Lecture 3

Karl Friedrich Gauss

(1777-1855) – German mathematician

Gauss’s LawGauss’s Law

Easiest case:

• The E-field is uniform

• The plane is perpendicular to the field

E E A

Electric Flux Electric Flux – Case 1– Case 1

Electric Flux

Easiest case:

• The E-field is uniform

• The plane is perpendicular to the field

E E A

Electric Flux Electric Flux – Case 1– Case 1

Electric Flux

Flux depends on how strong the E-field is and how big the area is.

E

E

A

A

most general case:

• The E-field is not uniform

• The surface is curvy and is not perpendicular to the field

Electric Flux Electric Flux – Case 2– Case 2

E

E

A

A

Electric Flux Electric Flux – Case 2– Case 2Imagine the surface A is a mosaic of little

tiny surfaces ΔA.

Pretend that each little ΔA is so small that it is essentially flat.

E

E

A

A

Electric Flux Electric Flux – Case 2– Case 2

Then, the flux through each little ΔA is just:

AEE

is a special vector. It points in the normal direction and has a magnitude that tells us the area of ΔA .

A

E

E

A

A

Electric Flux Electric Flux – Case 2– Case 2

So… to get the flux through the entire surface A, we just have to add up the contributions from each of the little ΔA’s that compose A.

n

nE AE

E

E

A

A

Electric Flux Electric Flux – Case 2– Case 2

n

nE AE

surface

E AdE

Electric Flux through an arbitrary surface caused by a spatially varying E-field.

• The vectors dAi point in different directions

– At each point, they are perpendicular to the surface

– By convention, they point outward

Electric Flux Electric Flux – Flux through a Closed Surface– Flux through a Closed Surface

Af_2404.swf

Electric Flux: General Definition

surface

E AdE

Electric Flux: General Definition

surface

E AdE

E-Flux through a surface depends on three things:

1. How strong the E-field is at each infinitesimal area.

2. How big the overall area A is after integration.

3. The orientation between the E-field and each infinitesimal area.

Electric Flux: General Definition

surface

E AdE

Flux can be negative, positive or zero!

-The sign of the flux depends on the convention you assign. It’s up to you, but once you choose, stick with it.

which little area experiences the most flux?

• The surface integral means the integral must be evaluated over the surface in question… more in a moment.

• The value of the flux will depend both on the field pattern and on the surface

• The units of electric flux are N.m2/C

Electric Flux Electric Flux – Calculating E-Flux– Calculating E-Flux

• The net electric flux through a surface is directly proportional to the number of electric field lines passing through the surface.

Electric FluxElectric Flux

Assume a uniform E-field pointing only in +x direction

Find the net electric flux through the surface of a cube of edge-length l, as shown in the diagram.

Flux through a CubeFlux through a Cube

Gauss’s Law is just a flux calculation

We’re going to build imaginary surfaces – called Gaussian surfaces – and calculate the E-flux.

Gauss’s Law only applies to closed surfaces.

Gauss’s Law directly relates electric flux to the charge distribution that creates it.

Gauss’s LawGauss’s Law

Gauss’s Law

0enclosed

surface

NET

qAdE

Gauss’s LawGauss’s Law

Gauss’s Law

0enclosed

surface

NET

qAdE

Gauss’s LawGauss’s Law

The net E-flux through a closed surface

Charge inside the surface

0enclosed

surface

NET

qAdE In other words…

1. Draw a closed surface around a some charge.

2. Set up Gauss’s Law for the surface you’ve drawn.

3. Use Gauss’s Law to find the E-field.

Gauss’s LawGauss’s Law

You get to choose the surface – it’s a purely imaginary thing.

Which surface – S1, S2 or S3 – experiences the most electric flux?

Let’s calculate the net flux through a Gaussian surface.

Assume a single positive point charge of magnitude q sits at the center of our imaginary Gaussian surface, which we choose to be a sphere of radius r.

surface

NET AdE

Gauss’s LawGauss’s Law – confirming Gauss’s Law – confirming Gauss’s Law

surface

NET AdE

Gauss’s LawGauss’s Law – confirming Gauss’s Law – confirming Gauss’s Law

At every point on the sphere’s surface, the electric field from the charge points normal to the sphere… why?

This helps make our calculation easy.

surfacesurface

NET EdAAdE cos

Gauss’s LawGauss’s Law – confirming Gauss’s Law – confirming Gauss’s Law

At every point on the sphere’s surface, the electric field from the charge points normal to the sphere… why?

This helps make our calculation easy.

surface

EdA

surface

NET EdA

Gauss’s LawGauss’s Law – confirming Gauss’s Law – confirming Gauss’s Law

Now we have:

But, because of our choice for the Gaussian surface, symmetry works in our favor.

The electric field due to the point charge is constant all over the sphere’s surface. So…

surface

NET dAE

Gauss’s LawGauss’s Law – confirming Gauss’s Law – confirming Gauss’s Law

This, we can work with.

We know how to find the magnitude of the electric field at the sphere’s surface.

Just use Coulomb’s law to calculate the E-field due to a point charge a distance r away from the charge.

surface

NET dAE

2argint r

qkE e

echpo

Gauss’s LawGauss’s Law – confirming Gauss’s Law – confirming Gauss’s Law

Thus:

surface

eNET dA

r

qk2

Gauss’s LawGauss’s Law – confirming Gauss’s Law – confirming Gauss’s Law

Thus:

surface

eNET dA

r

qk2

And, this surface integral is easy.

24 rdAsphere

Gauss’s LawGauss’s Law – confirming Gauss’s Law – confirming Gauss’s Law

Therefore:

But, we can rewrite Coulomb’s constant.

)4( 22

rr

qkeNET

04

1

ek

Gauss’s LawGauss’s Law – confirming Gauss’s Law – confirming Gauss’s Law

Therefore:

But, we can rewrite Coulomb’s constant.

)4( 22

rr

qkeNET

04

1

ek

0q

NET Thus, we have confirmed Gauss’s law:

A few more questions

• If the electric field is zero for all points on the surface, is the electric flux through the surface zero?

• If the electric flux is zero for a closed surface, can there be charges inside the surface?

• What is the flux through the surface shown? Why?

–+ Q – 3Q+

A spherical surface surrounds a point charge.

What happens to the total flux through the surface if:

(A) the charge is tripled,(B) the radius of the sphere is doubled,(C) the surface is changed to a cube, and(D) the charge moves to another location inside the surface?

Flux due to a Point ChargeFlux due to a Point Charge

Gauss’s Law can be used to

(1) find the E-field at some position relative to a known charge distribution, or

(2) to find the charge distribution caused by a known E-field.

In either case, you must choose a Gaussian surface to use.

Applying Gauss’s LawApplying Gauss’s Law

Choose a surface such that…

1. Symmetry helps: the E-field is constant over the surface (or some part of the surface)

2. The E-field is zero over the surface (or some portion of the surface)

3. The dot product reduces to EdA (the E-field and the dA vectors are parallel)

4. The dot product reduces to zero (the E-field and the dA vectors are perpendicular)

Applying Gauss’s LawApplying Gauss’s Law

An insulating solid sphere of radius a has a uniform volume charge density ρ and carries total charge Q.

(A) Find the magnitude of the E-field at a point outside the sphere

(B) Find the magnitude of the E-field at a point inside the sphere

Spherical Charge DistributionSpherical Charge Distribution

Spherical Charge DistributionSpherical Charge Distribution

Find the E-field a distance r from a line of positive charge of infinite length and constant charge per unit length λ.

Spherical Charge DistributionSpherical Charge Distribution

Find the E-field due to an infinite plane of positive charge with uniform surface charge density σ

Spherical Charge DistributionSpherical Charge Distribution

• In an insulator, excess charge stays put.

• Conductors have free electrons and, correspondingly, have different electrostatic characteristics.

• You will learn four critical characteristics of a conductor in electrostatic equilibrium.

• Electrostatic Equilibrium – no net motion of charge.

Conductors in Electrostatic EquilibriumConductors in Electrostatic Equilibrium

• Most conductors, on their own, are in electrostatic equilibrium.

• That is, in a piece of metal sitting by itself, there is no “current.”

Conductors in Electrostatic EquilibriumConductors in Electrostatic Equilibrium

Four key characteristics

1. The E-field is zero at all points inside a conductor, whether hollow or solid.

2. If an isolated conductor carries excess charge, the excess charge resides on its surface.

3. The E-field just outside a charged conductor is perpendicular to the surface and has magnitude σ/ε0, where σ is the surface charge density at that point.

4. Surface charge density is biggest where the conductor is most pointy.

Conductors in Electrostatic EquilibriumConductors in Electrostatic Equilibrium

Einside = 0• Place a conducting slab in an external

field, E.

• If the field inside the conductor were not zero, free electrons in the conductor would experience an electrical force.

• These electrons would accelerate.

• These electrons would not be in equilibrium.

• Therefore, there cannot be a field inside the conductor.

Conductors (cont.) Conductors (cont.) – Justifications– Justifications

Einside = 0• Before the external field is applied,

free electrons are distributed evenly throughout the conductor.

• When the external field is applied, charges redistribute until the magnitude of the internal field equals the magnitude of the external field.

• There is a net field of zero inside the conductor.

• Redistribution takes about 10-15s.

Conductors (cont.) Conductors (cont.) – Justifications– Justifications

Charge Resides on the Surface

• Choose a Gaussian surface inside but close to the actual surface

• The electric field inside is zero (prop. 1)

• There is no net flux through the gaussian surface

• Because the gaussian surface can be as close to the actual surface as desired, there can be no charge inside the surface

Conductors (cont.) Conductors (cont.) – Justifications– Justifications

Charge Resides on the Surface

• Since no net charge can be inside the surface, any net charge must reside on the surface

• Gauss’s law does not indicate the distribution of these charges, only that it must be on the surface of the conductor

Conductors (cont.) Conductors (cont.) – Justifications– Justifications

E-Field’s Magnitude and Direction

• Choose a cylinder as the Gaussian surface

• The field must be perpendicular to the surface– If there were a parallel

component to E, charges would experience a force and accelerate along the surface and it would not be in equilibrium

Conductors (cont.) Conductors (cont.) – Justifications– Justifications

E-Field’s Magnitude and Direction

• The net flux through the surface is through only the flat face outside the conductor

– The field here is perpendicular to the surface

• Applying Gauss’s law

Conductors (cont.) Conductors (cont.) – Justifications– Justifications

0A

EAE 0

E

E-Field’s Magnitude and Direction

E-Field’s Magnitude and DirectionConductors (cont.) Conductors (cont.) – Justifications– Justifications

E-Field’s Magnitude and Direction

• The field lines are perpendicular to both conductors

• There are no field lines inside the cylinder

A solid insulating sphere of radius a carries a uniformly distributed charge, Q.

A conducting shell of inner radius b and outer radius c is concentric and carries a net charge of -2Q.

Find the E-field in regions 1-4 using Gauss’s Law.

Sphere inside a Spherical ShellSphere inside a Spherical Shell

Example 1Consider a thin spherical shell of radius

14.0 cm with a total charge of 32.0 μC distributed uniformly on its surface. Find the electric field

(a) 10.0 cm and (b) 20.0 cm from the center of the

charge distribution.

9 6

2 2

8.99 10 32.0 107.19 MN C

0.200ekQEr

Example 2

A uniformly charged, straight filament 7.00 m in length has a total positive charge of 2.00 μC. An uncharged cardboard cylinder 2.00 cm in length and 10.0 cm in radius surrounds the filament at its center, with the filament as the axis of the cylinder. Using reasonable approximations, find (a) the electric field at the surface of the cylinder and (b) the total electric flux through the cylinder.

9 2 2 62 8.99 10 N m C 2.00 10 C 7.00 m20.100 m

ekEr

51.4 kN C, radially outwardE

cos 2 cos0E EA E r 4 25.14 10 N C 2 0.100 m 0.0200 m 1.00 646 N m CE

Example 3

A square plate of copper with 50.0-cm sides has no net charge and is placed in a region of uniform electric field of 80.0 kN/C directed perpendicularly to the plate. Find

(a) the charge density of each face of the plate and

(b) the total charge on each face.

4 12 7 28.00 10 8.85 10 7.08 10 C m

277.08 10 0.500 CQ A 71.77 10 C 177 nCQ

Example 4

A long, straight wire is surrounded by a hollow metal cylinder whose axis coincides with that of the wire. The wire has a charge per unit length of λ, and the cylinder has a net charge per unit length of 2λ. From this information, use Gauss’s law to find (a) the charge per unit length on the inner and outer surfaces of the cylinder and (b) the electric field outside the cylinder, a distance r from the axis.

0

2 3 6 3 radially outward

2e ek k

Er r r

in0 q inq

3