Lesson 7 Gauss’s 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 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

Section 1 Visualizing Gauss’s 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─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─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─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 what’s inside?

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

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.

Section 2 Charge Density

Charge Density Volume: ρ = Surface: σ = Linear: λ = Charge Volume Charge Area Charge Length

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 won’t usually need to think of density as a derivative.

Section 3 Gauss’s Law of Magnetism

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. N

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

Section 4 Gauss’s Law and Spherical Symmetry

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 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!

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. How can we get zero net field lines? This is the only way it can be done!

Conclusion: the static electric field inside a hollow charged sphere with a spherically symmetric charge distribution must be zero. The Electric Field inside a Hollow Sphere

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

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 Let’s 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 r Inside a spherically symmetric charge distribution, the static electric field is:

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

Example: Uniform Distribution

Section 5 Gauss’s Law and Conductors

Gauss’s Law and Conductors Take an arbitrarily shaped conductor with charges on the outside

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 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 Spheres with 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 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

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 doesn’t need to be a Cord….

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! Is it the insulating tires?

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?

Physicist’s Cow I d Cow Earth

Physicist’s Cow I d Cow Earth 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.

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 6 Integration

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 is valid when?

Charge and Density when ρ is uniform. If ρ 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 Integration Examples Square in x-y plane Cylinder Sphere

Fundamental Rule of Integration Examples Consider a very thin slice. Is constant on this slice?

Fundamental Rule of Integration Examples Consider a very thin slice. Is constant on this slice?

Fundamental Rule of Integration Examples Square in x-y plane Cylinder Sphere

Fundamental Rule of Integration Examples Square in x-y plane Cylinder Sphere

Fundamental Rule of Integration Examples Square in x-y plane Cylinder Sphere

Fundamental Rule of Integration Examples Square in x-y plane Cylinder Sphere

Fundamental Rule of Integration Examples Square in x-y plane Cylinder Sphere

Rules for Areas and Volumes of Slices Memorize These!!! Square in x-y plane Disk Cylinder Sphere

Rules for Areas and Volumes of Slices Memorize These!!! Square in x-y plane Disk Cylinder Sphere

Rules for Areas and Volumes of Slices Memorize These!!! Square in x-y plane Disk Cylinder Sphere

Rules for Areas and Volumes of Slices Memorize These!!! Square in x-y plane Disk Cylinder Sphere

Rules for Areas and Volumes of Slices Memorize These!!! Square in x-y plane Disk Cylinder Sphere

Rules for Areas and Volumes of Slices Memorize These!!! Square in x-y plane Disk Cylinder Sphere

Let’s 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 7 Gauss’s Law and Flux

Field Lines and Electric Field This is valid when 1).A is the area of a section of a perpendicular surface. 2)The electric field is constant on A.

Field Lines and Electric Field This is valid when 1)A is the area of a section of a perpendicular surface. 2)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: EA is called the electric flux. We write it as or just.

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

Electric Flux and Gauss’s Law This means that we can write Gauss’s 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 you’ll know what it really means…

An Area Vector We wish to define a vector area. To do this 1)we need a flat surface. 2)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.

A Few Facts about Flux 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 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?

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 Gauss’s law to find the electric fields in cases of high symmetry insdide and outside spheres inside and outside cylinders outside planes

Section 7 Gauss’s Laws in Integral Form

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. 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 With the integral for magnetic flux, this is:

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

Gauss’s Law of Electricity Practical Form Now let’s think about what this equation really means!

Gauss’s Law of Electricity Practical Form Electric field on Gaussian surface -- Must be the same everywhere on the surface!

Gauss’s Law of Electricity Practical 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 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)! Integral of the charge density over the volume enclosed by the Gaussian surface!

Section 9 Using Gauss’s Law to Find Fields

Problem 1: Spherical Charge Distribution Outside Basic Plan: 1)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 Distribution Outside

Problem 2: Spherical Charge Distribution Inside Basic Plan: 1)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 Distribution Inside

Problem 3: Cylindrical Charge Distribution Outside Basic Plan: 1)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 Outside 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 Distribution Outside

Problem 4: Cylindrical Charge Distribution Inside Basic Plan: 1)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 Distribution Inside

Infinite Sheets of Charge Basic Plan: 1)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 σ given) Note there is flux out both sides of the box!

Problem 6: Infinite Sheet of Charge (Conductor with σ on each surface) Note there is flux out both sides of the box, and the total charge density is 2σ!

Problem 6: A second way… Now there is flux out only one side of the box, but the total charge density inside is just σ!

Problem 7: A Capacitor There is flux out only one side of the box! The area of the plate is and the area of the box is.

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!