Week 3: Potential Energy and Potential
- The change in electrostatic potential energy moving a charge between two points in the field
of other charges is:
∆U(~x 0 →~x 1 ) =−
∫~x 1
~x 0
F~·d~x (124)
whereF~is the total force due to all other charges.
- The vector electrostatic force can be found from the the potential energy function by taking
its negativegradient:
F~=−∇~U (125)
- For charge density distributions with “compact support” (ones wecan draw a ball around,
basically) we by convention define the zero of the potential energyfunction to be at∞:
U(~x) =−
∫~x
∞
F~·d~x (126)
For point chargesq 1 andq 2 , it is just:
U(~x 1 ,~x 2 ) =
kq 1 q 2
|~x 1 −~x 2 |
(127)
- Since the potential energy is just a scalar and satisfies the superposition principle, we can
evalute the total energy of a system of point charges as:
Utot=
1
2
∑
i 6 =j
kqiqj
|~xi−~xj|
(128)
(there is a similar integral expression for continuous charge distributions we will address later)
where the 1/2 is to compensate for double counting in the sum.
- The electrostaticpotentialproduced by a chargeqis a one-body scalar field defined by:
V(~x) = limq
0 →^0
U(~x)
q 0
(129)
so that the potential of a point charge in coordinates centered onthe charge is just:
V(~r) =kq
r
(130)
- The potential is to the field as the potential energy is to the force,so:
V(~x) =−
∫
E~·d~x+V 0 (131)
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