Lesson 7 Gauss’s Law and Electric Fields

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Lesson 7 Gauss’s Law and Electric Fields. 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 Gauss’s Law of Electricity learn about volume, surface, and linear charge density

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### Lesson 7Gauss’s Law and Electric Fields

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 Gauss’s Law of Electricity
• learn about volume, surface, and linear charge density
• learn Gauss’s Law of Magnetism
• show by Gauss’s law and symmetry that the electric field inside a hollow sphere is zero

### 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

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?

### Electric Field Lines

+8

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?

### Electric Field Lines

+8

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?

### Electric Field Lines

─8

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?

### Electric Field Lines

─8

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?

### Electric Field Lines

0

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?

### Electric Field Lines

0

From the field lines coming out of this box, what can you tell about what’s inside?

### Electric Field Lines

The net charge inside must be +1 (if we draw 4 lines per unit of charge).

### Electric Field Lines

The net number of electric field lines passing through a Gaussian surface is proportional to the charge enclosed within the Gaussian surface.

Charge

Volume

Charge

Area

Charge

Length

Volume: ρ =

Surface: σ =

Linear: λ =

### 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:

### Charge Density

You need to understand what we mean by this equation, but we won’t usually need to think of density as a derivative.

### Section 3Gauss’s Law of Magnetism

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.

### Gauss’s Law and Magnetic Field Lines

N

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 there’s no reason we shouldn’t find a magnetic monopole as the electric and magnetic fields are both equally fundamental.

### Gauss’s Law and Magnetic Field Lines

What characteristic would a magnetic monopole field have?

### Gauss’s Law and Magnetic Field Lines

What characteristic would a magnetic monopole field have?

### Gauss’s Law and Magnetic Field Lines

So what can we conclude about the number of lines passing through a Gaussian surface?

### Gauss’s Law and Magnetic Field Lines

The net number of magnetic field lines passing through any Gaussian surface is zero.

### Section 4Gauss’s Law and Spherical Symmetry

The charge density, ρ, 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!

### Spherically Symmetric Charge Distribution

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.

### Inside a Hollow Sphere

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

### How can we get zero net field lines?

… but this violates symmetry!

### How can we get zero net field lines?

… 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.

Today, we will:
• learn how to use Gauss’s 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 aren’t

### Class 19

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

Let’s draw a spherical Gaussian surface at radius r.

### Spherically Symmetric Charge Distribution

r

Now we split the sphere into two parts – the part outside the Gaussian surface and the part inside the Gaussian surface.

### Spherically Symmetric Charge Distribution

r

r

The total electric field at r will be the sum of the electric fields from the two parts of the sphere.

### Spherically Symmetric Charge Distribution

r

r

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.

### Spherically Symmetric Charge Distribution

r

r

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.

### Spherically Symmetric Charge Distribution

r

Inside a spherically symmetric charge distribution, the static electric field is:

### Spherically Symmetric Charge Distribution

r

A uniformly charged sphere of radius R has a total charge Q. What is the electric field at r < R ?

### Example: Uniform Distribution

A uniformly charged sphere of radius R has a total charge Q. What is the electric field at r < R ?

### Example: Uniform Distribution

Since the charge density is uniform:

r

### Gauss’s Law and Conductors

+

+

+

+

+

+

+

+

The static electric field inside the conductor must be zero. – Draw a Gaussian surface inside the conductor.

### Gauss’s Law and Conductors

+

+

+

+

+

+

+

+

No field lines go through the Gaussian surface because E=0. Hence, the total enclosed charge must be zero.

+

+

+

+

+

+

+

+

### Gauss’s Law 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.

+

+

+

### Surface Charge and Conductors

+

+

+

+

+

+

+

+

+

+

+

+

+

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?

### Example: Surface Charge on a Spherical Conductor

On the outside, the potential is that of a point charge.

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

The smaller sphere has a larger charge density.

### Take Two Conducting Sphereswith the Same Voltage

+

+

+

+

+

+

+

+

+

+

+

+

+

+

The charge density is greater near the “pointy” end.

The electric field is also greater near the “pointy” end.

### Now Connect the Two Spheres

+

+

+

+

+

+

+

+

+

+

+

+

+

+

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.

### Edges on Conductors

What if there’s 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.

### A Hollow Conductor

+

+

+

+

+

+

+

+

Static charge moves to the outside surface of a conductor.

### Charge on a Conductor

+

+

+

+

+

+

+

+

Why is a car a safe place to be when lightning strikes?

### Lightning and Cars

Note: Any car will do – it doesn’t need to be a Cord….

Is it the insulating tires?

### Lightning and Cars

Is it the insulating tires?

### Lightning and Cars

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!

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.

### Physicist’s Cow

Cow

Earth

d

I

When d is bigger, the resistance along the ground between the cow’s feet is bigger, the voltage across the cow is bigger, and the current flowing through the cow is bigger.

### Physicist’s Cow

Cow

Earth

d

I

So the cow should keep her feet

close together!

### How should a cow stand to avoid injury when lightning strikes nearby?

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

### Gauss’s 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?

is valid when?

### Charge and Density

when ρ is uniform.

If ρis 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.

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.

### Integration

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 Integration

Square in x-y plane

Cylinder

Sphere

### Fundamental Rule of IntegrationExamples

Consider a very thin slice.

### Fundamental Rule of IntegrationExamples

Is constant on this slice?

Consider a very thin slice.

### Fundamental Rule of IntegrationExamples

Is constant on this slice?

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

### Fundamental Rule of IntegrationExamples

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

### Let’s Do Some Integrals

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

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?

### 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.

Gauss’s Law states that:

### Electric Flux

EA is called the electric flux. We write it as or just .

Gauss’s Law states that:

### Electric Flux

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!

Lets calculate the electric flux from a point charge

passing through a sphere of radius r.

### Electric Flux and a Point Charge

Gauss’s law says this is proportional to the charge enclosed in the sphere!

### Electric Flux and a Point Charge

This means that we can write Gauss’s Law of Electricity as

### Electric Flux and Gauss’s Law

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 you’ll know what it really means…

### A Few Facts about Flux

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.
• (Don’t 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.

### An Area Vector

First, Let’s 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 θ, 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

What if the surface (frame) isn’t flat, or the electric field isn’t uniform?

### A Few Facts about Flux

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.

### Area Vectors on a Gaussian Surface

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.

### A Few Facts about Flux

Recall that the normal to each small area is taken to be in the outward direction.

### A Few Facts about Flux

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:

Today, we will:
• learn how to use Gauss’s law to find the electric fields in cases of high symmetry
• insdide and outside spheres
• inside and outside cylinders
• outside planes

### Section 7Gauss’s Laws in Integral Form

The number of electric field lines passing through a Gaussian surface is proportional to the charge enclosed by the surface.

### Gauss’s Law of ElectricityIntegral Form

We can make this simple expression look much more impressive by replacing the flux and enclosed charge with integrals:

The number of magnetic field lines passing through a Gaussian surface is zero

### Gauss’s Law of MagnetismIntegral Form

With the integral for magnetic flux, this is:

### Gauss’s Law of ElectricityTee-Shirt Form

This can be written in many different ways. A popular form seen on many tee-shirts is:

### Gauss’s 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 Gauss’s law to use if you want to impress someone with how smart you are.

### Gauss’s Law of ElectricityPractical Form

This is the form of Gauss’s law you will use when you actually work problems.

### Gauss’s Law of ElectricityPractical Form

Now let’s think about what this

equation really means!

### Gauss’s Law of ElectricityPractical Form

Electric field on

Gaussian surface

-- Must be the same

everywhere on the

surface!

### Gauss’s 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)!

### Gauss’s 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)!

### 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 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 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 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.

### 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.

Note there is flux out both sides of the box!

### Problem 5: Infinite Sheet of Charge(Insulator with σgiven)

Note there is flux out both sides of the box, and the total charge density is 2σ!

### Problem 6: Infinite Sheet of Charge(Conductor with σon each surface)

Now there is flux out only one side of the box, but the total charge density inside is just σ!

### Problem 6: A second way…

The area of the plate is and the area of the box is .

There is flux out only one side of the box!

### Problem 7: A Capacitor

If you can do these seven examples, you can do every Gauss’s law problem I can give you! Know them well!