PHYS117B:%20Lecture%204

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PHYS117B Lecture 4

• Last lecture We used
• Coulombs law
• Principle of superposition
• To find the electric field of continuous charge
  distributions
• Today Im going to teach you the easy way!

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The key word is
SYMMETRY
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We will

• Recognize and use symmetry to determine the shape
  of electric fields
• Calculate electric flux through a surface
• Use Gausss law to calculate the electric field
  of symmetric charge distributions
• Use Gausss law to understand the properties of
  conductors in electrostatic equilibrium

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Suppose we know only 2 things about electric
fields

• An electric field points away from charges and
  towards negative charges
• An electric field exerts a force on a charged
  particle
• What can we deduce for the electric field of an
  infinitely long charged cylinder ?

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The charge distribution has cylindrical symmetry

• What does this mean ?
• There is a group of geometrical transformations
  that do not cause any physical change
• Lets try it (I have a cylinder here)

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Translate, rotate, reflect
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If you cant 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!

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Can the electric field of the cylinder look like
this ?
NO the reflection symmetry is not obeyed!
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Can the electric field of the cylinder look like
this ?
NO the reflection symmetry is not obeyed!
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What about this situation?
No The translational symmetry is not obeyed!
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What about this situation ?
Yes Finally something that obeys symmetry !
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Three basic symmetries planar, cylindrical,
spherical
We will make heavy use of these three basic
symmetries.
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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 ?

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The mystery box

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

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Imagine a situation like this

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

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Or you can have

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

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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

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We can define electric field flux
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If the field is non-uniform
This is trouble, I need to do an integral
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The surface is NOT flat ooh bother !
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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
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What good is the electric field flux ?Gausss
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 Gausss law determine the magnitude of E

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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 Gausss 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 gt no
  angles to deal with in the dot product!

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Lets do it for an infinitely long wire with
uniform linear charge density l

• Last time we used Coulombs law this is the HARD
  way
• Today or sweet simplicity well use Gausss
  law to get the same result !

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Field of a line of charge use Coulombs law,
superposition and symmetry !
line of charge length 2a and linear charge
density l
For an infinite line of charge i.e. xltlt a
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See now how to do it using symmetry and Gausss
law