physics 202, lecture 6 · physics 202, lecture 6 today’s topics electric potential (ch. 23)...
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Physics 202, Lecture 6Today’s Topics
Electric Potential (Ch. 23) Electric Potential For Various Charge Distributions
Point charges Continuous chargese.g Uniform Ring, Sphere, Shell
Equipotential surfaces, Electrostatic equilibrium of conductors
Expected from preview: EV relationship, equipotential lines, electrostatic
equilibrium of conductors… Electric potential for a charge distribution…
Review: Electric Potential Difference Electric Potential Energy: q In a Generic E. Field
Electric Potential Difference
VqdqUUUB
AAB Δ=•−=−=Δ ∫ sE
test sourcesystem energy
AB
B
AVVd
qUV −=•−=Δ≡Δ ∫ sE
source only
Electric Potential Difference For Single Point Charge (Lecture 5 Review)
VB-VA = ke(q/rB-q/rA)
A
B
rA
rB
q
Take V∞ =0
V = keq/r
Electric Potential ForContinuous Charge Distribution
For finite charge distribution, it is common to setV=0 at infinite.
If the charge distribution is known, V can be calculated simply by scalar integral.
dV=ke dq/r (V=0 @ ∞)
∫=rdqkV e
Example: Uniformly Charged Ring For a uniformly charged ring, show that the
potential along the central axis is
22 axQkV e
+=
∫
∫
∫
+=
+=
=
dqax
kax
dqkrdqkV
e
e
e
22
22
Solution
=Q
Uniformly Charged Ring: Electric Field
( )symmetry to due 0
cos
2/3223
2
=+
===
==
⊥
∫E
axxQk
rxQkdEE
rx
rdqkdEdE
eexx
ex θ
Find the electric field along the central axis.Approach 1: Superposition.
Approach 2: derivative of potential
( ) 2/322 axxQk
xVE e
x+
=∂∂−=
Example: Uniformly Charged Spherical Shell
For uniformly charged spherical shell.Again, use:
RrrQkE e
r >= 2
RrE <= 0
Tip: V is the same inside E=0 region
AB
B
AVVdV −=•−=Δ ∫ sE
R
Example: Uniformly Charged Sphere Show that for a uniformly charged
sphere, the electric potential is: R
Hint: It is more convenientto use:
since from Gauss’s law:
AB
B
AVVdV −=•−=Δ ∫ sE
RrrQkE e
r >= 2
RrRQrkE e
r <= 3
Exer
cise w
ith T
A
Coulomb’s Law: (lecture 2)
Gauss’s Law: (lecture 3,4)
Potential: (lecture 5,today)
E = k dq
r2r̂∫
EidA∫ =
qenclε0
V = k dqr∫
E = −∇V
Ex = − ∂V∂x ,Ey = − ∂V
∂y ,Ez = − ∂V∂z
Summary: Methods for Obtaining E, V
V = −
Eidl∫
Know all methods (and when to apply which)!
Conductors in Electrostatic EquilibriumWe have learned that:
Electric field inside conductor is zero (also zeroinside any empty cavity within the conductor)
All net charges reside on the surface Electric field on surface of conductor always
normal to the surface, magnitude σ/ε0
sharper edge larger field.
Since E=0 inside conductor, Potential is the same throughout the conductor:
Equipotential
Charge Distribution on Conductors:Field lines and Equipotential Surfaces
Review: Charge Distribution On Conductors (I)
|E| higher at smaller radius of curvature (more charge density)
|E| lower(less charge density)
High Voltage Electrostatic Generator:Van de Graaff Generator
104V
up to 3x106V(~50KV in today’s demo)
Example
A B
Two conductors are connected by a wire. How do the potentialsat the conductor surfaces compare?
a) VA > VB b) VA = VB c) VA < VB
What happens to the charge on conductor A after it isconnected to conductor B ?
a) QA increases b) QA decreases c) QA doesn’t change