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## PowerPoint Slideshow about ' Lesson 7 Gauss’s Law and Electric Fields' - ethan-patton

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### Gaussian Surface

### Counting Field Lines

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

### Gauss’s Law of Electricity

### Charge and Density

### Charge and Density

### Field Lines and Electric Field

### Field Lines and Electric Field

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

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

### Gauss’s Law of ElectricityPractical Form

### Gauss’s Law of ElectricityPractical Form

### Gauss’s Law of ElectricityPractical Form

### Gauss’s Law of ElectricityPractical Form

### Problem 1: Spherical Charge DistributionOutside

### Problem 2: Spherical Charge DistributionInside

### Problem 3: Cylindrical Charge DistributionOutside

### Problem 3: Cylindrical Charge DistributionOutside

### Problem 4: Cylindrical Charge DistributionInside

### Infinite Sheets of Charge

Today, we will:### Class 18

- 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

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

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

From the field lines coming out of this box, what can you tell about what’s inside?### 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.### Gauss’s Law of Electricity

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.

If magnetic field lines came out from point sources like electric field lines, then we would have a law that said:### Gauss’s Law and Magnetic Field Lines

The net number of magnetic field lines passing through a Gaussian surface is proportional to the magnetic charge inside.

N

But we have never found a magnetic monopole.### Gauss’s Law and Magnetic Field Lines

- 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.

All known magnetic fields have field lines that form closed loops.### Gauss’s Law and Magnetic Field Lines

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

The net number of magnetic field lines passing through any Gaussian surface is zero.### Gauss’s Law of Magnetism

The charge density, ρ, can vary with r only.### Spherically Symmetric Charge Distribution

Below, we assume that the charge density is greatest near the center of a sphere.

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!

2. We could have some radial lines come in and other radial lines go out…### 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:### Class 19

- 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

Electric field lines do not start or end outside charge distributions, but that can start or end inside charge distributions.### 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

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

The static electric field inside the conductor must be zero. – Draw a Gaussian surface inside the conductor.### Gauss’s Law and Conductors

+

+

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+

+

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?### Surface Charge and Conductors

-- The volume charge density inside the conductor must be zero and the surface charge density on the conductor must also be zero.

What if there are no charges on the outside and there is net charge on the surface of a conductor?### Surface Charge and Conductors

+

+

+

+

+

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The charge distributes itself so the field inside is zero and the surface is at the same electric potential everywhere.### Surface Charge and Conductors

+

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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 charge density is greater near the “pointy” end.### Now Connect the Two Spheres

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

+

+

+

+

+

+

+

+

+

+

+

+

+

+

Charge moves to sharp points on conductors.### Edges on Conductors

Electric field is large near sharp points.

Smooth, gently curved surfaces are the best for holding static charge.

Lightning rods are pointed.

There is no field surrounding the charge to hold the charges fixed, so the charges migrate and cancel each other out.### A Hollow Conductor

+

+

+

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+

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+

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

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. ### Lightning and Cars

Charge goes to the outside.

The electric field inside is small.

A car is essentially a hollow conductor. ### Lightning and Cars

Charge goes to the outside.

The electric field inside is small.

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

close together!

Today, we will:### Class 20

- 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

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?

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.### Integration

Our goal is to turn two- and three- dimensional integrals into one-dimensional integrals.

Identify the spatial variables on which the integrand depends.### Fundamental Rule of Integration

You must slice the volume (length or surface) into slices on which these variables are constant.

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

A cylinder of length L and radius R has a charge### Charge on a Cylinder

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?

A sphere of radius R has a charge density where is a constant. Find the total charge on the sphere.### Charge on a Sphere

How do you slice the sphere?

What is the volume of each slice?

- This is valid when
- .A is the area of a section of a perpendicular surface.
- The electric field is constant on A.

- 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 .

Flux is a mathematical expression for number of field lines passing through a surface!

Lets calculate the electric flux from a point charge### Electric Flux and a Point Charge

passing through a sphere of radius r.

Gauss’s law says this is proportional to the charge enclosed in the sphere!### Electric Flux and a Point Charge

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 ### An Area Vector

- 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.

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

only holds when the frame is flat and the field is uniform.### A Few Facts about Flux

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

1) We must take a small region of the surface dA that is essentially flat.### Area Vectors on a Gaussian Surface

2) We choose a unit vector perpendicular to the plane of dA going in an outward direction.

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.

Thus, the most general equation for flux through a surface is:### 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:### Class 21

- 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

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:

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

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.

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

Electric field on

Gaussian surface

-- Must be the same

everywhere on the

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)!

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)!

Basic Plan:

Choose a spherical Gaussian surface of radius r outside the charge distribution.

2)

3) Integrate the charge over the entire charge distribution.

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.

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.

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!

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.

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, 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 .### Problem 7: A Capacitor

There is flux out only one side of the box!

If you can do these seven examples, you can do every Gauss’s law problem I can give you! Know them well!### A Word to the Wise!

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