W9_parallel_resonance.eps

32 Week 1: Discrete Charge and the Electrostatic Field

atoms, the conductivity of metals, the motion of charged particles, and much, much more, we

infer that for any two stationary charges, theexperimentally verifiedelectrostatic force acting

oncharge 1due tocharge 2 is:

F~ 12 =keq 1 q 2 (~r^1 −~r^2 )

|~r 1 −~r 2 |^3

Note that it acts on a linefromcharge 2tocharge 1, is proportional to both charges, and is

inversely proportional to the distance that separates them squared.

• The Electrostatic Constantke
  The electrostatic constantkesets the scale; it is avery important number(as we shall see)
  
  
  • a genuine constant of nature as wasGfor the gravitational field. It is often expressed in
    terms of a related quantity called thepermittivity of free space,ǫ 0 , which is more useful for
    advanced treatments of electrodynamics. We will often/generallyusekeinstead in this course
    (because it is very easy to remember), but I would like you to know the relationship between
    this quantity andǫ 0 so that you can easily calculate the latter if you should ever need it or
    care.
  
  
  
  

ke=

1

4 πǫ 0

= 9× 109

N−m^2

C^2

This is accurate to something like 3 significant figures, which is plenty for our purposes. Note

also that you don’t have torememberthe units ofkeper se, you can figure them out by

just remembering Coulomb’s Law (which you have to know anyway). Newtons on the left,

coulombs squared on top and meters squared on the bottom on theright.

• Electrostatic Field
  The fundamental definition of electrostatic field produced by a chargeqat position~ris that
  it is the electrostatic force per unit charge on a small test chargeq 0 placed at each point in
  space~r 0 in the limit that the test charge vanishes:

E~= lim

q 0 → 0

F

q 0

or

E~(~r 0 ) =keq(~r^0 −~r)

|~r 0 −~r|^3

If we locate the chargeqat the origin and relabel~r 0 →~r, we obtain the following simple

expression for the electrostatic field of a point charge:

E~(~r) =keq

r^2

ˆr

• Superposition Principle
  Given a collection of charges located at various points in space, the total electric field at a
  point is the sum of the electric fields of the individual charges:

E~(~r) =

∑

i

keqi(~r−~ri)

|~r−~ri|^3

To find the electrostatic field produced by a charge density distribution, we use the superpo-

sition principle inintegralform: