CapacitanceCapacitance

Chapter 24Chapter 24

Two flat parallel plates are Two flat parallel plates are dd = 0.40 = 0.40 cm apart. The potential difference cm apart. The potential difference between the plates is 360 V. The between the plates is 360 V. The electric field at the point P at the electric field at the point P at the

center is approximately center is approximately A.A. 90 kN/C. 90 kN/C.

B.B. 180 N/C. 180 N/C.

C.C. 0.9 kN/C. 0.9 kN/C.

D.D. Zero. Zero.

E.E. 3.6 3.6 10 1055 N/C N/C

CapacitanceCapacitance

The figure shows the basic elements The figure shows the basic elements of of anyany capacitor—two isolated capacitor—two isolated conductors of any shape. No matter conductors of any shape. No matter what their geometry, flat or not, we what their geometry, flat or not, we call these conductors call these conductors plates.plates.

What is it?What is it?

Because the plates are conductors, Because the plates are conductors, they are equipotential surfaces; all they are equipotential surfaces; all points on a plate are at the same points on a plate are at the same electric potential. Moreover, there electric potential. Moreover, there is a potential difference between is a potential difference between the two plates. the two plates.

we representwe represent

the absolute value of this potential the absolute value of this potential

difference with difference with VV rather than with rather than with

the the ΔΔVV we used in previous notation. we used in previous notation.

Capacitance IICapacitance II The charge The charge qq and the potential and the potential

difference difference VV for a capacitor are for a capacitor are proportional to each other; that is, proportional to each other; that is,

V

C1F1 The SI unit of

capacitance is farad (F)

A capacitor of capacitance A capacitor of capacitance C C holds a holds a charge charge QQ when the potential when the potential

difference across the plates is difference across the plates is V.V. If If the charge the charge QQ on the plates is doubled on the plates is doubled

to 2to 2Q, Q, A.A. the capacitance becomes (1/2)the capacitance becomes (1/2)V. V.

B.B. the capacitance becomes 2the capacitance becomes 2C. C.

C.C. the potential changes to (1/2)the potential changes to (1/2)V. V.

D.D. the potential changes to 2the potential changes to 2VV. .

E.E. the potential does not change. the potential does not change.

If a capacitor of capacitance 2.0 µF is If a capacitor of capacitance 2.0 µF is given a charge of 1.0 mC, the given a charge of 1.0 mC, the potential difference across the potential difference across the

capacitor is capacitor is

A.A. 0.50 kV. 0.50 kV.

B.B. 2.0 V. 2.0 V.

C.C. 2.0 µV. 2.0 µV.

D.D. 0.50 V. 0.50 V.

E.E. None of these is correct. None of these is correct.

ChargingCharging In (a), a battery B, a switch S, an In (a), a battery B, a switch S, an

uncharged capacitor C, and uncharged capacitor C, and interconnecting wires form a circuit. interconnecting wires form a circuit. The same circuit is shown in the The same circuit is shown in the schematic diagramschematic diagram ( (b)b), in which the , in which the symbols for a battery, a switch, and a symbols for a battery, a switch, and a capacitor represent those devices. capacitor represent those devices. The battery maintains potential The battery maintains potential difference difference VV between its terminals. between its terminals. The terminal of higher potential is The terminal of higher potential is labeled + and is often called the labeled + and is often called the positivepositive terminal; the terminal of terminal; the terminal of lower potential is labeled – and is lower potential is labeled – and is often called the often called the negativenegative terminal. terminal.

Finding Finding CC

Calculating CapacitancePICTURE Make a sketch of the

capacitor that has a charge of +Q on one conductor and a charge of -Q on the other conductor.

SOLVE

1. Determine the electric field , usually by using Gauss’s law.

2. Determine the magnitude of the potential difference V between the two conductors by integrating (Equation 23-2a).

3. The capacitance is equal to C = Q/V.

24-1824-18

Two isolated conducting spheres of Two isolated conducting spheres of equal radius equal radius RR have charges + have charges +QQ and and --QQ respectively. Their centers are respectively. Their centers are separated by a distance separated by a distance dd that is that is large compared to their radius. large compared to their radius. Estimate the capacitance of this Estimate the capacitance of this unusual capacitor. unusual capacitor.

Other shapesOther shapes

Cylindrical CapacitorCylindrical Capacitor

Spherical CapacitorSpherical Capacitor

If the area of the plates of a If the area of the plates of a parallel-plate capacitor is halved, parallel-plate capacitor is halved,

the capacitance is the capacitance is A.A. not changed. not changed.

B.B. doubled. doubled.

C.C. halved. halved.

D.D. increased by a factor of 4. increased by a factor of 4.

E.E. decreased by a factor of 1/4. decreased by a factor of 1/4.

Doubling the potential Doubling the potential difference across a difference across a

capacitor capacitor A.A. doubles its capacitance. doubles its capacitance.

B.B. halves its capacitance. halves its capacitance.

C.C. quadruples the charge stored on quadruples the charge stored on the capacitor. the capacitor.

D.D. halves the charge stored on the halves the charge stored on the capacitor. capacitor.

E.E. does not change the capacitance of does not change the capacitance of the capacitor. the capacitor.

Energy storedEnergy stored

Potential Energy per unit volume in Potential Energy per unit volume in EE

Which of the following statements Which of the following statements isis false false? ?

A.A. In the process of charging a capacitor, an In the process of charging a capacitor, an electric field is produced between its plates. electric field is produced between its plates.

B.B. The work required to charge a capacitor The work required to charge a capacitor can be thought of as the work required to can be thought of as the work required to create the electric field between its plates. create the electric field between its plates.

C.C. The energy density in the space between The energy density in the space between the plates of a capacitor is directly the plates of a capacitor is directly proportional to the electric field. proportional to the electric field.

D.D. The potential difference between the plates The potential difference between the plates of a capacitor is directly proportional to the of a capacitor is directly proportional to the electric field. electric field.

E.E. None of these is false. None of these is false.

(a) The potential difference between the plates of a 3.00-μF capacitor is 100 V. How much energy is stored in the capacitor? (b) How much additional energy is required to increase the potential difference between the plates from 100 V to 200 V?

24-1924-19

45.0mJ15.0mJ

If you increase the charge on a If you increase the charge on a parallel-plate capacitor from 3 µC to 9 parallel-plate capacitor from 3 µC to 9 µC and increase the plate separation µC and increase the plate separation

from 1 mm to 3 mm, but keep all other from 1 mm to 3 mm, but keep all other properties the same, the energy properties the same, the energy

stored in the capacitor changes by a stored in the capacitor changes by a factor of factor of

A.A. 27. 27.

B.B. 9. 9.

C.C. 3. 3.

D.D. 8. 8.

E.E. 1/3. 1/3.

24-2424-24

A solid metal sphere has radius of 10.0 cm A solid metal sphere has radius of 10.0 cm and a concentric metal spherical shell has and a concentric metal spherical shell has an inside radius of 10.5 cm. The solid an inside radius of 10.5 cm. The solid sphere has a charge 5.00 nC. (sphere has a charge 5.00 nC. (aa) Estimate ) Estimate the energy stored in the electric field in the the energy stored in the electric field in the region between the spheres. (region between the spheres. (bb) Estimate ) Estimate the capacitance of this two-sphere system. the capacitance of this two-sphere system. ((cc) Estimate the total energy stored in the ) Estimate the total energy stored in the electric field from ½Qelectric field from ½Q22/C and compare it /C and compare it to your answer in Part (to your answer in Part (aa). ).

24-524-5

A parallel-plate capacitor is A parallel-plate capacitor is connected to a battery. The space connected to a battery. The space between the two plates is empty. If between the two plates is empty. If the separation between the the separation between the capacitor plates is tripled while the capacitor plates is tripled while the capacitor remains connected to the capacitor remains connected to the battery, what is the ratio of the final battery, what is the ratio of the final stored energy to the initial stored stored energy to the initial stored energy? energy? 1/3 .

24-624-6

If the capacitor of Problem 5 is If the capacitor of Problem 5 is disconnected from the battery disconnected from the battery before the separation between the before the separation between the plates is tripled, what is the ratio of plates is tripled, what is the ratio of the final stored energy to the initial the final stored energy to the initial stored energy? stored energy?

3 .

24-4824-48

Model Earth as a conducting sphere. Model Earth as a conducting sphere. ((aa) What is its self-capacitance? () What is its self-capacitance? (bb) ) Assume the magnitude of the Assume the magnitude of the electric field at Earth’s surface is electric field at Earth’s surface is 150 V/m. What charge density does 150 V/m. What charge density does this correspond to? Express this this correspond to? Express this value in fundamental charge units value in fundamental charge units ee per square centimeter. per square centimeter. 829*10

^3 e/cm^2

A circuit consists of a capacitor, a battery, A circuit consists of a capacitor, a battery, and a switch, all connected in series. and a switch, all connected in series. Initially, the switch is open and the Initially, the switch is open and the

capacitor is uncharged. The switch is then capacitor is uncharged. The switch is then closed and the capacitor charges. While the closed and the capacitor charges. While the

capacitor is charging, how does the net capacitor is charging, how does the net charge within the battery change? charge within the battery change?

A.A. It increases. It increases.

B.B. It decreases. It decreases.

C.C. It stays the same It stays the same

Several different capacitors are Several different capacitors are hooked across a DC battery in hooked across a DC battery in parallel. The charge on each parallel. The charge on each

capacitor is capacitor is A.A. directly proportional to its directly proportional to its

capacitance. capacitance.

B.B. inversely proportional to its inversely proportional to its capacitance. capacitance.

C.C. independent of its capacitance. independent of its capacitance.

Circuits: parallelCircuits: parallel

When a potential difference When a potential difference VV is applied across several is applied across several capacitors connected in capacitors connected in parallel, that potential parallel, that potential difference difference VV is applied is applied across each capacitor. The across each capacitor. The total charge total charge qq stored on the stored on the capacitors is the sum of the capacitors is the sum of the charges stored on all the charges stored on all the capacitors. capacitors.

Several different capacitors are Several different capacitors are hooked across a DC battery in hooked across a DC battery in

parallel. The voltage across each parallel. The voltage across each capacitor is capacitor is

A.A. directly proportional to its directly proportional to its capacitance. capacitance.

B.B. inversely proportional to its inversely proportional to its capacitance. capacitance.

C.C. independent of its capacitance. independent of its capacitance.

Several different capacitors are Several different capacitors are hooked across a DC battery in hooked across a DC battery in

series. The charge on each series. The charge on each capacitor is capacitor is

A.A. directly proportional to its capacitance. directly proportional to its capacitance.

B.B. inversely proportional to its inversely proportional to its capacitance. capacitance.

C.C. independent of its capacitance. independent of its capacitance.

Circuits: SeriesCircuits: Series

When a potential difference When a potential difference VV is applied across several is applied across several capacitors connected in capacitors connected in series, the capacitors have series, the capacitors have identical charge identical charge qq. The sum . The sum of the potential differences of the potential differences across all the capacitors is across all the capacitors is equal to the applied potential equal to the applied potential difference difference VV. .

Several different capacitors Several different capacitors are hooked across a DC are hooked across a DC

battery in series. The voltage battery in series. The voltage across each capacitor is across each capacitor is

A.A. directly proportional to its directly proportional to its capacitance. capacitance.

B.B. inversely proportional to its inversely proportional to its capacitance. capacitance.

C.C. independent of its capacitance. independent of its capacitance.

CalculationsCalculations

Parallel: voltage the same on all Parallel: voltage the same on all capacitorscapacitors

Series: Charge the same on all Series: Charge the same on all capacitorscapacitors

Do all examples in section 24-3!!

24-3224-32 For the circuit shown in For the circuit shown in Figure 24-36, ,

the capacitors were each discharged the capacitors were each discharged before being connected to the voltage before being connected to the voltage source. Find (source. Find (aa) the equivalent ) the equivalent capacitance of the combination, (capacitance of the combination, (bb) the ) the charge stored on the positively charge stored on the positively charged plate of each capacitor, (charged plate of each capacitor, (cc) the ) the voltage across each capacitor, and (voltage across each capacitor, and (dd) ) the energy stored in each capacitor. the energy stored in each capacitor.

24-3524-35

Five identical capacitors of capacitance Five identical capacitors of capacitance CC0 0

are connected in a so-called bridge are connected in a so-called bridge network, as shown in network, as shown in Figure 24-38. (. (aa) ) What is the equivalent capacitance What is the equivalent capacitance between points between points aa and and bb? (? (bb) Find the ) Find the equivalent capacitance between points equivalent capacitance between points aa and and bb if the capacitor at the center is if the capacitor at the center is replaced by a capacitor that has a replaced by a capacitor that has a capacitance of 10capacitance of 10CC00. .

DielectricsDielectrics

So far we have done capacitors that So far we have done capacitors that are made out of vacuum rather than are made out of vacuum rather than air, or grease, or some other air, or grease, or some other material.material.

Magnitude of the bound charge

The bound charge density is always less than or equal to the free charge density on the capacitor plates, and it is zero if κ = 1, which is the case when there is no dielectric. For a conducting slab, κ = infinity.

HW 24-56HW 24-56

A 12-A 12-μμF capacitor and a capacitor of unknown F capacitor and a capacitor of unknown capacitance are both charged to 2.00 kV. capacitance are both charged to 2.00 kV. After charging, the two capacitors are After charging, the two capacitors are disconnected from the voltage source. The disconnected from the voltage source. The capacitors are then connected to each other—capacitors are then connected to each other—positive plate to negative plate and negative positive plate to negative plate and negative plate to positive plate. The final voltage plate to positive plate. The final voltage across the terminals of the 12-across the terminals of the 12-μμF capacitor is F capacitor is 1.00 kV. (1.00 kV. (aa) What is the capacitance of the ) What is the capacitance of the second capacitor? (second capacitor? (bb) How much energy was ) How much energy was dissipated when the connection was made? dissipated when the connection was made?

4.0 micro F 24J

24-6124-61 You are a laboratory assistant in a physics You are a laboratory assistant in a physics

department that has budget problems. department that has budget problems. Your supervisor wants to construct Your supervisor wants to construct inexpensive parallel plate capacitors for inexpensive parallel plate capacitors for use in introductory laboratory use in introductory laboratory experiments. The design uses experiments. The design uses polyethylene, which has a dielectric polyethylene, which has a dielectric constant of 2.30, between two sheets of constant of 2.30, between two sheets of aluminum foil. The area of each sheet of aluminum foil. The area of each sheet of foil is 400 cmfoil is 400 cm22 and the thickness of the and the thickness of the polyethylene is 0.300 mm. Find the polyethylene is 0.300 mm. Find the capacitance of this arrangement. capacitance of this arrangement.

C=k epsilon A/d

2.72nF

24-8424-84

A rectangular parallel-plate capacitor A rectangular parallel-plate capacitor that has a length that has a length aa and a width and a width bb has a has a dielectric that has a width dielectric that has a width bb partially partially inserted a distance inserted a distance xx between the between the plates, as shown in plates, as shown in Figure 24-49. (. (aa) ) Find the capacitance as a function of Find the capacitance as a function of xx. Neglect edge effects. (. Neglect edge effects. (bb) Show that ) Show that your answer gives the expected results your answer gives the expected results for for xx = 0 and = 0 and xx = = aa. .