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Chapter 20
Electric Potential
and
Capacitance
Electrical Potential Energy



When a point charge, qo, is placed in an
electric field, it experiences a force
 F  q E
o
The force is conservative
To find the work associated with the
force, look at ds, an infinitesimal
displacement vector that is oriented
tangent to a path through space
Electric Potential Energy, cont

The work done by the electric field is
Fe  ds  qoE  ds

As this work is done by the field, the
potential energy of the charge-field
system is changed by DU
For a finite displacement of the charge
from A to B,

B
DU  UB  UA  qo  E  ds
A
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
Electric Potential

The potential energy per unit charge,
U/qo, is the electric potential



The potential is independent of the value of
qo
The potential has a value at every point in
an electric field
U
The electric potential is V 
qo
Electric Potential, cont

The potential is a scalar quantity


The potential is a property of only the field


Since energy is a scalar
Not the charge-field system
As a charged particle moves in an electric
field, it will experience a change in
potential
B
DU
DV 
   E  ds
A
qo
Electric Potential, final

We often take the value of the potential
to be zero at some convenient point in
the field


Sometimes called ground
Electric potential is a scalar
characteristic of an electric field,
independent of any charges that may be
placed in the field
Potential and Potential Energy

The potential is characteristic of the field only



It is independent of the charge placed in the field
The difference in potential is proportional to the
difference in potential energy
Potential energy is characteristic of the
charge-field system

Due to an interaction between the field and a
charged particle placed in the field
Work and Electric Potential

The electric potential at an arbitrary point due
to source charges equals the work required
by an external agent to bring a test charge
from infinity to that point divided by the
charge on the test particle


Assumes a charge moves in an electric field
without any change in its kinetic energy
The work performed on the charge is
W = DV = q DV
Units

1 V = 1 J/C



V is a Volt
It takes one Joule of work to move a 1
Coulomb charge through a potential
difference 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 change
with position of the electric potential
Electron-Volts


Another unit of energy that is commonly used
in atomic and nuclear physics is the electronvolt
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 potential
difference of 1 volt

1 eV = 1.60 x 10-19 J
Potential Difference
in a Uniform Field

The equations for electric potential can
be simplified if the electric field is
uniform:
B
B
A
A
VB  VA  DV   E  ds  E  ds  Ed

The negative sign indicates that the
electric potential at B is lower than at
point A
Energy and the
Direction of Electric Field


When the electric field
is directed downward,
point B is at a lower
potential than point A
When a positive test
charge moves from A
to B, the charge-field
system loses potential
energy
More About Directions

A system consisting of a positive charge and
an electric field loses electric potential
energy when the charge moves in the
direction of the field


An electric field does work on a positive charge
when the charge moves in the direction of the
electric field
The charged particle gains kinetic energy
equal to the potential energy lost by the
charge-field system

Another example of Conservation of Energy
Directions, cont


If qo is negative, then DU is 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
Equipotentials



Point B is at a lower
potential than point A
Points B and C are at
the same potential
The name
equipotential surface
is given to any surface
consisting of a
continuous distribution
of points having the
same electric potential
Charged Particle
in a Uniform Field, Example




A positive charge is
released from rest and
moves 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 the
direction of the field
Potential and Point Charges


A positive point
charge produces a
field directed radially
outward
The potential
difference between
points A and B will
be
1 1
VB  VA  keq   
 rB rA 
Potential and
Point Charges, cont



The electric potential is independent of
the path between points A and B
It is customary to choose a reference
potential of V = 0 at rA = 
Then the potential at some point r is
q
V  ke
r
Electric Potential
with Multiple Charges

The electric potential due to several
point charges is the sum of the
potentials due to each individual charge


This is another example of the
superposition principle
The sum is the algebraic sum
qi
V  ke 
i ri

V = 0 at r = 
Potential Energy
of Multiple Charges


Consider two
charged particles
The potential energy
of the system is
q1q2
U  ke
r12
More About U
of 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,
U is negative and work is done to keep
the charges apart
U with Multiple Charges, final

If there are more than two charges, then
find U for each pair of charges and add
them


Add them algebraically
The total electric potential energy of a
system of point charges is equal to the
work required to bring the charges, one
at a time, from an infinite separate to
their final positions
Potential of a Charge


Now remove one
charge
The potential due to
charge q2 is
U keq2
V 
q1
r12

The equipotential
surfaces for an isolated
point charge are a
family of spheres
concentric with the
charge
Finding E From V

Assume, to start, that E has only an x
component
dV
E  ds becomes E x dx and E x  
dx

Similar statements would apply to the y and z
components
Equipotential surfaces must always be
perpendicular to the electric field lines
passing through them

E and V for 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
E and V for 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
E and V for 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
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
Ez as partial derivatives
V
Ex  
x
V
Ey  
y
V
Ez  
z
Electric Potential for a
Continuous Charge Distribution

Consider a small
charge element dq


Treat it as a point
charge
The potential at
some point due to
this charge element
is
dq
dV  ke
r
V for a Continuous
Charge Distribution, cont

To find the total potential, you need to
integrate to include the contributions
from all the elements
dq
V  ke 
r

This value for V uses the reference of
V = 0 when P is infinitely far away from
the charge distributions
Problem Solving Strategies –
Electric Potentials

Conceptualize


Think about the charges or the charge
distribution
Image the type of potential they would
create


This establishes a mental representation
Use any symmetry in the arrangement of
the charges to help you visualize the
potential
Problem Solving Strategies –
Electric Potentials

Categorize


Individual charges or a distribution?
Analyze





Scalar, so no components
Superposition principle is algebraic sum
Signs are important
Changes in potential are what is important
The point where V = 0 is arbitrary

But usually at a point infinitely far from the charges
Problem Solving Strategies –
Electric Potentials, cont

Analyze, cont



For a group of individual charges, use the
superposition principle
For a continuous charge distribution,
integrate over the entire distribution
If E is known, the line integral of E  ds
can be evaluated
Problem Solving Strategies –
Electric Potentials, final

Finalize



Check to see if your result is consistent
with the mental representation
Be sure the result reflects any symmetry
you noted
Image varying parameters to see if the
mathematical result changes in a
reasonable way
V for a Uniformly
Charged Ring

P is located on the
perpendicular
central axis of the
uniformly charged
ring

The ring has a radius
a and a total charge
Q
dq
V  ke 

r
k eQ
2
2
x a
V for a Uniformly
Charged Sphere



A solid sphere of
radius R and total
charge Q
Q
For r > R, V  k e
r
For r < R,

keQ 2
2
VD  VC 
R

r
2R 3
keQ 
r2 
VD 
3  2 
3R 
R 

V for a Uniformly
Charged Sphere, Graph

The curve for VD is
for the potential
inside the curve



It is parabolic
It joins smoothly with
the curve for VB
The curve for VB is
for the potential
outside the sphere

It is a hyperbola
V Due to a
Charged Conductor




Consider two points on
the surface of the
charged conductor as
shown
E is always
perpendicular to to the
displacement ds
Therefore, E  ds = 0
Therefore, the potential
difference between A
and B is also zero
V Due to a
Charged Conductor, cont

V is constant everywhere on the surface of a
charged conductor in equilibrium



DV = 0 between any two points on the surface
The surface of any charged conductor in
electrostatic equilibrium is an equipotential
surface
Because the electric field is zero inside the
conductor, we conclude that the electric
potential is constant everywhere inside the
conductor and equal to the value at the
surface
E Compared to V



The electric potential is
a function of r
The electric field is a
function of r2
The effect of a charge
on the space
surrounding it


The charge sets up a
vector electric field which
is related to the force
The charge sets up a
scalar potential which is
related to the energy
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
of curvature and
reaches very high
values at sharp points
Cavity in a Conductor



Assume an
irregularly shaped
cavity is inside a
conductor
Assume no charges
are inside the cavity
The electric field
inside the conductor
is must be zero
Cavity in a Conductor, cont


The electric field inside does not
depend on the charge distribution on
the outside surface of the conductor
For all paths between A and B,
VB  VA    E  ds  0

A cavity surrounded by conducting walls
is a field-free region as long as no
charges are inside the cavity
Capacitors


Capacitors are devices that store
electric charge
The capacitor is the first example of a
circuit element

A circuit generally consists of a number of
electrical components (called circuit
elements) connected together by
conducting wires forming one or more
closed loops
Definition of Capacitance

The capacitance, C, of a capacitor is
defined as the ratio of the magnitude of
the charge on either conductor to the
potential difference between the
conductors
Q
C
DV

The SI unit of capacitance is a farad (F)
Makeup of a Capacitor

A capacitor consists of
two conductors



When the conductors are
charged, they carry
charges of equal
magnitude and opposite
directions
A potential difference
exists between the
conductors due to the
charge
The capacitor stores
charge
More About Capacitance





Capacitance will always be a positive quantity
The capacitance of a given capacitor is
constant
The capacitance is a measure of the
capacitor’s ability to store charge
The Farad is a large unit, typically you will
see microfarads (mF) and picofarads (pF)
The capacitance of a device depends on the
geometric arrangement of the conductors
Capacitance –
Isolated Sphere


Assume a spherical charged conductor
Assume V = 0 at infinity
Q
Q
R
C


 4 o R
DV keQ / R ke

Note, this is independent of the charge
and the potential difference
Parallel Plate Capacitor


Each plate is
connected to a
terminal of the
battery
If the capacitor is
initially uncharged,
the battery
establishes an
electric field in the
connecting wires
Capacitance – Parallel Plates

The charge density on the plates is
s = Q/A



A is the area of each plate, which are equal
Q is the charge on each plate, equal with
opposite signs
The electric field is uniform between the
plates and zero elsewhere
Capacitance –
Parallel Plates, cont.

The capacitance is proportional to the
area of its plates and inversely
proportional to the plate separation
o A
Q
Q
Q
C



DV Ed Qd /  o A
d
Parallel Plate Assumptions


The assumption that the electric field is uniform is
valid in the central region, but not at the ends of the
plates
If the separation between the plates is small
compared with the length of the plates, the effect of
the non-uniform field can be ignored
Energy in a
Capacitor – Overview



Consider the circuit to
be a system
Before the switch is
closed, the energy is
stored as chemical
energy in the battery
When the switch is
closed, the energy is
transformed from
chemical to electric
potential energy
Energy in a
Capacitor – Overview, cont


The electric potential energy is related
to the separation of the positive and
negative charges on the plates
A capacitor can be described as a
device that stores energy as well as
charge
Capacitance of a
Cylindrical Capacitor



From Gauss’ Law,
the field between
the cylinders is
E = 2 ke l / r
DV = -2 ke l ln (b/a)
The capacitance
becomes
Q
C

DV 2ke ln b
a


Circuit Symbols




A circuit diagram is a
simplified
representation of an
actual circuit
Circuit symbols are
used to represent the
various elements
Lines are used to
represent wires
The battery’s positive
terminal is indicated by
the longer line
Capacitors in Parallel

When capacitors are
first connected in
the circuit, electrons
are transferred from
the left plates
through the battery
to the right plate,
leaving the left plate
positively charged
and the right plate
negatively charged
Capacitors in Parallel, 2



The flow of charges ceases when the voltage
across the capacitors equals that of the
battery
The capacitors reach their maximum charge
when the flow of charge ceases
The total charge is equal to the sum of the
charges on the capacitors


Q = Q 1 + Q2
The potential difference across the capacitors
is the same

And each is equal to the voltage of the battery
Capacitors in Parallel, 3

The capacitors can
be replaced with
one capacitor with a
capacitance of Ceq

The equivalent
capacitor must have
exactly the same
external effect on the
circuit as the original
capacitors
Capacitors in Parallel, final


Ceq = C1 + C2 + …
The equivalent capacitance of a parallel
combination of capacitors is the
algebraic sum of the individual
capacitances and is larger than any of
the individual capacitances
Capacitors in Series

When a battery is
connected to the
circuit, electrons are
transferred from the
left plate of C1 to the
right plate of C2
through the battery
Capacitors in Series, 2


As this negative charge accumulates on
the right plate of C2, an equivalent
amount of negative charge is removed
from the left plate of C2, leaving it with
an excess positive charge
All of the right plates gain charges of
–Q and all the left plates have charges
of +Q
Capacitors in
Series, 3


An equivalent capacitor
can be found that
performs the same
function as the series
combination
The potential differences
add up to the battery
voltage
Capacitors in Series, final
Q  Q1  Q2 
DV  V1  V2 
1
1
1



Ceq C1 C2

The equivalent capacitance of a series
combination is always less than any
individual capacitor in the combination
Summary and Hints

Be careful with the choice of units



In SI, capacitance is in F, distance is in m and the
potential differences in V
Electric fields can be in V/m or N/c
When two or more capacitors are connected
in parallel, the potential differences across
them are the same


The charge on each capacitor is proportional to its
capacitance
The capacitors add directly to give the equivalent
capacitance
Summary and Hints, cont

When two or more capacitors are
connected in series, they carry the
same charge, but the potential
differences across them are not the
same

The capacitances add as reciprocals and
the equivalent capacitance is always less
than the smallest individual capacitor
Equivalent
Capacitance, Example



The 1.0mF and 3.0mF are in parallel as are the 6.0mF and
2.0mF
These parallel combinations are in series with the
capacitors next to them
The series combinations are in parallel and the final
equivalent capacitance can be found
Energy Stored in a Capacitor


Assume the capacitor is being charged
and, at some point, has a charge q on it
The work needed to transfer a charge
from one plate to the other is
q
dW  DVdq  dq
C

The total work required is
W 
Q
0
q
Q2
dq 
C
2C
Energy, cont




The work done in charging the capacitor
appears as electric potential energy U
Q2 1
1
U
 QDV  C( DV )2
2C 2
2
This applies to a capacitor of any geometry
The energy stored increases as the charge
increases and as the potential difference
increases
In practice, there is a maximum voltage
before discharge occurs between the plates
Energy, final



The energy can be considered to be
stored in the electric field
For a parallel plate capacitor, the energy
can be expressed in terms of the field
as U = ½ (oAd)E2
It can also be expressed in terms of the
energy density (energy per unit volume)
uE = ½ o E2
Capacitors with Dielectrics

A dielectric is an insulating material that,
when placed between the plates of a
capacitor, increases the capacitance


Dielectrics include rubber, plastic, or waxed paper
With a dielectric, C = κCo


The capacitance is multiplied by the factor κ when
the dielectric completely fills the region between
the plates
For a parallel plate capacitor, this becomes C = =
κεo(A/d)
Dielectrics, cont


In theory, d could be made very small to
create a very large capacitance
In practice, there is a limit to d


d is limited by the electric discharge that could
occur though the dielectric medium separating the
plates
For a given d, the maximum voltage that can
be applied to a capacitor without causing a
discharge depends on the dielectric strength
of the material
Dielectrics, final

Dielectrics provide the following
advantages



Increase in capacitance
Increase the maximum operating voltage
Possible mechanical support between the
plates


This allows the plates to be close together
without touching
This decreases d and increases C
Dielectrics – An Atomic View


The molecules that
make up the
dielectric are
modeled as dipoles
The molecules are
randomly oriented in
the absence of an
electric field
Dielectrics –
An Atomic View, cont



An external electric
field is applied
This produces a
torque on the
molecules
The molecules
partially align with
the electric field
Dielectrics –
An Atomic View, final


An external field can
polarize the dielectric
whether the molecules
are polar or nonpolar
The charged edges of
the dielectric act as a
second pair of plates
producing an induced
electric field in the
direction opposite the
original electric field
Table of
Some Dielectric Values
Types of Capacitors – Tubular


Metallic foil may be
interlaced with thin
sheets of paper or
Mylar
The layers are rolled
into a cylinder to
form a small
package for the
capacitor
Types of Capacitors –
Oil Filled


Common for high
voltage capacitors
A number of
interwoven metallic
plates are immersed
in silicon oil
Types of Capacitors – Variable



Variable capacitors
consist of two
interwoven sets of
metallic plates
One plate is fixed and
the other is moveable
The capacitor generally
vary between 10 and
500 pF
Types of Capacitors –
Electrolytic


Is used to store
large amounts of
charge at relatively
low voltages
The electrolyte is a
solution that
conducts electricity
by virtue of motion
of ions contained in
the solution
The Atmosphere as a
Capacitor



A negative charge
occurs on the
Earth’s surface
Positive charges are
distributed through
the atmosphere
This separation of
charge can be
modeled as a
capacitor
Atmosphere, cont.


The charge distribution on the surface is
assumed to be spherically symmetric
The effective distance between the
plates is 5 km


Based on modeling the charge distribution
in the atmosphere
The capacitance of this system is
approximately 0.9 F