W9_parallel_resonance.eps

Week 1: Discrete Charge and the Electrostatic Field 51

which is the version of Coulomb’s Law that we will most often use in the problems – find field

first, then find force if necessary. Used in nearly all of the problems in this context.

• The Superposition Principle for the Electric Field:

E~(~r) =

∑

i

keqi(~r−~ri)

|~r−~ri|^3

or, for a continuous distribution of charge:

E~(~r) =ke

∫

ρ(~r 0 )(~r−~r 0 )d^3 r 0

|~r−~r 0 |^3

One can also integrate over sheets or lines of charge, using theircharge densities:

ρ = dq

dV

σ =

dq

dA

λ = dq

dx

Needed in problems 2, 3.

• We should keep in mind thatcharge is conserved. The net charge of objects cannot change;
  charge can only move around, not be created or destroyed. A basic concept.


• The electric dipole moment of a pair of equal and opposite point charges of magnitudeq
  separated by a vector~lis:
  ~p=q~l
  We sometimes need theideaof quadrupole moments and monopole moments in this chapter.
  Needed in problems 2, 3, 5, 6, 9.


• The force on a dipole in a uniform electric field is:

F~= 0

(more generally it isF~=−∇~(−~p·E~). The torque on a dipole in a uniform field is:

~τ=~p×E~

Needed in problems 2, 3, 5, 6, 9.

• Yes, we use Newton’s Second Law:
  F~=m~a
  (problems 3, 4, 8 and 11); Newton’s Second Law for torque:

τ=Iα

(problem 9); our knowledge of the Simple Harmonic Oscillator equationand its solutions:

d^2 x

dt^2

+ω^2 x= 0

(problems 9 and 11); and gravity near the Earth’s surface:

F~g=−mgyˆ

(down, in problems 7 and 8); and the ideas associated with stable versus unstable equilibrium

in problem 3.

Our knowledge of Newton’s Laws, rotation and oscillation and gravitynear the earth’s surface

from the Mechanics part of this course is essential in this part as well!