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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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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 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

Class 18

gaussian surface

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

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

gauss s 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.

Gauss’s Law of Electricity

charge density
Charge

Volume

Charge

Area

Charge

Length

Volume: ρ =

Surface: σ =

Linear: λ =

Charge Density

charge density1
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.

gauss s 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.

Gauss’s Law and Magnetic Field Lines

N

gauss s law and magnetic field lines1
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

gauss s law and magnetic field lines4
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?

Gauss’s Law and Magnetic Field Lines

spherically symmetric charge distribution
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

inside a hollow sphere1
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

inside a hollow sphere2
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

inside a hollow sphere3
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

slide42
How can we get zero net field lines?

3. Or we could just have no electric field at all inside the hollow sphere.

slide43
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!

slide44
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 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

spherically symmetric charge distribution3
Electric field lines do not start or end outside charge distributions, but that can start or end inside charge distributions.

Spherically Symmetric Charge Distribution

spherically symmetric charge distribution5
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

spherically symmetric charge distribution6
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

spherically symmetric charge distribution10
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

spherically symmetric charge distribution11
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

example uniform distribution1
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 conductors1
The static electric field inside the conductor must be zero. – Draw a Gaussian surface inside the conductor.

Gauss’s Law and Conductors

+

+

+

+

+

+

+

+

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

surface charge and conductors1
What if there are no charges on the outside and there is net charge on the surface of a conductor?

+

+

+

+

+

Surface Charge and Conductors

+

+

+

+

+

+

+

+

+

+

+

surface charge and conductors2
The charge distributes itself so the field inside is zero and the surface is at the same electric potential everywhere.

+

+

+

Surface Charge and Conductors

+

+

+

+

+

+

+

+

+

+

+

+

+

example surface charge on a spherical conductor1
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).

now connect the two spheres
The charge density is greater near the “pointy” end.

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

Now Connect the Two Spheres

+

+

+

+

+

+

+

+

+

+

+

+

+

+

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.

Edges on Conductors

a hollow conductor4
There is no field surrounding the charge to hold the charges fixed, so the charges migrate and cancel each other out.

A Hollow Conductor

+

+

+

+

+

+

+

+

lightning and cars
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….

lightning and cars2
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!

lightning and cars3
A car is essentially a hollow conductor.

Charge goes to the outside.

The electric field inside is small.

Lightning and Cars

lightning and cars4
A car is essentially a hollow conductor.

Charge goes to the outside.

The electric field inside is small.

Lightning and Cars

physicist s cow1
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

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

Class 20

gauss s law of electricity1

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?

charge and density

Charge and Density

is valid when?

charge and density1

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.

integration1
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

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

fundamental rule of integration1
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

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

field lines and electric field

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 field1

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
Gauss’s Law states that:

Electric Flux

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

electric flux1
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!

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

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.
  • (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

a few facts about flux3
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

a few facts about flux5
only holds when the frame is flat and the field is uniform.

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

A Few Facts about Flux

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.

Area Vectors on a Gaussian Surface

a few facts about flux7
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 flux8
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:

class 21
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

Class 21

gauss s law of electricity 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:

gauss s law of magnetism integral form
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 electricity tee shirt form

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 electricity tee shirt form1

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 electricity practical form

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 electricity practical form1

Gauss’s Law of ElectricityPractical Form

Now let’s think about what this

equation really means!

gauss s law of electricity practical form2

Gauss’s Law of ElectricityPractical Form

Electric field on

Gaussian surface

-- Must be the same

everywhere on the

surface!

gauss s law of electricity practical form3

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 electricity practical form4

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 distribution outside

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 distribution inside

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 distribution outside

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 distribution outside1

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 distribution inside

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

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.

a word to the wise
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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