Transcript No Slide Title

Chapter 24: Electric Potential
Electric Potential energy
Work done by a force acting on an object moving along a path:
dl
a
F
b
 
  F  dl
b
Wab
a
If the force is a conservative force, the work done can be
expressed as a change in potential energy:
Wa b  U a  U b
Work-Energy Theorem: the change in kinetic energy equals
the total work done on the particle.
If only conservative forces are present:
K b  K a  Wab  U a  U b  K a  U a  K b  U b
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A charge in a uniform electric field















a
q
dy







b
 
 
F  dl  qE  dl  qE (dy)
  yb
  F  dl   qE (dy)  qE ( ya  yb )
b
Wa b
a
ya
U  qEy
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The work done on a charge q’ in the presence of another charge q
 
q' q
F  dl  k 2 dr
r
b
  rb q ' q
Wa b   F  dl   k 2 dr
r
a
ra
1 1
q' q
 k
 kq' q  
r ra
 ra ra 
 Ua Ub
rb
for several source charges
q' q
U (r )  k
conventionU (r  )  0
r
 q1 q2 q3

qi
U q '  kq'       kq' 
i ri
 r1 r2 r3

qi q j
U total  
all pairs rij
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Potential
Potential is potential energy per unit charge
U
orU  q 'V
q'
units : 1V  1Volt  1J / C
V
Potential is often referred to as “voltage”
In terms of “work per charge”
Wa b U a  U b

 Va  Vb
q'
q'
In terms of source charges
qi
V  k
ri
dq
or V  k 
r
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Potential and potential differences from the Electric Field
 
  q ' E  dl
b
Wa b
a
 
Va  Vb   E  dl
b
a
 
Vab  Va  Vb    E  dl
a
b
 
dU
dV   E  dl ~ Fx  
dx
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Electric Potential (continued): Examples
What is the speed of an electron accelerated from rest across a potential difference of
100V? What is the speed of a proton accelerated under the same conditions?
Vab
An electric dipole oriented vertically at the origin consists of two point charges,
+/- 12.0 nC placed 10 cm apart. What is the potential at a point located 12cm
from the dipole in the horizontal direction? What is the potential energy
associated with a +4.0 nC charge placed at this point?
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Spherical Charged Conductor
Outside: looks like a point charge
Inside: field is zero (dV =-E ·dl = 0)
q
V k
r
q
Vk
R
at the surface of the sphere
:
q
E k 2
R
E
V
Vmax=Emax R => Larger E with smaller R
Dielectric Strength = maximum electric field strength an insulator can withstand
before Dielectric Breakdown (Insulator becomes a conductor).
=> High voltage terminals have large radii of curvature
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





















Potential between parallel plates
a
Uniform field: U = qEy
d
=> V = Ey (VyVb)
q
y
Vab = VaVb = Ed
b
or
E = Vab /d
(True for uniform fields only, although this can provide an
estimate of field strength)
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Line charge and (long) conducting cylinder
2k
Er 
r
 
Va  Vb   E  dl
b
a
2k
r
dr  2k ln r rb
a
r
ra
rb

 2k ln ra  ln rb 
ra
 2k ln
rb
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Equipotential surfaces
A surface in space on which the potential is the same for
every point.
Surfaces of constant voltage.
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Potential Gradient
a
 
 
Va  Vb   E  dl    E  dl
b
a
b
 
 
Va  Vb   dV    E  dl  dV   E  dl
a
a
b
b
dV  El dl
V
V
V
Ey  
Ez  
x
y
z

V ˆ V ˆ V   ˆ 

ˆ
E   i
j
k
   i 
x
x   x
 x
Ex  


E  V
ˆj   kˆ  V
x
x 
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Millikan oil-drop experiment
Experimental investigation into the quantization of charge
small drops of oil, with small amounts
of excess charge.
F = qE
naive: balance -> q = mg/E
F = mg
need mass! (or drop radius + density)
Balance gravity, electric force and air
resistance on drop in motion:
F f  6rv F f '  6rv'
4 3
Fg  mg  r 
3
FE  qE
v
Ff
F = qE
Ff
’
v’
F = mg
F = mg
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Millikan’s (and later) results
thousands of measurements => all (positive or negative)
integer multiples of e!
=> charge is quantized!!
Modern Quantum Chromodynamics:
quarks have fractional (+/- 1/3e +/- 2/3e) charges, but
never appear alone (“confinement”) net charge of all
observed objects are integer multiples of e.
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Cathode Ray tube
V2

d
L
V1
2eV1
electron gun v x 
me
eE eV2
between plates a y 

me me d
v y  a yt
L  vxt
eV2 L
vy 
me d v x
vy
eV2 L eV2 Lme
V2 L
tan   


2
v x me d v x
me d 2eV1 d 2eV
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