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AP Physics Chapter 16Electric Potential, Energy, and
Capacitance
Chapter 16: Electric Potential, Energy, and Capacitance
16.1 Electric Potential and Potential Difference16.2 Equipotential Surfaces and the Electric
Field16.3 Capacitance16.4 Omitted16.5 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 GOD D I V I D E D B Y ZE R O
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,
ORthe change in electric potential energy per unit positive charge.
ΔV = W = ΔUe where qo is the positive test charge qo qo
• 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 where qo is the positive test charge and qo qo 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 Potential Difference qo qo 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.
Positive charges, when released, tend to move toward regions of low potential, and negative charges tend to move toward regions of high potential.
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.
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?
• 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.
Potential Difference Due to Point 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).
V = k ∑ i qi ri
Note: on the gold sheet, Coulomb’s law constant is written as 1/4o.
• 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
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 electric potential difference (voltage) definition
qo qo
ΔV = Ed electric potential difference between parallel plates
V = kq electric potential due to a point charge (V = 0 at r = ) r
U = U12 + U23 + U13 + … electric potential energy of a configuration of point charges
Ue = kq1q2 electric potential energy (two charges)r
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:
1.The SI unit of electric potential difference is the
a) jouleb) newtonc) newton-meterd) joule per coulomb
Answer: d
Check for Understanding:
4. The electrostatic potential energy of two point charges
a) is inversely proportional to their separation distanceb) is a vector quantityc) is always positived) 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 regionb) move toward the higher potential regionc) 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.
Activity: Learn by Drawing, p. 522.
Animation:http://regentsprep.org/Regents/physics/phys03/aequilines/default.htm
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 electric field ∆x max from potential On Gold Sheet
Electric Field from Potential
E = - ∆ V electric field ∆x max 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 = Nm 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?
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 relationship between potential and electric field ∆x max
Check for Understanding
1)Equipotential surfaces are those surfaces on which
a) the potential is constantb) the electric field is zeroc) the potential is zero
Answer: a
2) Equipotential Surfaces are
a) parallel to the electric fieldb) perpendicular to the electric fieldc) 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 plateb) the ones near the negative platec) 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 equipotentialb) the next lowest equipotentialc) 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) - a quantitative measure of how effective a capacitor is in storing charge.
C = Q definition of capacitance V
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
Metal plates
+Q
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).
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:
C = o A capacitance of a parallel-plate d 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)
• o is related to Coulomb’s constant by:
k = 1 = 9.00 x 109 N·m2/C2
4o
On Blue Sheet
On Blue Sheet
On Gold Sheet
Capacitance
• A plot of charge vs. voltage for a charging capacitor is a straight line with slope C.
Q = CV ( y = mx + b)
• 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 = Q2 = ½ CV2 energy storage 2C 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.
Voltage
Q(charge)
slope = Capacitance
On Gold Sheet
On Gold Sheet
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)What is the capacitance?b)What is the charge on the plates?c)How much energy is stored in the capacitor?d)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 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.
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 definition of capacitance V
C = o A capacitance of a parallel-plate d capacitor (in air)
Uc = ½ QV = Q2 = ½ CV2 energy in a 2C charged capacitor
1 = 1 + 1 + 1 +... equivalent series capacitanceCs C1 C2 C3
Cp = C1 + C2 + C3 + … equivalent parallel capacitance
Check for Understanding
1.Capacitance has units of a. faradsb. joulesc. coulombs per voltd. 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 distanceb. increase the plate areac. evacuate the space between the platesd. 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 samea. voltageb. chargec. energy storage
Answer: b
4. Capacitors in parallel have the samea. voltageb. chargec. 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
