

only. Below, we assume that the charge density is

greatest near the center of a sphere.

Spherically Symmetric Charge Distribution

Outside the distribution, the field lines will go

radially outward and will be uniformly

distributed.

Spherically Symmetric Charge Distribution

The field is the same as if all the charge were

located at the center of the sphere!

Inside a Hollow Sphere

Now consider a hollow sphere of inside radius r

with a spherically symmetric charge distribution.

Inside a Hollow Sphere

There will be electric field lines outside the

sphere and within the charged region. The field

lines will point radially outward because of

symmetry. But what about inside?

Inside a Hollow Sphere

Draw a Gaussian surface inside the sphere. What

is the net number of electric field lines that

pass through the Gaussian surface?

Inside a Hollow Sphere

The total number of electric field lines from the

hollow sphere that pass through the Gaussian

surface inside the sphere is zero because there

is no charge inside.

How can we get zero net field lines?

1. We could have some lines come in and go out

again

but this violates symmetry!

How can we get zero net field lines?

2. We could have some radial lines come in and

other radial lines go out

but this violates symmetry, too!

How can we get zero net field lines?

3. Or we could just have no electric field at all

inside the hollow sphere.

How can we get zero net field lines?

3. Or we could just have no electric field at all

inside the hollow sphere.

This is the only way it can be done!

The Electric Field inside a Hollow Sphere

Conclusion the static electric field inside a

hollow charged sphere with a spherically

symmetric charge distribution must be zero.

Class 19

- Today, we will
- learn how to use Gausss law and symmetry to

find the electric field inside a spherical charge

distribution - show that all the static charge on a conductor

must reside on its outside surface - learn why cars are safe in lightning but cows

arent

Spherically Symmetric Charge Distribution

Electric field lines do not start or end outside

charge distributions, but that can start or end

inside charge distributions.

Spherically Symmetric Charge Distribution

What is the electric field inside a spherically

symmetric charge distribution?

Spherically Symmetric Charge Distribution

Inside the distribution, it is difficult to draw

field lines, as some field lines die out as we

move inward. We need to draw many, many field

lines to keep the distribution uniform as we move

inward.

Spherically Symmetric Charge Distribution

But we do know that if we drew enough lines, the

distribution would be radial and uniform in every

direction, even inside the sphere.

Spherically Symmetric Charge Distribution

Lets draw a spherical Gaussian surface at radius

r.

r

Spherically Symmetric Charge Distribution

Now we split the sphere into two parts the part

outside the Gaussian surface and the part inside

the Gaussian surface.

r

r

Spherically Symmetric Charge Distribution

The total electric field at r will be the sum of

the electric fields from the two parts of the

sphere.

r

r

Spherically Symmetric Charge Distribution

Since the electric field at r from the hollow

sphere is zero, the total electric field at r is

that of the core, the part of the sphere within

the Gaussian surface.

r

r

Spherically Symmetric Charge Distribution

Outside the core, the electric field is the same

as that of a point charge that has the same

charge as the total charge inside the Gaussian

surface.

r

Spherically Symmetric Charge Distribution

Inside a spherically symmetric charge

distribution, the static electric field is

r

Example Uniform Distribution

A uniformly charged sphere of radius R has a

total charge Q. What is the electric field at r lt

R ?

Example Uniform Distribution

A uniformly charged sphere of radius R has a

total charge Q. What is the electric field at r lt

R ?

Since the charge density is uniform

r

Example Uniform Distribution

Section 5Gausss Law and Conductors

Gausss Law and Conductors

Take an arbitrarily shaped conductor with charges

on the outside.

Gausss Law and Conductors

The static electric field inside the conductor

must be zero. Draw a Gaussian surface inside

the conductor.

Gausss Law and Conductors

No field lines go through the Gaussian surface

because E0. Hence, the total enclosed charge

must be zero.

Gausss Law and Conductors

The same must be true of all Gaussian surfaces

inside the conductor.

Surface Charge and Conductors

What if there are no charges on the outside and

the net charge of the conductor is zero? -- The

volume charge density inside the conductor must

be zero and the surface charge density on the

conductor must also be zero.

Surface Charge and Conductors

What if there are no charges on the outside and

there is net charge on the surface of a conductor?

Surface Charge and Conductors

The charge distributes itself so the field inside

is zero and the surface is at the same electric

potential everywhere.

Example Surface Charge on a Spherical Conductor

A spherical conductor of radius R has a voltage

V. What is the total charge? What is surface

charge density?

Example Surface Charge on a Spherical Conductor

A spherical conductor of radius R has a voltage

V. What is the total charge? What is surface

charge density?

On the outside, the potential is that of a point

charge.

On the surface, the voltage is V(R).

Take Two Conducting Sphereswith the Same Voltage

The smaller sphere has a larger charge density.

Now Connect the Two Spheres

The charge density is greater near the pointy

end. The electric field is also greater near the

pointy end.

Edges on Conductors

Charge moves to sharp points on

conductors. Electric field is large near sharp

points. Smooth, gently curved surfaces are the

best for holding static charge. Lightning rods

are pointed.

A Hollow Conductor

What if theres a hole in the conductor?

A Hollow Conductor

Draw a Gaussian surface around the hole.

A Hollow Conductor

There is no net charge inside the Gaussian

surface.

A Hollow Conductor

Is there surface charge on the surface of the

hole?

A Hollow Conductor

There is no field surrounding the charge to hold

the charges fixed, so the charges migrate and

cancel each other out.

Charge on a Conductor

Static charge moves to the outside surface of a

conductor.

Lightning and Cars

Why is a car a safe place to be when lightning

strikes?

Note Any car will do it doesnt need to be a

Cord.

Lightning and Cars

Is it the insulating tires?

Lightning and Cars

Is it the insulating tires?

If lightning can travel 1000 ft through the air

to get to your car, it can go another few inches

to go from your car to the ground!

Lightning and Cars

A car is essentially a hollow conductor. Charge

goes to the outside. The electric field inside is

small.

Lightning and Cars

A car is essentially a hollow conductor. Charge

goes to the outside. The electric field inside is

small.

How should a cow stand to avoid injury when

lightning strikes nearby?

Physicists Cow

Cow

Earth

d

I

Physicists Cow

When d is bigger, the resistance along the ground

between the cows feet is bigger, the voltage

across the cow is bigger, and the current flowing

through the cow is bigger.

Cow

Earth

d

I

How should a cow stand to avoid injury when

lightning strikes nearby?

So the cow should keep her feet close together!

Class 20

- Today, we will
- learn how integrate over linear, surface, and

volume charge densities to find the total charge

on an object - learn that flux is the mathematical quantity

that tells us how many field lines pass through a

surface

Section 6Integration

Gausss Law of Electricity

- The net number of electric field lines passing

through a Gaussian surface is proportional to the

enclosed charge. - But, how do we find the enclosed charge?

Charge and Density

- is valid when?

Charge and Density

- when ? is uniform.
- If ? is not uniform over the whole volume, we

find some small volume dV where it is uniform.

Then - If we add up all the little bits of dq, we get

the entire charge, q.

Integration

The best way to review integration is to work

through some practical integration problems.

Integration

The best way to review integration is to work

through some practical integration problems. Our

goal is to turn two- and three- dimensional

integrals into one-dimensional integrals.

Fundamental Rule of Integration

Identify the spatial variables on which the

integrand depends. You must slice the volume

(length or surface) into slices on which these

variables are constant.

Fundamental Rule of Integration

When integrating densities to find the total

charge, the density must be a constant on the

slice or we cannot write

Fundamental Rule of IntegrationExamples

Square in x-y plane Cylinder Sphere

Fundamental Rule of IntegrationExamples

Consider a very thin slice.

Is constant on this slice?

Fundamental Rule of IntegrationExamples

Consider a very thin slice.

Is constant on this slice?

Fundamental Rule of IntegrationExamples

Square in x-y plane Cylinder Sphere

Fundamental Rule of IntegrationExamples

Square in x-y plane Cylinder Sphere

Fundamental Rule of IntegrationExamples

Square in x-y plane Cylinder Sphere

Fundamental Rule of IntegrationExamples

Square in x-y plane Cylinder Sphere

Fundamental Rule of IntegrationExamples

Square in x-y plane Cylinder Sphere

Rules for Areas and Volumes of SlicesMemorize

These!!!

Square in x-y plane Disk Cylinder Sphere

Rules for Areas and Volumes of SlicesMemorize

These!!!

Square in x-y plane Disk Cylinder Sphere

Rules for Areas and Volumes of SlicesMemorize

These!!!

Square in x-y plane Disk Cylinder Sphere

Rules for Areas and Volumes of SlicesMemorize

These!!!

Square in x-y plane Disk Cylinder Sphere

Rules for Areas and Volumes of SlicesMemorize

These!!!

Square in x-y plane Disk Cylinder Sphere

Rules for Areas and Volumes of SlicesMemorize

These!!!

Square in x-y plane Disk Cylinder Sphere

Lets Do Some Integrals

Charge on a Cylinder

A cylinder of length L and radius R has a

charge density where is a

constant and z is the distance from one end of

the cylinder. Find the total charge on the

cylinder. How do you slice the cylinder? What is

the volume of each slice?

Charge on a Cylinder

Charge on a Sphere

A sphere of radius R has a charge density

where is a constant. Find the total

charge on the sphere. How do you slice the

sphere? What is the volume of each slice?

Charge on a Sphere

Section 7Gausss Law and Flux

Field Lines and Electric Field

- This is valid when
- .A is the area of a section of a perpendicular

surface. - The electric field is constant on A.

Field Lines and Electric Field

- This is valid when
- A is the area of a section of a perpendicular

surface. - The electric field is constant on A.

-- But E is a constant on A only in a few cases

of high symmetry spheres, cylinders, and planes.

Electric Flux

Gausss Law states that

EA is called the electric flux. We write it as

or just .

Electric Flux

Gausss Law states that

EA is called the electric flux. We write it as

or just . Flux is a mathematical

expression for number of field lines passing

through a surface!

Electric Flux and a Point Charge

Lets calculate the electric flux from a point

charge passing through a sphere of radius r.

Electric Flux and a Point Charge

Gausss law says this is proportional to the

charge enclosed in the sphere!

Electric Flux and Gausss Law

This means that we can write Gausss Law of

Electricity as

A Few Facts about Flux

For our purposes, we will (almost) always

calculate flux through a section of perpendicular

surface where the field is constant. So we will

evaluate flux simply as

A Few Facts about Flux

But we do need to find a more general expression

for flux so youll know what it really means

An Area Vector

- We wish to define a vector area. To do this
- we need a flat surface.
- the direction is perpendicular to the plane of

the area. - (Dont worry about the fact there are two

choices of direction that are both perpendicular

to the area up and down in the figure below.) - 3) the magnitude of vector is the area.

A Few Facts about Flux

First, Lets consider the flux passing through a

frame oriented perpendicular to the field.

A Few Facts about Flux

If we tip the frame by an angle ?, the angle

between the field and the normal to the frame,

there are fewer field lines passing through the

frame.

A Few Facts about Flux

Or, using the vector area of the loop, we may

write

A Few Facts about Flux

only holds when the frame is flat and

the field is uniform. What if the surface

(frame) isnt flat, or the electric field isnt

uniform?

Area Vectors on a Gaussian Surface

1) We must take a small region of the surface dA

that is essentially flat. 2) We choose a unit

vector perpendicular to the plane of dA going in

an outward direction.

A Few Facts about Flux

The flux through this small region is

A Few Facts about Flux

To find the total flux, we simply add up all the

contributions from every little piece of the

surface.

Recall that the normal to each small area is

taken to be in the outward direction.

A Few Facts about Flux

Thus, the most general equation for flux through

a surface is

If we take the flux through a Gaussian surface,

we usually write the integral sign with a circle

through it to emphasize the fact that the

integral is over a closed surface

Class 21

- Today, we will
- learn how to use Gausss law to find the

electric fields in cases of high symmetry - insdide and outside spheres
- inside and outside cylinders
- outside planes

Section 7Gausss Laws in Integral Form

Gausss Law of ElectricityIntegral Form

The number of electric field lines passing

through a Gaussian surface is proportional to the

charge enclosed by the surface.

We can make this simple expression look much

more impressive by replacing the flux and

enclosed charge with integrals

Gausss Law of MagnetismIntegral Form

The number of magnetic field lines passing

through a Gaussian surface is zero

With the integral for magnetic flux, this is

Gausss Law of ElectricityTee-Shirt Form

This can be written in many different ways. A

popular form seen on many tee-shirts is

Gausss Law of ElectricityTee-Shirt Form

This can be written in many different ways. A

popular form seen on many tee-shirts is

This is a good form of Gausss law to use if you

want to impress someone with how smart you are.

Gausss Law of ElectricityPractical Form

This is the form of Gausss law you will use when

you actually work problems.

Gausss Law of ElectricityPractical Form

Now lets think about what this equation really

means!

Gausss Law of ElectricityPractical Form

Electric field on Gaussian surface -- Must be the

same everywhere on the surface!

Gausss Law of ElectricityPractical Form

Electric field on Gaussian surface -- Must be the

same everywhere on the surface!

Area of the entire Gaussian surface Must be a

perpendicular surface (an element of a field

contour)!

Gausss Law of ElectricityPractical Form

Integral of the charge density over the

volume enclosed by the Gaussian surface!

Electric field on Gaussian surface -- Must be the

same everywhere on the surface!

Area of the entire Gaussian surface Must be a

perpendicular surface (an element of a field

contour)!

Section 9Using Gausss Law to Find Fields

Problem 1 Spherical Charge DistributionOutside

- Basic Plan
- Choose a spherical Gaussian surface of radius r

outside the charge distribution. - 2)
- 3) Integrate the charge over the entire charge

distribution.

Problem 1 Spherical Charge DistributionOutside

Problem 1 Spherical Charge DistributionOutside

Problem 2 Spherical Charge DistributionInside

- Basic Plan
- Choose a spherical Gaussian surface of radius r

inside the charge distribution. - 2)
- 3) Integrate the charge over the inside of the

Gaussian surface only.

Problem 2 Spherical Charge DistributionInside

Problem 2 Spherical Charge DistributionInside

Problem 3 Cylindrical Charge DistributionOutside

- Basic Plan
- Choose a cylindrical Gaussian surface of radius r

and length L outside the charge distribution. - 2)
- 3) Integrate the charge over the entire charge

distribution.

Problem 3 Cylindrical Charge DistributionOutside

- Basic Plan
- 4) Note that there are no field lines coming out

the ends of the cylinder, so there is no flux

through the ends!

Problem 3 Cylindrical Charge DistributionOutside

Problem 3 Cylindrical Charge DistributionOutside

Problem 4 Cylindrical Charge DistributionInside

- Basic Plan
- Choose a cylindrical Gaussian surface of radius r

and length L inside the charge distribution. - 2)
- 3) Integrate the charge over the inside of the

Gaussian surface only.

Problem 4 Cylindrical Charge DistributionInside

Problem 4 Cylindrical Charge DistributionInside

Infinite Sheets of Charge

- Basic Plan
- Choose a box with faces parallel to the plane as

a Gaussian surface. Let A be the area of each

face. - 2) Find the charge inside the box. No integration

is needed.

Problem 5 Infinite Sheet of Charge(Insulator

with s given)

Note there is flux out both sides of the box!

Problem 6 Infinite Sheet of Charge(Conductor

with s on each surface)

Note there is flux out both sides of the box, and

the total charge density is 2s!

Problem 6 A second way

Now there is flux out only one side of the box,

but the total charge density inside is just s!

Problem 7 A Capacitor

The area of the plate is and the area of the

box is .

There is flux out only one side of the box!

A Word to the Wise!

If you can do these seven examples, you can do

every Gausss law problem I can give you! Know

them well!