Transcript Chapter 1 Units and Problem Solving

AP Physics
Chapter 16
Electric Potential, Energy, and
Capacitance
Chapter 16: Electric Potential, Energy,
and Capacitance
16.1
16.2
16.3
16.4
16.5
Electric Potential and Potential
Difference
Equipotential Surfaces and the
Electric Field
Capacitance
Omitted
Capacitors in Series and in Parallel
Homework for Chapter 16
• Read Chapter 16
• HW 16.A: p.537-539: 4,8-13, 16,17, 20-23, 47,48, 51-53, 55-57.
• HW 16.B: p.540-541: 64-70, 84-87, 89, 90.
B L A C K H O L E S A R E WH ERE
D D I V I D E D B Y ZE
RO
GO
If your car is traveling at the speed of light and
you turn your headlights on, what happens?
-Steven Wright
16.1: Electric Potential and
Potential Difference
Electric Potential Energy Difference
a) Moving a positive charge qo against the electric field requires positive work and
increases the electric potential energy.
• The force required to move the charge is equal to the electric force: Fe = qo E
• The work done by the force: Fe d = qo E d
• The increase in the charge’s electric potential energy is equal to the work done
on the charge: ΔUe = UB – UA = qo E d
• The SI unit of electric potential energy is the joule (J).
b) Moving a mass m against the gravitational field requires positive work and
increases the gravitational potential energy: ΔUg = UB – UA = mgh
Electric Potential Difference
** This is not the same as Electric Potential Energy Difference**
• The electric potential difference (voltage) between two points is the work per
unit positive charge done by an external force in moving charge between these
two points,
OR
the change in electric potential energy per unit positive charge.
ΔV
=
W = ΔUe
qo
qo
where qo is the positive test charge
• The SI Unit of electric potential difference is: joule/coulomb (J/C) or volt (V).
• Electric potential difference is commonly shorted from ΔV to just V.
• Potential difference is defined per unit charge, so it does not depend on the
amount of charge moved (potential energy difference does).
On Gold Sheet
Potential Difference for a Uniform Field Between Two Parallel Plates
• Assume we would like to move a positive test charge, qo ,from the negative to
the positive plate. This move would be against the electric field, would require
work, and would increase the charge’s potential energy. The potential difference
between the plates is:
ΔV
=
W = ΔUe
qo
qo
where qo is the positive test charge and
ΔUe is the potential energy gained
• In a uniform electric field E, the potential difference in moving the charge through
a straight line distance d is:
ΔV
= ΔUe = qo E d = E d
qo
qo
Potential Difference
Parallel Plates
• When talking about electric potential, we always must define the reference
value. Only changes in electric potential (voltage) are meaningful.
ex: The negative plate is commonly assigned a value of zero, so voltage
will be positive.
Accelerating a Charge
a) Moving a proton from the negative to the positive plate increases the
proton’s potential energy.
b) When it is released from the positive plate, the proton accelerates back
toward the negative plate, gaining kinetic energy and losing electric potential
energy.
c) The work done to move a proton between any two points in the electric field
is INDEPENDENT OF PATH.
Positive charges, when released, tend to move toward regions of low potential,
and negative charges tend to move toward regions of high potential.
Example 16.1: An electron initially at rest, is accelerated through an electric
potential difference of 50.0 V.
a) What is the kinetic energy of the electron?
b) What is the speed of the electron?
Example 16.2: A 12-V battery maintains the electric potential difference between
tow parallel metal plates separated by 0.10 m. What is the electric field between
the plates?
Potential Difference Due to Point Charges
• A positive point charge creates
an electric field.
• The electric potential
increases as you move closer
to the positive charge.
• Since rB < rA, B is at a higher
potential than A, and the
potential difference is positive.
Electric potential increases (+ΔV) as we get closer to positive charges or farther
away from negative charges.
Electric potential decreases (-ΔV) as we get farther away from positive charges
or nearer to negative charges.
On Gold Sheet
Electric Potential
V = kq electric potential due to a point charge
r ( V = 0 at r = )
• Unlike electric field, electric potential is a scalar quantity. When adding
potentials due to point charges, just add them algebraically (including the + or –
signs).
On Gold Sheet
V = k ∑ i qi
ri
Note: on the gold sheet, Coulomb’s law constant is written as 1/4o.
• Another difference between electric field and electric potential is notable:
electric field is proportional to 1/r2
electric potential is proportional to 1/r
On Gold Sheet
Electric Potential Energy of Various Charge Configurations
a) A positive point charge q1 is fixed in space and a second positive charge q2 is
pushed toward it from a very large distance (r = ) to a distance r12.
There is an increase in potential energy because positive work must be done
to bring the mutually repelling charges closer together.
Electric Potential Energy of Various Charge Configurations
•
For more than two charges, the system’s
electric potential energy is the sum of the
energies of each pair.
•
Electric Potential Energy follows the Law
of Conservation of Energy. For example,
consider two protons that are held near
each other at rest. Work was done to
bring them close, and stored in the form
of electric potential energy. If we release
them, they will fly apart. The electric
potential energy will be converted to
kinetic energy. This is very similar to two
masses connected by a compressed
spring.
Example 16.3: A charge of 5.0 nC is at (0,0) and a second charge of -2.0 nC is at
(3.0m, 0m). If the potential is taken to be zero at infinity,
a) what is the electric potential at point P (0, 4.0) m?
b) what is the potential energy of a 1.0 nC charge at point P?
c) what is the work required to bring a charge of 1.0 nC charge from infinity to
point P?
d) what is the total potential energy of the three charge system?
Summary:
• The electric potential difference between two points is the work per unit positive
charge done by an external force in moving charge between those two points.
• Electric potential difference is the change in electric potential energy per unit
positive charge.
• Voltage is synonymous with electric potential difference.
ΔV
=
W = ΔUe
qo
qo
electric potential difference (voltage) definition
ΔV = Ed
electric potential difference between parallel plates
V = kq
r
electric potential due to a point charge (V = 0 at r = )
U = U12 + U23 + U13 + …
electric potential energy of a configuration of point
charges
Ue = kq1q2
r
electric potential energy (two charges)
Check for Understanding:
1.The SI unit of electric potential difference is the
a) joule
b) newton
c) newton-meter
d) joule per coulomb
Answer: d
2. What is the difference between electrostatic potential energy and electrostatic
potential?
Answer: Potential is the potential energy per unit charge: V = Ue / qo
3. What is the difference between electric potential difference and voltage?
Answer: no difference.
Check for Understanding:
4. The electrostatic potential energy of two point charges
a) is inversely proportional to their separation distance
b) is a vector quantity
c) is always positive
d) has units of newton per coulomb
Answer: a, since potential energy is inversely proportional to the
distance between the charges.
5. An electron is released in a region where there is a varying electric
potential. The electron will
a) move toward the lower potential region
b) move toward the higher potential region
c) remain at rest
Answer: b, because the electron has a negative charge
I lived in a house that ran on static electricity...
If you wanted to run the blender, you had to
rub balloons on your head. If you wanted to
cook, you had to pull off a sweater real quick.
-Steven Wright
16.2: Equipotential Surfaces and
the Electric Field
Construction of Equipotential Surfaces Between Parallel Plates
• Consider a positive charge moving from A to A’
perpendicular to an electric field.
• Since the electric field is perpendicular to the
displacement, no work is done by the field.
• If no work was done, then the electric potential
energy of the charge must not have changed.
• We can conclude all points on path I have the
same electric potential energy, and therefore the
same potential.
• We can extend this to all points on the plane
parallel to the plates containing path I.
• Such a plane, is called an equipotential surface,
or simply• an equipotential.
• Since path II starts and stops on the same equipotential, no work was done.
Equipotential Surfaces Between Parallel Plates
• Once you move to a higher potential (A to B), you can
stay on that new equipotential by moving
perpendicularly to the electric field. (B to B’).
• The change in potential is independent of path, since
the same change occurs via path I as via path II.
Since no work is required to move a charge along an equipotential surface,
then it must be generally true that equipotential surfaces are always at right
angles to the electric field.
Gravitational Potential Energy as an Analogy
• Raising an object away from the earth results in an increase in the object’s
potential energy.
• At a given height, its potential energy is constant as long as it remains on that
equipotential surface.
Topographic Maps: A Gravitational Analogy for Equipotential Surfaces
• Consider a symmetrical hill with slices at different elevations. Each slice is
a plane of constant gravitational potential.
• Here is an overhead view, or topographic
map, of the hill.
• The contours, where the planes intersect the
surface, represent gravitational equipotentials.
Topographic Maps: A Gravitational Analogy for Equipotential Surfaces
• The potential V around a
point charge forms a
symmetrical hill.
• V is constant at fixed
distances from q.
• Electrical equipotentials around a point charge
are spherical, or in two dimensions, circular
slices.
Equipotentials of an Electric Dipole
• Equipotentials are perpendicular to
electric field lines.
• Notice that V1 > V2, since equipotential
surface 1 is closer to the positive charge
than is surface 2.
Animation:
http://regentsprep.org/Regents/physics/phy
s03/aequilines/default.htm
Activity: Learn by Drawing, p. 522.
Electric Field From Potential
• In a uniform electric field, such as one
between two parallel plates, the potential
difference between any two equipotential
planes, separated by a distance ∆x is
∆ V = E ∆x
• For a given travel distance ∆x, movement
perpendicular to the equipotentials vields
maximum potential gain.
• By finding the direction of maximum
potential change, we are finding the direction
opposite that of the E field.
E=- ∆V
∆x
max
electric field
from potential
On Gold Sheet
Electric Field from Potential
E=- ∆V
∆x
max
electric field
from potential
• The minus sign indicates that E is in the direction opposite that in which V
increases most rapidly, or in the direction V decreases most rapidly.
• The units of electric field are volts per meter (V/m). This is dimensionally
equivalent to N/C, which we learned about in Chapter 15. (Prove it!)
Answer: V = J/C = J = N·m = N
m
m
C·m
C·m
C
Example 16.4: Two parallel plates, separated by 0.10 m, are connected to a 6.0 V
battery. An electron is released from rest at the negative plate.
a) What is the E-field between the battery plates?
b) What is the speed of the electron when it arrives at the positive plate?
The Electron Volt
electron volt (eV)
- the kinetic energy acquired by an electron
accelerated through a potential difference of exactly 1 V.
1 eV = 1.60 x 10-19 J
• The electron volt is a convenient way to express typical energies on the atomic
scale.
• The electron volt is a unit of energy, not voltage:
• You may also encounter
e ∆V = q ∆V = ∆Ue
keV: kiloelectron volts, meaning 1000
eV
• MeV: megaelectron volts, meaning 106
eV
• GeV: gigaelectron volts, meaning 109
eV
** Warning – electron volt is not an SI unit. You must convert back to joules before
you can use the number in a formula**
Summary
• Equipotential surfaces (equipotentials) are surfaces on which a charge has a
constant electric potential (V), and constant electric potential energy (Ue).
• Equipotentials are perpendicular to the electric field at all points.
• It takes no work to move a charge along an equipotential.
• E is in the direction opposite that in which V increases most rapidly, or in the
direction in which V decreases most rapidly.
• An electron volt (eV) is the kinetic energy gained by an electron accelerated
from rest through a potential difference of 1 V.
1 eV = 1.60 x 10-19 J
E=- ∆V
∆x
relationship between potential and electric field
max
Check for Understanding
1) Equipotential surfaces are those surfaces on which
a) the potential is constant
b) the electric field is zero
c) the potential is zero
Answer: a
2) Equipotential Surfaces are
a) parallel to the electric field
b) perpendicular to the electric field
c) at any angle with respect to the electric field
Answer: b
Check for Understanding
3) Between charged parallel plates, which equipotentials have a higher potential:
a) the ones near the positive plate
b) the ones near the negative plate
c) the ones near the middle?
Answer: a
4) At a given point on an equipotential surface, the electric field points directly
to the
a) next highest equipotential
b) the next lowest equipotential
c) parallel to the equipotential surface
Answer: b
• HW 16.A: p.537-539: 4,8-13, 16,17, 20-23, 47,48, 51-53, 55-57.
I heard that in relativity theory space and time
are the same thing.
Einstein discovered this when he kept showing
up three miles late for his meetings.
-Steven Wright
16.3: Capacitance
Capacitance
• A capacitor consists of two conductors.
• Capacitors store charge, and therefore electric energy, in the form of an electric field.
• capacitance (C)
is in storing charge.
- a quantitative measure of how effective a capacitor
C=Q
V
definition of capacitance
where Q is the magnitude of the charge on either plate (the plates have
equal but opposite charge), and V is the potential difference across the plates
• Note, from this point on we will use V for ∆V; it means potential difference.
assorted capacitors
Capacitor and Circuit Diagram
• Two metal plates are charged by a battery to a charge Q = CV, where C is the
capacitance.
• Work is done in charging the capacitor, and energy is stored in the electric field.
• Notice the symbols used for a battery (V) and a capacitor (C).
+Q
Metal plates
parallel lines
are equal in
length
the longer line of the battery
symbol is the positive terminal
Capacitance
• The battery works as a pump to remove electrons from the positive plate and
transfer them through the wire to the negative plate.
•The battery charges the capacitor until the potential difference between the
plates is equal to the voltage of the battery.
• When the battery is disconnected from the capacitor, the electric potential
energy is stored in the electric field. This stored energy can then be used to do
work.
• The SI unit of capacitance is coulomb per volt (C/V), or farad (F).
It is more common to see the microfarad ( 1 F = 10-6 F)
or the picofarad (1 pF = 10-12 F)
•The farad was named for the English scientist Michael Faraday (1791-1867), an
early investigator of electrical phenomena who first introduced the concept of the
electric field.
Capacitance
• Capacitance depends only on the size, shape and spacing of the plate
arrangement, as well at the material between the plates (dielectric).
• A common capacitor is the parallel plate capacitor. It consists of two metal plates
of area A and separated by a distance d. The formula is:
On Gold Sheet
C = o A
d
capacitance of a parallel-plate
capacitor (in air)
• o is the permittivity of free space. It is a fundamental constant which describes
the electrical properties of a vacuum. Its value in air is essentially the same.
o
•
= 8.85 x 10-12 C2/(N·m2)
On Blue Sheet
o is related to Coulomb’s constant by:
k=
1
= 9.00 x 109 N·m2/C2
4o
On Blue Sheet
Capacitance
• A plot of charge vs. voltage for a charging capacitor is a straight line with slope C.
slope =
Capacitance
On Gold Sheet
Q
(charge)
Q = CV
( y = mx + b)
Voltage
• The work done by the battery is stored in the capacitor as potential energy, Uc.
• This work is the area under the curve. Therefore, Uc = ½ Q V.
The equivalent forms of this equation are:
Uc =
½ QV =
On Gold Sheet
Q2
2C
=
½ CV2 energy storage
in a capacitor
• The form ½ CV2 is usually the most practical, since the capacitance and the
applied voltage are often known or can be measured most easily.
Example 16.5: A parallel-plate capacitor consists of plates of area 1.5 x 10-4 m2
and separated by 2.0 mm. The capacitor is connected to a 12 volt battery.
a)
b)
c)
d)
What is the capacitance?
What is the charge on the plates?
How much energy is stored in the capacitor?
What is the electric field between the plates?
If you’re not part of the solution, you’re part
of the precipitate.
-Steven Wright
16.5: Capacitors in Series and in
Parallel
• Capacitors can be connected in two basic ways: in series or in parallel.
• In series capacitors are connected “head to tail”.
• In parallel capacitors have all their “heads” hooked together, and all
their “tails” hooked together.
Capacitors in Series
• All capacitors in series have the same charge.
• The sum of the voltage drops is equal to the voltage of the battery.
• The total capacitance is equivalent to Cs, or the equivalent series capacitance.
• Cs is always less than that of the smallest capacitor in the combination.
Capacitors in Series
• When capacitors are wired in series the charge is the same on all the plates.
Q = Q1 = Q2 = Q3 = …
• The voltage drop across all the capacitors must be equal to the voltage across the
battery.
• Therefore, the sum of the individual voltage drops across the capacitors is equal to
the voltage of the battery. V = V1 + V2 + V3 +…
•The individual voltages are related to the individual charges by
V1 = Q, V2 = Q, V3 = Q, … and V = Q
C1
C2
C3
Cs
• Substituting Q/C for V:
Q = Q + Q + Q +…
Cs
C1
C2
C3
• Dividing both sides of the equation by Q:
equivalent series capacitance
1 = 1 + 1 + 1 +...
Cs
C1
C2 C 3
Capacitors in Parallel
• When the capacitors are in parallel, the voltages across the capacitors are
the same.
• The total charge is equal to the sum of the charges on the individual
capacitors.
• The total capacitance is equivalent to Cp.
• Cp is always larger than that of the largest capacitor in the combination.
Capacitors in Parallel
• When capacitors are wired in parallel the voltages across the capacitors are the
same, each equal to the voltage of the battery
V = V1 = V2 = V3 =…
• The total stored charge is equal to the sum of the charges of the individual
capacitors.
Qtotal = Q1 + Q2 + Q3 +…
•The individual charges are given by Q1 = C1V, Q2 = C2V, … and Qtotal = CpV
• Substituting CV for Q:
CpV = C1V + C2V + C3V +…
• Divide both sides of the equation by V:
equivalent parallel capacitance Cp = C1 + C2 + C3 + …
On Gold Sheet
Example 16.7: Capacitors C1 and C2 are in parallel. This combination is in series
with C3. The positive terminal of a 12.0 V battery is connected to C3.
C1 = 6.00 F, C2 = 8.00 F, C3 = 14.0 F
a) What is the equivalent capacitance?
b) What is the charge of each capacitor?
c) What is the voltage across each capacitor?
Summary
•A capacitor stores charge, and therefore electric energy, in the form of an electric
field.
•Capacitance is a quantitative measure of how effective a capacitor is in storing
charge.
•The equivalent series capacitance is always less than that of the smallest
capacitor of the series combination.
•The equivalent parallel capacitance is always larger than that of the largest
capacitor in the parallel combination.
Summary of Equations
C=Q
V
definition of capacitance
C = o A
d
capacitance of a parallel-plate
capacitor (in air)
Uc =
½ QV =
Q2
2C
=
½ CV2
energy in a
charged capacitor
1 = 1 + 1 + 1 +...
Cs
C1
C2 C3
equivalent series capacitance
Cp = C1 + C2 + C3 + …
equivalent parallel capacitance
Check for Understanding
1. Capacitance has units of
a. farads
b. joules
c. coulombs per volt
d. both a. and c.
Answer: d.
2. To increase the capacitance and the energy-storage capability of a parallel-plate
capacitor, we can
a. increase the plate separation distance
b. increase the plate area
c. evacuate the space between the plates
d. none of the above
Answer: b, as the capacitance is directly proportional to the plate area,
C = o A
d
Check for Understanding
3. Capacitors in series have the same
a. voltage
b. charge
c. energy storage
Answer: b
4. Capacitors in parallel have the same
a. voltage
b. charge
c. energy storage
Answer: a
Check for Understanding
5. Under what conditions would two capacitors in series have the same voltage?
Answer: When they are equal in capacitance.
HW 16.B: p.540-541: 64-70, 84-87, 89,90.
Chapter 16 Formulas
Chapter 16 Formulas