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1/30/12 1
Chapter 23
- Electrostatic Potential Energy of a system of fixed point charges is equal to the work that must be done by an external agent to assemble the system, bringing each charge in from an infinite distance. q 1 Point 1 Point 2 1/30/12 2
Example: Electrostatic Potential Energy
q 1 Point 2 q 2 Point 1 If q 1 & q 2 have the same sign, U is positive because positive work by an external agent must be done to push against their mutual repulsion. If q 1 & q 2 have opposite signs, U is negative because negative work by an external agent must be done to work against their mutual attraction. 1/30/12 3
Example: Three Point Charges
q 1 r^ Point 2 1, Point 1 q 2 r1, r2, Point 3 q 3 1/30/12 4 Electrostatic Potential Energy
- We can conclude that the total work required to assemble the three charges is the electrostatic potential energy U of the system of three point charges: 2 , 3 3 2 1 , 3 3 1 1 , 2 2 1 2 3 r kq q r kq q r kq q U = Wtotal = W + W = + +
The electrostatic potential energy of a
system of point charges is the work needed
to bring the charges from an infinite
separation to their final position
1/30/12 5
Calculate Electric Field from the Potential
Electric field always points in the direction of
steepest descent of V (steepest slope) and tis
magnitude is the slope.
Potential from a Negative Point Charge Potential from a Positve Point Charge -V(r ) y x V(r ) y x 1/30/12 6
Calculating the Electric Field from the Potential
Field
E = −
∇ V = −
∂ V
∂ x
i^ ˆ +
∂ V
∂ y
ˆ j +
∂ V
∂ z
⎛ k ˆ
Ex = −
∂ V
∂ x
, Ey = −
∂ V
∂ y
, and Ez = −
∂ V
∂ z
1/30/12 7
Example: Calculating the Electric Field from the
Potential Field
What is the electric field at any point on the central
axis of a uniformly charged disk given the potential?
1/30/12 8
Potential due to a Group of Point Charges
q 1
q 2
q 3 q 4
r 1
r 2
r 3
r 4
X
V ( r ) = Vn ( r ) =
n = 1 4 πε o
N
qn
n = 1^ rn
N
" 1/30/12 13 Calculate Potential on the central axis of a charged disk (another way) 1/30/12 14 Calculate Potential due to an infinite sheet
1/30/12 15 E due to an infinite line charge
Corona discharge around a high voltage power line,
which roughly indicates the electric field lines.
1/30/12 16 Equipotentials
Definition: locus of points with the same potential.
- General Property: The electric field is always perpendicular to an equipotential surface.
1/30/12 17 Equipotentials: Examples
infinite positive
charge sheet
Point charge^ electric dipole
1/30/12 18 + + + + + + + + + + + + + + +
Locally
Gauss:
at electrostatic
^ equilibrium E ⊥ = σ ε 0 E || = 0
in electrostatic equilibrium
all of this metal is an equipotential;
i.e., it is all at the same voltage
Equipotential Lines on a Metal Surface 1/30/12 19 Potential inside & outside a conducting sphere
Common mistake: thinking that potential must be zero
inside because electric field is zero inside.
Vref = 0
at r = ∞.
1/30/12 20 Summary
- If you know the functional behavior of the potential V
at any point, you can calculate the electric field.
- The electric potential for a continuous charge
distribution can be calculated by breaking the
distribution into tiny pieces of dq and then integrating
over the whole distribution.
- Finally no work needs to be done if you move a charge
on an equipotential, since it would be moving
perpendicular to the electric field.
- The charge concentrates on a conductor on surfaces
with smallest radius of curvature.
