W9_parallel_resonance.eps

106 Week 3: Potential Energy and Potential

withV 0 and arbitrary constant of integration, used to set a suitable zeroof the potential

energy. For compact charge distributions:

V(~x) =−

∫~x

∞

E~·d~x (132)

and

E~=−∇~V (133)

• The potential of a charge distribution can obviously be evaluated bysuperposition:

Vtot(~x) =

∑

i

kqi

|~x−~xi|

(134)

or

Vtot(~x) =

∫

kdq 0

|~x−~x 0 |

=

∫

kρ(~x 0 )d^3 r 0

|~x−~x 0 |

(135)

• Conductors at electrostatic equilibrium areequipotential. We can therefore speak of thepo-
  tential differencebetween two conductors in electrostatic equilibrium where it doesn’tmatter
  what path we use to go from one conductor to the other. This also means that if we charge
  one isolated conductor to some potential and then connect it to another isolated conductor,
  charge will flow until the two conductors (now one) are at thesamepotential, a process called
  charge sharing.


• In a strong enough electric field,dielectric breakdownoccurs and insulators “suddenly” become
  conductors (e.g. lightning in air). Strong fields are often induced in the vicinity of a sharp
  conducting point, causing a slowercorona effectdischarge that is the basis for lightning rods.

This completes the chapter/week summary. The sections below illuminate these basic facts and
illustrate them with examples.

3.1: Electrostatic Potential Energy

The electrostatic force isconservative. That is, the work done moving a charge between any two
points in an electrostatic field is independent of the path taken. Forconservative forces we can
define thechange in potential energyto be the negative work done by the electrostatic force moving
between two points:

∆U(~x 0 →~x 1 ) =−

∫~x 1

~x 0

F~·d~x (136)

The corresponding relation between the potential energy thus defined and the force is (as usual):

F~=−∇~U (137)

Consequently we see that we could equally well define the electrostatic potential energy in terms of
anindefiniteintegral and anarbitrary constant of integration:

∆U(~x) =−

∫

F~·d~x+U 0 (138)

that effectively sets the point where the potential energy is zero.

By convention, for charge densities that havecompact support– ones that one can draw a ball of
finite radius (however large that radius might be) so that itcompletely containsall of the charge –
we define the potential energy to be zero at∞, just as we did for the gravitational potential energy:

∆U(~x) =−

∫~x

∞

F~·d~x (139)