W9_parallel_resonance.eps

64 Week 2: Continuous Charge and Gauss’s Law

• Gauss’s Law for the Electric Field
  Gauss’s Law is written:
  ∮

S/V

E~·nˆdA= 4πk

∫

V

ρ dV =

Qin S

ǫ 0

or in words, the flux of the electric field through a closed surfaceSequals the total charge

insideSdivided byǫ 0 , the permittivity of the electric field.

Gauss’s law can be used to easily evaluate the electric field for chargedensity distributions that

have the symmetry of a coordinate system, but its real importance is that it is one ofMaxwell’s

Equations, the fundamental laws of nature that govern charge and the electromagnetic field.

• Gauss’s Law and Properties of Conductors
  One can easily use Gauss’s Law to prove the following properties of conductorsin electrostatic
  equilibrium. Note well that these propertiesonlyapply in equilibrium when no charge is
  actually moving.
  
  
  • The electric field vanishes inside a conductorin electrostatic equilibrium(really vanishes
    across the first few layers of atoms, not at a mathematical surface, but we will consider
    changes on the scale of a few angstroms as being “instantly” and treat it as a perfect
    surface).
  
  
  • All non-neutral charge distributed on a conductorin electrostatic equilibriummust reside
    on the surface.
  
  
  • The electric field at thesurfaceof a conductorin electrostatic equilibriummust begin or
    terminate on the conductorperpendicularto the surface. There can be no field component
    parallel to the surface of a conductor.
  
  
  • Since the field at the surface of a conductor isE~⊥only andzeroinside, if we consider
    an infinitesimally thin Gaussian pillbox with inner surface in the conductor and outer
    surface just outside, we can easily show that:
  
  
  
  

E~⊥= 4πkeσ=σ

ǫ 0

The field at the surface is directly proportional to the surface charge density!

2.1: The Field of Continuous Charge Distributions

In natural matter, charges are very, very small compared to the length scales we can directly perceive.
An atom is order of 1 ̊A (10−^10 meters) in size where a nucleus is order of 1 fermi (10−^15 meters) in
size. An electron is a pointlike particle with no physical extent at all. Ina tiny piece of solid matter

• one only 10−^6 meters cubed, say – there are around (10^4 )^3 = 10^12 atoms, and each atom is made
  up of 2 to 200 electricchargesin its electron cloud and nucleus, and this is still only a chunkone
  micronin size!

Clearly, if we want to evaluate the electric field produced by a macroscopic piece of matter,
we’re going to have to do something other thanjust sumover theE~ifields produced by all of these
charges. Instead weaverageover the amount of charge inside all of the tiny micron-scale blocks that
might make up a large object. For each block there is a certainnet charge∆Q, in the block of size
(volume) ∆V. We can use this to define theaverage charge densityof the object:

ρ=

∆Q

∆V

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