Chapter 24
Capacitance and
Dielectrics
Lecture 1
Dr. Armen Kocharian
Capacitors
„
„
Capacitors are devices that store
electric charge
Examples of where capacitors are used
include:
„
„
„
radio receivers
filters in power supplies
energy-storing devices in electronic flashes
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=
ΔV
„
The SI unit of capacitance is the farad (F)
Makeup of a Capacitor
„
A capacitor consists of
two conductors
„
„
„
These conductors are
called plates
When the conductor is
charged, the plates carry
charges of equal
magnitude and opposite
directions
A potential difference
exists between the plates
due to the charge
C=Q/Vab Constant
+Q
conductor
a
some random path
conductor
b
-Q
„
„
„
If Q doubles (triples, quadruples...), the field
doubles (triples, quadruples...)
Then Vab also doubles (triples, quadruples...)
But C=Q/Vab remains the same
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 (μF) and picofarads (pF)
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
Parallel Plate Capacitor, cont
„
„
„
This field applies a force on electrons in the
wire just outside of the plates
The force causes the electrons to move onto
the negative plate
This continues until equilibrium is achieved
„
„
The plate, the wire and the terminal are all at the
same potential
At this point, there is no field present in the
wire and the movement of the electrons
ceases
Parallel Plate Capacitor, final
„
„
„
The plate is now negatively charged
A similar process occurs at the other
plate, electrons moving away from the
plate and leaving it positively charged
In its final configuration, the potential
difference across the capacitor plates is
the same as that between the terminals
of the battery
Capacitance – Isolated
Sphere
„
„
Assume a spherical charged conductor
Assume V = 0 at infinity
Q
Q
R
C=
=
=
= 4πεoR
ΔV keQ / R ke
„
Note, this is independent of the charge
and the potential difference
Capacitance – Parallel Plates
„
The charge density on the plates is
σ = 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 distance between the
plates
εo A
Q
Q
Q
C=
=
=
=
ΔV Ed Qd / εo 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’s Law,
the field between
the cylinders is
E = 2keλ / r
ΔV = -2keλ ln (b/a)
The capacitance
becomes
Q
C=
=
ΔV 2ke ln ( b / a )
Capacitance of a Spherical
Capacitor
b
b
b
dr
⎡ 1⎤
⎛ 1 1⎞
Vb −Va = −∫ Er dr = − keQ∫ 2 =keQ ⎢ ⎥ = keQ⎜ − ⎟
r
⎣ r ⎦a
⎝b a⎠
a
a
The potential
difference will be
⎛ 1 1⎞
ΔV = keQ ⎜ − ⎟
⎝b a⎠
„ The capacitance will
be
1
Q
ab
=
=
C=
ΔV
⎛ 1 1 ⎞ ke ( b − a )
ke ⎜ − ⎟
⎝a b⎠
„
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
Connecting capacitors
together
Two ways of connecting capacitors together:
Vb
Va
in parallel
Vb
Va
in series
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
„
„
Qtotal = Q1 + 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 greater
than any of the individual capacitors
„
Essentially, the areas are combined
Equivalent Capacitance,
Example
„
„
„
The 1.0-μF and 3.0-μF capacitors are in parallel as are the
6.0-μF and 2.0-μF capacitors
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
Capacitors in parallel
The potential difference across the two capacitors is the same
Q1 = C1Vab and Q2 = C2 Vab
Therefore, Q=Q1+Q2 = (C1 + C2) Vab
This is equivalent to
equivalent capacitance
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 + …
ΔV = 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
Capacitors in series
These two plates are
connected
The two connected plates effectively form a single conductor
Thus, the two connected plates have equal and opposite charge
Capacitors in series (cont.)
Va
Q
-Q
Q
-Q Vb
Remember, definition:
Thus, this is entirely equivalent to
Va
-Q
Q
Vb
Ceq
ui
q
e
a
p
a
tc
n
e
val
ce
n
a
cit
For more than two capacitors in
parallel or in serees the results
generalize to
Problem-Solving Hints
„
Be careful with the choice of units
„
„
„
In SI, capacitance is in farads, distance is in meters
and the potential differences are in volts
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
Problem-Solving 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
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 = ΔVdq = 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=
= Q ΔV = C ( Δ V ) 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 (εoAd)E2
It can also be expressed in terms of the
energy density (energy per unit volume)
uE = ½ εoE2
Example
C2
C1
C3
Find the equivalent capacitance
of this network.
The trick here is to take it one step at a time
C1 and C3 are in series. So this circuit is equivalent to
C4
C3
Then, this is equivalent to
Ceq
C4
C3
C1
C2
Another example
Find the equivalent capacitance
of this network.
Again, take it in steps. C1 and C2 are in series.
So this is equivalent to
C4
C3
C5
C4
C5
C3
Now this looks a little different than what we have seen.
But it is just three capacitors in parallel. We can redraw it as
C3
C4
C5
which is equivalent to
Ceq
Energy stored in a capacitor
„
„
„
A capacitor stores potential energy
By conservation of energy, the stored
energy is equal to the work done in
charging up the capacitor
Our goal now is to calculate this work,
and thus the amount of energy stored in
the capacitor
„
„
Once the capacitor is charged
Let q and v be the charge and potential of the
capacitor at some instant while it is being charged
„
„
q<Q and v<V, but still v=q/C
If we want to increase the charge from q Æ q+dq,
we need to do an amount of work dW
„
The total work done in charging up the capacitor is
„
Potential energy stored in the capacitor is
Energy in the electric field
„
„
„
If a capacitor is charged, there is an electric
field between the two conductors
We can think of the energy of the capacitor
as being stored in the electric field
For a parallel plate capacitor, ignoring edge
effects, the volume over which the field is
active is Axd
„
Then, the energy per unit volume (energy density) is
„
But the capacitance and electric field are given by
„
„
Putting it all together:
This is the energy density (energy per unit volume)
associated with an electric field
„
Derived it for parallel plate capacitor, but valid in general
Problem
„
Capacitors C1 = 6 μF, C2 = 3 μF
and ΔV= 20 V are given. Capacitor
C1 are first charged by closing
switch S1 . Switch S1 then is
opened and the charged capacitor
is connected to uncharged
capacitor C2 by closing switch S2
(C1>C2) Find the initial charge
acquired by C1 and the final charge
on each capacitor.
Example
Q1i = Q initial charge of C1 = C1V
Q1f = final charge of C1
Q2f = final charge of C2
Charge Qtotal = Q1i + Q2i
Q
C=
ΔV
After we close the switches, this charge
Will distribute itself partially on C1 and
partiallyon C2, but with Qtotal = Q1f + Q2f
Q
6.00 × 10−6 =
20.0
Q = 120 μ C
Q1 + Q2 = Q
ΔV1 = ΔV
Q1 Q2
=
C1 C2
Q1 =120μC−Q2
Q
ΔV =
C
120− Q 2 Q 2
=
C1
C2
( 3.00 )(120 − Q2 ) = ( 6.00 ) Q2
360
= 40.0 μ C
Q2 =
9.00
Q1 = 120μC −40.0μC = 80.0μC
Example
„
C1 and C2 (C1>C2) are both charged to potential V, but with
opposite polarity. They are removed from the battery, and
are connected as shown. Then we close the two switches
Find Vab after the switches have been closed
Q1i = initial charge of C1 = C1V
Q2i = initial charge of C2 = - C2V
ÆCharge Qtotal = Q1i + Q2i = (C1-C2)V
-
+
After we close the switches, this charge will
distribute itself partially on C1 and partially
on C2, but with Qtotal = Q1f + Q2f
+Q1f
+Q2f
-Q1f
-Q2f
Qtotal = Q1i + Q2i = (C1-C2)V=Q1f + Q2f
Q1f = C1 Vab
Q2f = C2 Vab
Æ
Q1f + Q2f = (C1 + C2) Vab
Then, equating the two boxed equations
Now calculate the energy before and after
„
„
Ebefore = ½ C1 V2 + ½ C2 V2 = ½ (C1 + C2) V2
Eafter = ½ Ceq Vab, where Ceq is the equivalent
capacitance of the circuit after the switches have been
closed
„ C1 and C2 are in parallel
Æ
Ceq = C1 + C2
Æ Eafter = ½ (C1 + C2) Vab
What happens to conservation of energy????
It turns out that some of the energy is radiated as electromagnetic waves!!
Some Uses of Capacitors
„
Defibrillators
„
„
„
When fibrillation occurs, the heart produces a
rapid, irregular pattern of beats
A fast discharge of electrical energy through the
heart can return the organ to its normal beat
pattern
In general, capacitors act as energy
reservoirs that can be slowly charged and
then discharged quickly to provide large
amounts of energy in a short pulse
Capacitors with Dielectrics
„
A dielectric is a nonconducting material
that, when placed between the plates of
a capacitor, increases the capacitance
„
„
Dielectrics include rubber, plastic, and
waxed paper
For a parallel-plate capacitor, C = κCo =
κεo(A/d)
„
The capacitance is multiplied by the factor
κ when the dielectric completely fills the
region between the plates
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
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 highvoltage capacitors
A number of
interwoven metallic
plates are immersed
in silicon oil
Types of Capacitors –
Electrolytic
„
„
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
Types of Capacitors –
Variable
„
„
„
„
Variable capacitors
consist of two
interwoven sets of
metallic plates
One plate is fixed and
the other is movable
These capacitors
generally vary between
10 and 500 pF
Used in radio tuning
circuits
Capacitor types
„
„
„
Capacitors are often classified by the materials
used between electrodes
Some types are air, paper, plastic film, mica,
ceramic, electrolyte, and tantalum
Often you can tell them apart by the packaging
Plastic Film Capacitor
Ceramic Capacitor
Tantalum Capacitor
Electrolyte Capacitor
Electric Dipole
„
„
„
An electric dipole consists
of two charges of equal
magnitude and opposite
signs
The charges are
separated by 2a
The electric dipole
moment (p) is directed
along the line joining the
charges from –q to +q
Electric Dipole, 2
„
„
The electric dipole moment has a
magnitude of p = 2aq
Assume the dipole is placed in a
uniform external field, E
„
„
E is external to the dipole; it is not the field
produced by the dipole
Assume the dipole makes an angle θ
with the field
Electric Dipole, 3
„
„
„
Each charge has a
force of F = Eq
acting on it
The net force on the
dipole is zero
The forces produce
a net torque on the
dipole
Electric Dipole, final
„
„
„
The magnitude of the torque is:
τ = 2Fa sin θ = pE sin θ
The torque can also be expressed as the
cross product of the moment and the field:
τ=pxE
The potential energy can be expressed as a
function of the orientation of the dipole with
the field:
Uf – Ui = pE(cos θi – cos θf) →
U = - pE cos θ = - p · E
Polar vs. Nonpolar Molecules
„
„
„
Molecules are said to be polarized when a
separation exists between the average
position of the negative charges and the
average position of the positive charges
Polar molecules are those in which this
condition is always present
Molecules without a permanent polarization
are called nonpolar molecules
Water Molecules
„
„
„
A water molecule is
an example of a
polar molecule
The center of the
negative charge is
near the center of
the oxygen atom
The x is the center
of the positive
charge distribution
Polar Molecules and Dipoles
„
„
The average positions of the positive
and negative charges act as point
charges
Therefore, polar molecules can be
modeled as electric dipoles
Induced Polarization
„
„
„
A symmetrical molecule
has no permanent
polarization (a)
Polarization can be
induced by placing the
molecule in an electric
field (b)
Induced polarization is the
effect that predominates in
most materials used as
dielectrics in capacitors
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,
2
„
„
„
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,
3
„
„
The degree of alignment of the
molecules with the field depends on
temperature and the magnitude of the
field
In general,
„
„
the alignment increases with decreasing
temperature
the alignment increases with increasing
field strength
Dielectrics – An Atomic View,
4
„
„
„
If the molecules of the dielectric are
nonpolar molecules, the electric field
produces some charge separation
This produces an induced dipole
moment
The effect is then the same as if the
molecules were polar
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
Induced Charge and Field
„
„
„
The electric field due to the
plates is directed to the right
and it polarizes the dielectric
The net effect on the
dielectric is an induced
surface charge that results
in an induced electric field
If the dielectric were
replaced with a conductor,
the net field between the
plates would be zero
Geometry of Some Capacitors
Chapter 26
Capacitance and
Dielectrics
Quick Quiz 26.1
A capacitor stores charge Q at a potential difference ΔV. If
the voltage applied by a battery to the capacitor is doubled to
2ΔV:
(a) the capacitance falls to half its initial value and the
charge remains the same
(b) the capacitance and the charge both fall to half their
initial values
(c) the capacitance and the charge both double
(d) the capacitance remains the same and the charge doubles
Quick Quiz 26.1
Answer: (d). The capacitance is a property of the physical
system and does not vary with applied voltage. According to
Equation 26.1, if the voltage is doubled, the charge is
doubled.
Quick Quiz 26.2
Many computer keyboard buttons are constructed of capacitors, as
shown in the figure below. When a key is pushed down, the soft
insulator between the movable plate and the fixed plate is
compressed. When the key is pressed, the capacitance
(a) increases
(b) decreases
(c) changes in a way that we
cannot determine because the
complicated electric circuit
connected to the keyboard
button may cause a change
in ΔV.
Quick Quiz 26.2
Answer: (a). When the key is pressed, the plate separation is
decreased and the capacitance increases. Capacitance
depends only on how a capacitor is constructed and not on
the external circuit.
Quick Quiz 26.3
Two capacitors are identical. They can be connected in
series or in parallel. If you want the smallest equivalent
capacitance for the combination, you should connect them in
(a) series
(b) parallel
(c) Either combination has the same capacitance.
Quick Quiz 26.3
Answer: (a). When connecting capacitors in series, the
inverses of the capacitances add, resulting in a smaller
overall equivalent capacitance.
Quick Quiz 26.4
Consider the two capacitors in question 3 again. Each
capacitor is charged to a voltage of 10 V. If you want the
largest combined potential difference across the
combination, you should connect them in
(a) series
(b) parallel
(c) Either combination has the same potential difference.
Quick Quiz 26.4
Answer: (a). When capacitors are connected in series, the
voltages add, for a total of 20 V in this case. If they are
combined in parallel, the voltage across the combination is
still 10 V.
Quick Quiz 26.5
You have three capacitors and a battery. In which of the
following combinations of the three capacitors will the
maximum possible energy be stored when the combination
is attached to the battery?
(a) series
(b) parallel
(c) Both combinations will store the same amount of energy.
Quick Quiz 26.5
Answer: (b). For a given voltage, the energy stored in a
capacitor is proportional to C: U = C(ΔV)2/2. Thus, you
want to maximize the equivalent capacitance. You do this by
connecting the three capacitors in parallel, so that the
capacitances add.
Quick Quiz 26.6
You charge a parallel-plate capacitor, remove it from the
battery, and prevent the wires connected to the plates from
touching each other. When you pull the plates apart to a
larger separation, do the following quantities increase,
decrease, or stay the same? (a) C; (b) Q; (c) E between the
plates; (d) ΔV ; (e) energy stored in the capacitor.
Quick Quiz 26.6
Answer: (a) C decreases (Eq. 26.3). (b) Q stays the same
because there is no place for the charge to flow. (c) E
remains constant (see Eq. 24.8 and the paragraph following
it). (d) ΔV increases because ΔV = Q/C, Q is constant (part
b), and C decreases (part a). (e) The energy stored in the
capacitor is proportional to both Q and ΔV (Eq. 26.11) and
thus increases. The additional energy comes from the work
you do in pulling the two plates apart.
Quick Quiz 26.7
Repeat Quick Quiz 26.6, but this time answer the questions
for the situation in which the battery remains connected to
the capacitor while you pull the plates apart.
Quick Quiz 26.7
Answer: (a) C decreases (Eq. 26.3). (b) Q decreases. The
battery supplies a constant potential difference ΔV; thus,
charge must flow out of the capacitor if C = Q /ΔV is to
decrease. (c) E decreases because the charge density on the
plates decreases. (d) ΔV remains constant because of the
presence of the battery. (e) The energy stored in the
capacitor decreases (Eq. 26.11).
Quick Quiz 26.8
If you have ever tried to hang a picture or a mirror, you
know it can be difficult to locate a wooden stud in which to
anchor your nail or screw. A carpenter’s stud-finder is
basically a capacitor with its plates arranged side by side
instead of facing one another, as shown in the figure below.
When the device is moved over a stud, the capacitance will:
(a) increase
(b) decrease
Quick Quiz 26.8
Answer: (a). The dielectric constant of wood (and of all
other insulating materials, for that matter) is greater than 1;
therefore, the capacitance increases (Eq. 26.14). This
increase is sensed by the stud-finder's special circuitry,
which causes an indicator on the device to light up.
Quick Quiz 26.9
A fully charged parallel-plate capacitor remains connected
to a battery while you slide a dielectric between the plates.
Do the following quantities increase, decrease, or stay the
same? (a) C; (b) Q; (c) E between the plates; (d) ΔV.
Quick Quiz 26.9
Answer: (a) C increases (Eq. 26.14). (b) Q increases.
Because the battery maintains a constant ΔV, Q must
increase if C increases. (c) E between the plates remains
constant because ΔV = Ed and neither ΔV nor d changes.
The electric field due to the charges on the plates increases
because more charge has flowed onto the plates. The
induced surface charges on the dielectric create a field that
opposes the increase in the field caused by the greater
number of charges on the plates (see Section 26.7). (d) The
battery maintains a constant ΔV.
The positive charge is the end
view of a positively charged
glass rod. A negatively charged
particle moves in a circular arc
around the glass rod. Is the work
done on the charged particle by
the rod’s electric field positive,
negative or zero?
1. Positive
2. Negative
3. Zero
The positive charge is the end
view of a positively charged
glass rod. A negatively charged
particle moves in a circular arc
around the glass rod. Is the work
done on the charged particle by
the rod’s electric field positive,
negative or zero?
1. Positive
2. Negative
3. Zero
Rank in order, from largest to smallest, the potential
energies Ua to Ud of these four pairs of charges.
Each + symbol represents the same amount of charge.
1.
2.
3.
4.
5.
Ua = Ub > Uc = Ud
Ua = Uc > Ub = Ud
Ub = Ud > Ua = Uc
Ud > Ub = Uc > Ua
Ud > Uc > Ub > Ua
Rank in order, from largest to smallest, the potential
energies Ua to Ud of these four pairs of charges.
Each + symbol represents the same amount of charge.
1.
2.
3.
4.
5.
Ua = Ub > Uc = Ud
Ua = Uc > Ub = Ud
U b = Ud > Ua = Uc
Ud > Ub = Uc > Ua
Ud > Uc > Ub > Ua
A proton is released
from rest at point B,
where the potential
is 0 V. Afterward,
the proton
1. moves toward A with an increasing speed.
2. moves toward A with a steady speed.
3. remains at rest at B.
4. moves toward C with a steady speed.
5. moves toward C with an increasing speed.
A proton is released
from rest at point B,
where the potential
is 0 V. Afterward,
the proton
1. moves toward A with an increasing speed.
2. moves toward A with a steady speed.
3. remains at rest at B.
4. moves toward C with a steady speed.
5. moves toward C with an increasing speed.
Rank in order, from
largest to smallest, the
potentials Va to Ve at the
points a to e.
1.
2.
3.
4.
5.
Va = Vb = Vc = Vd = Ve
Va = Vb > Vc > Vd = Ve
Vd = Ve > Vc > Va = Vb
Vb = Vc = Ve > Va = Vd
Va = Vb = Vd = Ve > Vc
Rank in order, from
largest to smallest, the
potentials Va to Ve at the
points a to e.
1.
2.
3.
4.
5.
Va = Vb = Vc = Vd = Ve
Va = Vb > Vc > Vd = Ve
Vd = Ve > Vc > Va = Vb
Vb = Vc = Ve > Va = Vd
Va = Vb = Vd = Ve > Vc
Rank in order, from largest to
smallest, the potential
differences ∆V12, ∆V13, and
∆V23 between points 1 and 2,
points 1 and 3, and points 2
and 3.
1.
2.
3.
4.
5.
∆V12 > ∆V13 = ∆V23
∆V13 > ∆V12 > ∆V23
∆V13 > ∆V23 > ∆V12
∆V13 = ∆V23 > ∆V12
∆V23 > ∆V12 > ∆V13
Rank in order, from largest to
smallest, the potential
differences ∆V12, ∆V13, and
∆V23 between points 1 and 2,
points 1 and 3, and points 2
and 3.
1.
2.
3.
4.
5.
∆V12 > ∆V13 = ∆V23
∆V13 > ∆V12 > ∆V23
∆V13 > ∆V23 > ∆V12
∆V13 = ∆V23 > ∆V12
∆V23 > ∆V12 > ∆V13
Chapter 29
Reading Quiz
What are the units of potential difference?
1. Amperes
2. Potentiometers
3. Farads
4. Volts
5. Henrys
What are the units of potential difference?
1. Amperes
2. Potentiometers
3. Farads
4. Volts
5. Henrys
New units of the electric field were introduced in
this chapter. They are:
1. V/C.
2. N/C.
3. V/m.
4. J/m2.
5. W/m.
New units of the electric field were introduced in
this chapter. They are:
1. V/C.
2. N/C.
3. V/m.
4. J/m2.
5. W/m.
The electric potential inside a capacitor
1. is constant.
2. increases linearly from the negative to
the positive plate.
3. decreases linearly from the negative to
the positive plate.
4. decreases inversely with distance from
the negative plate.
5. decreases inversely with the square of the
distance from the negative plate.
The electric potential inside a capacitor
1. is constant.
2. increases linearly from the negative to
the positive plate.
3. decreases linearly from the negative to
the positive plate.
4. decreases inversely with distance from
the negative plate.
5. decreases inversely with the square of the
distance from the negative plate.