Download - PC Chapter 25
-
8/14/2019 PC Chapter 25
1/58
Chapter 25
Electric Potential
-
8/14/2019 PC Chapter 25
2/58
Electrical Potential Energy When a test charge is placed in an
electric field, it experiences a force F = qoE
The force is conservative
ds is an infinitesimal displacement
vector that is oriented tangent to a path
through space
-
8/14/2019 PC Chapter 25
3/58
Electric Potential Energy, cont The work done by the electric field is
F.ds = qoE.ds
As this work is done by the field, the
potential energy of the charge-field
system is changed by U= -qoE.ds
For a finite displacement of the charge
from A to B,B
B A oA
U U U q d
E s
-
8/14/2019 PC Chapter 25
4/58
Electric Potential Energy, final Because qoE is conservative, the line
integral does not depend on the path
taken by the charge
This is the change in potential energy of
the system
-
8/14/2019 PC Chapter 25
5/58
Electric Potential The potential energy per unit charge, U/qo,
is the electric potential The potential is independent of the value ofqo The potential has a value at every point in an
electric field
The electric potential iso
UV q
-
8/14/2019 PC Chapter 25
6/58
Electric Potential, cont. The potential is a scalar quantity
Since energy is a scalar
As a charged particle moves in an
electric field, it will experience a change
in potentialB
Ao
UV dq E s
-
8/14/2019 PC Chapter 25
7/58
Electric Potential, final The difference in potential is the
meaningful quantity
We often take the value of the potentialto be zero at some convenient point inthe field
Electric potential is a scalarcharacteristic of an electric field,independent of any charges that maybe placed in the field
-
8/14/2019 PC Chapter 25
8/58
Work and Electric Potential Assume a charge moves in an electric
field without any change in its kinetic
energy The work performed on the charge is
W= V= qV
-
8/14/2019 PC Chapter 25
9/58
Units 1 V = 1 J/C
V is a volt
It takes one joule of work to move a 1-coulomb charge through a potentialdifference of 1 volt
In addition, 1 N/C = 1 V/m This indicates we can interpret the electric
field as a measure of the rate of changewith position of the electric potential
-
8/14/2019 PC Chapter 25
10/58
Electron-Volts Another unit of energy that is commonly used
in atomic and nuclear physics is the electron-
volt One electron-volt is defined as the energy a
charge-field system gains or loses when a
charge of magnitude e (an electron or a
proton) is moved through a potentialdifference of 1 volt 1 eV = 1.60 x 10-19 J
-
8/14/2019 PC Chapter 25
11/58
Potential Difference in a
Uniform Field The equations for electric potential can
be simplified if the electric field is
uniform:
The negative sign indicates that the
electric potential at point B is lower thanat pointA
B B
B AA A
V V V d E d Ed E s s
-
8/14/2019 PC Chapter 25
12/58
Energy and the Direction of
Electric Field When the electric field
is directed downward,
point B is at a lowerpotential than pointA
When a positive test
charge moves fromA
to B, the charge-fieldsystem loses potential
energy
-
8/14/2019 PC Chapter 25
13/58
More About Directions A system consisting of a positive charge and
an electric field loses electric potential energywhen the charge moves in the direction of thefield An electric field does work on a positive charge
when the charge moves in the direction of theelectric field
The charged particle gains kinetic energyequal to the potential energy lost by thecharge-field system Another example of Conservation of Energy
-
8/14/2019 PC Chapter 25
14/58
Directions, cont. Ifqo is negative, then Uis positive
A system consisting of a negative
charge and an electric field gains
potential energy when the charge
moves in the direction of the field In order for a negative charge to move in
the direction of the field, an external agent
must do positive work on the charge
-
8/14/2019 PC Chapter 25
15/58
Equipotentials Point B is at a lower
potential than pointA
PointsA and Care at thesame potential
The name equipotential
surface is given to any
surface consisting of a
continuous distribution of
points having the same
electric potential
-
8/14/2019 PC Chapter 25
16/58
Charged Particle in a Uniform
Field, Example A positive charge is
released from rest andmoves in the direction
of the electric field The change in potential
is negative The change in potential
energy is negative The force and
acceleration are in thedirection of the field
-
8/14/2019 PC Chapter 25
17/58
Potential and Point Charges A positive point
charge produces a
field directed radiallyoutward
The potential
difference between
pointsA and B willbe
1 1B A e
B A
V V k qr r
-
8/14/2019 PC Chapter 25
18/58
Potential and Point Charges,
cont. The electric potential is independent of
the path between pointsA and B
It is customary to choose a reference
potential ofV= 0 at rA =
Then the potential at some point ris
e
qV k
r
-
8/14/2019 PC Chapter 25
19/58
Electric Potential of a Point
Charge The electric potential
in the plane around
a single pointcharge is shown
The red line shows
the 1/rnature of the
potential
-
8/14/2019 PC Chapter 25
20/58
-
8/14/2019 PC Chapter 25
21/58
Electric Potential of a Dipole The graph shows
the potential (y-axis)
of an electric dipole The steep slope
between the
charges represents
the strong electricfield in this region
-
8/14/2019 PC Chapter 25
22/58
Potential Energy of Multiple
Charges Consider two
charged particles
The potential energyof the system is
1 2
12
e
q qU k
r
-
8/14/2019 PC Chapter 25
23/58
Active Figure 25.10
(SLIDESHOW MODE ONLY)
-
8/14/2019 PC Chapter 25
24/58
More About Uof Multiple
Charges If the two charges are the same sign, U
is positive and work must be done to
bring the charges together If the two charges have opposite signs,
Uis negative and work is done to keep
the charges apart
-
8/14/2019 PC Chapter 25
25/58
Uwith Multiple Charges, final If there are more than
two charges, then findUfor each pair ofcharges and add them
For three charges:
The result isindependent of the
order of the charges
1 3 2 31 2
12 13 23e
q q q qq qU k
r r r
-
8/14/2019 PC Chapter 25
26/58
Finding E From V Assume, to start, that E has only anx
component
Similar statements would apply to the yand zcomponents
Equipotential surfaces must always beperpendicular to the electric field linespassing through them
x dVEdx
-
8/14/2019 PC Chapter 25
27/58
E and Vfor an Infinite Sheet
of Charge The equipotential lines
are the dashed blue
lines The electric field lines
are the brown lines
The equipotential lines
are everywhere
perpendicular to the
field lines
-
8/14/2019 PC Chapter 25
28/58
E and Vfor a Point Charge The equipotential lines
are the dashed blue
lines The electric field lines
are the brown lines
The equipotential lines
are everywhere
perpendicular to the
field lines
-
8/14/2019 PC Chapter 25
29/58
E and Vfor a Dipole The equipotential lines
are the dashed blue
lines The electric field lines
are the brown lines
The equipotential lines
are everywhere
perpendicular to the
field lines
-
8/14/2019 PC Chapter 25
30/58
Electric Field from Potential,
General In general, the electric potential is a
function of all three dimensions
Given V(x, y, z) you can find Ex, Ey and
Ezas partial derivatives
x y z
V V V
E E E x y z
-
8/14/2019 PC Chapter 25
31/58
Electric Potential for a
Continuous Charge Distribution Consider a small
charge element dq
Treat it as a pointcharge
The potential at
some point due to
this charge elementis
e
dqdV k
r
-
8/14/2019 PC Chapter 25
32/58
Vfor a Continuous Charge
Distribution, cont. To find the total potential, you need to
integrate to include the contributions
from all the elements
This value forVuses the reference ofV=
0 when Pis infinitely far away from the
charge distributions
e
dqV k
r
-
8/14/2019 PC Chapter 25
33/58
Vfor a Uniformly Charged
Ring Pis located on the
perpendicular central
axis of the uniformlycharged ring The ring has a radius
a and a total charge Q
2 2
ee
k QdqV k
r x a
-
8/14/2019 PC Chapter 25
34/58
Vfor a Uniformly Charged
Disk The ring has a
radius a and surface
charge density of
1
2 2 22 eVk x a x
-
8/14/2019 PC Chapter 25
35/58
Vfor a Finite Line of Charge A rod of line has a
total charge ofQ
and a linear chargedensity of
2 2
lnek Q a
Va
l l
l
-
8/14/2019 PC Chapter 25
36/58
Vfor a Uniformly Charged
Sphere A solid sphere of
radius Rand total
charge Q Forr> R,
Forr< R,
e
QV k
r
2 2
32e
D C
k QV V R r R
-
8/14/2019 PC Chapter 25
37/58
Vfor a Uniformly Charged
Sphere, Graph The curve forVD is
for the potentialinside the curve It is parabolic It joins smoothly with
the curve forVB
The curve forVB isfor the potentialoutside the sphere It is a hyperbola
-
8/14/2019 PC Chapter 25
38/58
VDue to a Charged
Conductor Consider two points on
the surface of thecharged conductor as
shown E is always
perpendicular to thedisplacement ds
Therefore, E ds = 0 Therefore, the potential
difference betweenAand B is also zero
-
8/14/2019 PC Chapter 25
39/58
VDue to a Charged
Conductor, cont. Vis constant everywhere on the surface of a
charged conductor in equilibrium V= 0 between any two points on the surface
The surface of any charged conductor inelectrostatic equilibrium is an equipotentialsurface
Because the electric field is zero inside theconductor, we conclude that the electricpotential is constant everywhere inside theconductor and equal to the value at thesurface
-
8/14/2019 PC Chapter 25
40/58
E Compared to V The electric potential is
a function ofr The electric field is a
function ofr2 The effect of a charge
on the spacesurrounding it: The charge sets up a
vector electric field whichis related to the force
The charge sets up ascalar potential which isrelated to the energy
-
8/14/2019 PC Chapter 25
41/58
Irregularly Shaped Objects The charge density is high
where the radius of
curvature is small And low where the radius of
curvature is large
The electric field is large
near the convex points
having small radii ofcurvature and reaches
very high values at sharp
points
-
8/14/2019 PC Chapter 25
42/58
Irregularly Shaped Objects,
cont. The field lines are
still perpendicular to
the conductingsurface at all points
The equipotential
surfaces are
perpendicular to thefield lines
everywhere
-
8/14/2019 PC Chapter 25
43/58
Cavity in a Conductor Assume an
irregularly shaped
cavity is inside aconductor
Assume no charges
are inside the cavity
The electric field
inside the conductor
must be zero
-
8/14/2019 PC Chapter 25
44/58
Cavity in a Conductor, cont The electric field inside does not depend on
the charge distribution on the outside surfaceof the conductor
For all paths betweenA and B,
A cavity surrounded by conducting walls is afield-free region as long as no charges areinside the cavity
0B
B AA
V V d E s
-
8/14/2019 PC Chapter 25
45/58
Corona Discharge If the electric field near a conductor is
sufficiently strong, electrons resulting
from random ionizations of airmolecules near the conductor
accelerate away from their parent
molecules These electrons can ionize additional
molecules near the conductor
-
8/14/2019 PC Chapter 25
46/58
Corona Discharge, cont. This creates more free electrons The corona discharge is the glow that
results from the recombination of thesefree electrons with the ionized airmolecules
The ionization and corona dischargeare most likely to occur near very sharppoints
-
8/14/2019 PC Chapter 25
47/58
Millikan Oil-Drop Experiment
Experimental Set-Up
-
8/14/2019 PC Chapter 25
48/58
Millikan Oil-Drop Experiment Robert Millikan measured e, the
magnitude of the elementary charge on
the electron He also demonstrated the quantized
nature of this charge
Oil droplets pass through a small holeand are illuminated by a light
-
8/14/2019 PC Chapter 25
49/58
Active Figure 25.27
(SLIDESHOW MODE ONLY)
-
8/14/2019 PC Chapter 25
50/58
Oil-Drop Experiment, 2 With no electric field
between the plates,
the gravitationalforce and the drag
force (viscous) act
on the electron
The drop reachesterminal velocity with
FD = mg
-
8/14/2019 PC Chapter 25
51/58
Oil-Drop Experiment, 3 When an electric field
is set up between theplates The upper plate has a
higher potential
The drop reaches anew terminal velocity
when the electricalforce equals the sumof the drag force andgravity
-
8/14/2019 PC Chapter 25
52/58
Oil-Drop Experiment, final The drop can be raised and allowed to
fall numerous times by turning the
electric field on and off After many experiments, Millikan
determined:
q = ne where n = 1, 2, 3, e = 1.60 x 10-19 C
-
8/14/2019 PC Chapter 25
53/58
Van de Graaff
Generator Charge is delivered continuously to
a high-potential electrode by meansof a moving belt of insulating
material The high-voltage electrode is a
hollow metal dome mounted on aninsulated column
Large potentials can be developedby repeated trips of the belt
Protons accelerated through suchlarge potentials receive enoughenergy to initiate nuclear reactions
-
8/14/2019 PC Chapter 25
54/58
Electrostatic Precipitator An application of electrical discharge in
gases is the electrostatic precipitator
It removes particulate matter from
combustible gases The air to be cleaned enters the duct and
moves near the wire
As the electrons and negative ions created
by the discharge are accelerated toward the
outer wall by the electric field, the dirtparticles become charged
Most of the dirt particles are negatively
charged and are drawn to the walls by the
electric field
-
8/14/2019 PC Chapter 25
55/58
Application Xerographic
Copiers The process of xerography is used for
making photocopies
Uses photoconductive materials A photoconductive material is a poor
conductor of electricity in the dark but
becomes a good electric conductor whenexposed to light
-
8/14/2019 PC Chapter 25
56/58
-
8/14/2019 PC Chapter 25
57/58
Application Laser Printer The steps for producing a document on a laser
printer is similar to the steps in the xerographicprocess Steps a, c, and d are the same The major difference is the way the image forms on
the selenium-coated drum A rotating mirror inside the printer causes the beam of the
laser to sweep across the selenium-coated drum The electrical signals form the desired letter in positive
charges on the selenium-coated drum Toner is applied and the process continues as in the
xerographic process
-
8/14/2019 PC Chapter 25
58/58