1/16/2008 J.Velkovska 1 PHYS117B: Lecture 4 Last lecture: We used Coulomb’s law Principle of superposition To find the electric field of continuous.

1/16/2008 J.Velkovska 1

PHYS117B: Lecture 4

Last lecture: We used Coulomb’s law Principle of superposition

To find the electric field of continuous charge distributions

Today: I’m going to teach you the easy way!


The key word is

SYMMETRY


We will:

Recognize and use symmetry to determine the shape of electric fields

Calculate electric flux through a surface Use Gauss’s law to calculate the electric field

of symmetric charge distributions Use Gauss’s law to understand the properties

of conductors in electrostatic equilibrium


Suppose we know only 2 things about electric fields:1. An electric field points away from + charges

and towards negative charges

2. An electric field exerts a force on a charged particle

What can we deduce for the electric field of an infinitely long charged cylinder ?


The charge distribution has cylindrical symmetry What does this mean ?

There is a group of geometrical transformations that do not cause any physical change

Let’s try it (I have a cylinder here)


Translate, rotate, reflect


If you can’t tell that the charge distribution (the cylinder) was transformed geometrically, then the electric field should not change either ! The symmetry of the electric field MUST

match the symmetry of the charge distribution!


Can the electric field of the cylinder look like this ?


Can the electric field of the cylinder look like this ?


What about this situation?


What about this situation ?


Three basic symmetries: planar, cylindrical, spherical

We will make heavy use of these three basic symmetries.


We learned how to use symmetry to determine the direction of the electric field, if we know the geometry of the charge distribution.

Can we reverse the problem ? Can we use the symmetry of the electric field configuration to determine the geometry of the charge distribution that causes this field ?


The mystery box

Take a test charge, move it around, measure the force on it, figure out the field configuration => deduce the symmetry of the charge distribution inside


Imagine a situation like this:

No field around the box, or same “amount of field” going in and out


Or you can have:

Same box, same geometry of the field, but“more field” going out of the box


We need some way to measure “how much” field goes in or out of the box To measure the volume of water that passes

through a loop per unit time, we use FLUX: the dot product of the velocity vector and the area vector gives volume/time


We can define electric field flux:


If the field is non-uniform:

This is trouble, I need to do an integral


The surface is NOT flat : ooh bother !


Relax ! We are going to deal with two EASY situations: the field is UNIFORM in both.

The field is EVERYWHERE tangent the surface: FLUX = 0 !

The field is EVERYWHERE perpendicular to the surface:

FLUX = EA


What good is the electric field flux ?Gauss’s law gives a relation between the electric field flux through a closed surface and the charge that is enclosed in that surface.

From symmetry: determine the shape (direction in every point in space) of the electric field

From Gauss’s law: determine the magnitude of E


What surface are we talking about ? This is NOT a physical surface It is an IMAGINARY surface If we want to make life simple, we have to figure out

what type of surface to choose, so that the integral in Gauss’s law is EASY to do

We need to choose a surface that has the same symmetry as the charge distribution

Then the field will be either tangent to the surface or perpendicular to the surface => no angles to deal with in the dot product!


Let’s do it for an infinitely long wire with uniform linear charge density Last time we used Coulomb’s law: this is the

HARD way Today: or sweet simplicity – we’ll use

Gauss’s law to get the same result !


Field of a line of charge: use Coulomb’s law, superposition and symmetry !

2 2

2ˆ

a kE x

x x a

For an infinite line of charge i.e. x<< a

0

2ˆ ˆ2

kE x x

x x

line of charge: length 2a and linear charge density


See now how to do it using symmetry and Gauss’s law

enclQ L

0 0

2

top bot wall

wallA E

rLE

0

ˆ2

E rr

1/16/2008 J.Velkovska 1 PHYS117B: Lecture 4 Last lecture: We used Coulomb’s law Principle of superposition To find the electric field of continuous.

Documents

electric field flux

electric field configuration

electric field points

butmore field

electric fieldfrom gausss

charge distribution

test charge

velkovskathe surface