No Slide Title
Download
Report
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
Wab
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 Wab U a U b K a U a K b U b
p202c24: 1
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
p202c24: 2
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
p202c24: 3
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
p202c24: 4
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
p202c24: 5
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?
p202c24: 6
Spherical Charged Conductor
Outside: looks like a point charge
Inside: field is zero (dV =-E ·dl = 0)
q
V k
r
q
Vk
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
p202c24: 7
Potential between parallel plates
a
Uniform field: U = qEy
d
=> V = Ey (VyVb)
q
y
Vab = VaVb = Ed
b
or
E = Vab /d
(True for uniform fields only, although this can provide an
estimate of field strength)
p202c24: 8
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
p202c24: 9
Equipotential surfaces
A surface in space on which the potential is the same for
every point.
Surfaces of constant voltage.
p202c24: 10
p202c24: 11
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
p202c24: 12
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 6rv F f ' 6rv'
4 3
Fg mg r
3
FE qE
v
Ff
F = qE
Ff
’
v’
F = mg
F = mg
p202c24: 13
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.
p202c24: 14
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
p202c24: 15