CAPACITANCE

Akhtar Mahmood
03334281759
Akhtar Mahmood
03334281759
Akhtar Mahmood
03334281759
Akhtar Mahmood
03334281759
 Akhtar Mahmood
10
03334281759
5 A solid metal sphere, of radius r, is insulated from its surroundings. The sphere has For
charge +Q. Examiner’s
This charge is on the surface of the sphere but it may be considered to be a point charge at Use

its centre, as illustrated in Fig. 5.1.

+Q

Fig. 5.1

(a) (i) Define capacitance.

..................................................................................................................................

............................................................................................................................ [1]

(ii) Show that the capacitance C of the sphere is given by the expression

C = 4πε0r.

[1]

(b) The sphere has radius 36 cm.

Determine, for this sphere,

(i) the capacitance,

capacitance = ............................................ F [1]

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(ii) the charge required to raise the potential of the sphere from zero to 7.0 × 105 V. For
Examiner’s
Use

charge = ........................................... C [1]

(c) Suggest why your calculations in (b) for the metal sphere would not apply to a plastic
sphere.

..........................................................................................................................................

..........................................................................................................................................

..........................................................................................................................................

.................................................................................................................................... [3]

(d) A spark suddenly connects the metal sphere in (b) to the Earth, causing the potential of
the sphere to be reduced from 7.0 × 105 V to 2.5 × 105 V.

Calculate the energy dissipated in the spark.

energy = ............................................ J [3]

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4 (a) Define capacitance. For
Examiner’s
.......................................................................................................................................... Use

.................................................................................................................................... [1]

(b) An isolated metal sphere of radius R has a charge +Q on it.

The charge may be considered to act as a point charge at the centre of the sphere.

Show that the capacitance C of the sphere is given by the expression

C = 4!!0R

where !0 is the permittivity of free space.

[1]

(c) In order to investigate electrical discharges (lightning) in a laboratory, an isolated metal

sphere of radius 63 cm is charged to a potential of 1.2 × 106 V.

At this potential, there is an electrical discharge in which the sphere loses 75% of its
energy.

Calculate

(i) the capacitance of the sphere, stating the unit in which it is measured,

capacitance = ................................................ [3]

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(ii) the potential of the sphere after the discharge has taken place. For
Examiner’s
Use

potential = ............................................. V [3]

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Akhtar Mahmood
03334281759
Akhtar Mahmood
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 14 Akhtar Mahmood
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6 (a) State two different functions of capacitors in electrical circuits.

1. ...............................................................................................................................................

...................................................................................................................................................

2. ...............................................................................................................................................

...................................................................................................................................................
[2]

(b) Three uncharged capacitors of capacitances C1, C2 and C3 are connected in series with a
battery of electromotive force (e.m.f.) E and a switch, as shown in Fig. 6.1.

C1 C2 C3

plate P
charge +q

Fig. 6.1

When the switch is closed, there is a charge + q on plate P of the capacitor of capacitance C1.

Show that the combined capacitance C of the three capacitors is given by the expression
1 1 1 1
= + + .
C C1 C2 C3

[3]

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(c) A student has available four capacitors, each of capacitance 20 μF.

Draw circuit diagrams, one in each case, to show how the student may connect some or all of
the capacitors to produce a combined capacitance of:

(i) 60 μF

[1]

(ii) 15 μF.

[1]

[Total: 7]

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4 (a) State two functions of capacitors in electrical circuits. For
Examiner’s
1. ...................................................................................................................................... Use

..........................................................................................................................................

2. ......................................................................................................................................

..........................................................................................................................................
[2]

(b) Three uncharged capacitors of capacitance C1, C2 and C3 are connected in series, as
shown in Fig. 4.1.

plate A

C1 C2 C3

Fig. 4.1

A charge of +Q is put on plate A of the capacitor of capacitance C1.

(i) State and explain the charges that will be observed on the other plates of the
capacitors.
You may draw on Fig. 4.1 if you wish.

..................................................................................................................................

..................................................................................................................................

............................................................................................................................. [2]

(ii) Use your answer in (i) to derive an expression for the combined capacitance of the
capacitors.

[2]

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(c) A capacitor of capacitance 12 μF is charged using a battery of e.m.f. 9.0 V, as shown in For
Fig. 4.2. Examiner’s
Use

S1 S2

12 μF 20 μF
9.0 V

Fig. 4.2

Switch S1 is closed and switch S2 is open.

(i) The capacitor is now disconnected from the battery by opening S1.
Calculate the energy stored in the capacitor.

energy = ............................................. J [2]

(ii) The 12 μF capacitor is now connected to an uncharged capacitor of capacitance

20 μF by closing S2. Switch S1 remains open.
The total energy now stored in the two capacitors is 1.82 × 10–4 J.

Suggest why this value is different from your answer in (i).

..................................................................................................................................

............................................................................................................................. [1]

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4 (a) State two functions of capacitors connected in electrical circuits. For
Examiner’s
1. ..................................................................................................................................... Use

..........................................................................................................................................

2. .....................................................................................................................................

..........................................................................................................................................
[2]

(b) Three capacitors are connected in parallel to a power supply as shown in Fig. 4.1.

C1

C2

C3

Fig. 4.1

The capacitors have capacitances C1, C2 and C3. The power supply provides a potential
difference V.

(i) Explain why the charge on the positive plate of each capacitor is different.

..................................................................................................................................

..................................................................................................................................

.............................................................................................................................. [1]

(ii) Use your answer in (i) to show that the combined capacitance C of the three
capacitors is given by the expression

C = C1 + C2 + C3.

[2]

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(c) A student has available three capacitors, each of capacitance 12 μF. For
Draw circuit diagrams, one in each case, to show how the student connects the three Examiner’s
capacitors to provide a combined capacitance of Use

(i) 8 μF,

[1]

(ii) 18 μF.

[1]

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7 (a) (i) Define capacitance.

...........................................................................................................................................

.......................................................................................................................................[1]

(ii) Use the expression for the electric potential due to a point charge to show that an isolated
metal sphere of diameter 25 cm has a capacitance of 1.4 × 10–11 F.

[2]

(b) Three capacitors of capacitances 2.0 μF, 3.0 μF and 4.0 μF are connected as shown in Fig. 7.1
to a battery of e.m.f. 9.0 V.

4.0 μF

3.0 μF

2.0 μF

9.0 V

Fig. 7.1

Determine

(i) the combined capacitance of the three capacitors,

capacitance = ..................................................... μF [1]

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(ii) the potential difference across the capacitor of capacitance 3.0 μF,

potential difference = ...................................................... V [2]

(iii) the positive charge stored on the capacitor of capacitance 2.0 μF.

charge = .................................................... μC [2]

[Total: 8]

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7 (a) Define capacitance.

...................................................................................................................................................

...............................................................................................................................................[1]

(b) Three capacitors of capacitances C1, C2 and C3 are initially uncharged. They are then
connected in series to a battery, as shown in Fig. 7.1.

C1 C2 C3

Fig. 7.1

The battery applies a potential difference V across the three capacitors.

Show that the combined capacitance C of the capacitors is given by

1 1 1 1
= + + .
C C1 C2 C3

[2]

(c) A battery of e.m.f. 12 V and negligible internal resistance is connected to a network of two
capacitors and a resistor, as shown in Fig. 7.2.

200 F

A B
12 V

600 F

Fig. 7.2

The capacitors have capacitances of 200 μF and 600 μF. The switch has two positions,
A and B.

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(i) The switch is moved to position A.

Calculate

1. the combined capacitance of the two capacitors,

combined capacitance = ..................................................... μF [1]

2. the charge on the 600 μF capacitor,

charge = ....................................................... C [1]

3. the potential difference across the 600 μF capacitor.

potential difference = ....................................................... V [1]

(ii) The switch is now moved from position A to position B.

Calculate the potential difference across the 600 μF capacitor when it has discharged
50% of its initial energy.

potential difference = ....................................................... V [3]

[Total: 9]

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6 Two capacitors P and Q, each of capacitance C, are connected in series with a battery of e.m.f.
9.0 V, as shown in Fig. 6.1.

switch S
9.0 V
X Y
P T
R
C C

Fig. 6.1

A switch S is used to connect either a third capacitor T, also of capacitance C, or a resistor R, in

parallel with capacitor P.

(a) Switch S is in position X.

Calculate

(i) the combined capacitance, in terms of C, of the three capacitors,

capacitance = ......................................................... [2]

(ii) the potential difference across capacitor Q. Explain your working.

potential difference = ..................................................... V [2]

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(b) Switch S is now moved to position Y.
State what happens to the potential difference across capacitor P and across capacitor Q.

capacitor P: ..............................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

capacitor Q: .............................................................................................................................

...................................................................................................................................................

...................................................................................................................................................
[4]

[Total: 8]

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 12 Akhtar Mahmood For
Examiner’s
03334281759 Use
5 (a) State one function of capacitors in simple circuits.

..........................................................................................................................................

......................................................................................................................................[1]

(b) A capacitor is charged to a potential difference of 15 V and then connected in series with
a switch, a resistor of resistance 12 kΩ and a sensitive ammeter, as shown in Fig. 5.1.

12 kΩ

Fig. 5.1

The switch is closed and the variation with time t of the current I in the circuit is shown in
Fig. 5.2.

1.5

I/mA

1.0

0.5

0
0 5 10 15 t /s 20

Fig. 5.2
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Examiner’s
Use
(i) State the relation between the current in a circuit and the charge that passes a
point in the circuit.

..................................................................................................................................

..............................................................................................................................[1]

(ii) The area below the graph line of Fig. 5.2 represents charge.
Use Fig. 5.2 to determine the initial charge stored in the capacitor.

charge = ............................................ µC [4]

(iii) Initially, the potential difference across the capacitor was 15 V.

Calculate the capacitance of the capacitor.

capacitance = ............................................ µF [2]

(c) The capacitor in (b) discharges one half of its initial energy. Calculate the new potential
difference across the capacitor.

potential difference = ...............................................V [3]

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 15 Akhtar Mahmood
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5 The variation with potential difference V of the charge Q on one of the plates of a capacitor is
shown in Fig. 5.1.

1.8

1.6
Q / 10–4 C
1.4

1.2

1.0

0.8

0.6

0.4

0.2

0
0 2 4 6 8 10 12
V/V

Fig. 5.1

The capacitor is connected to an 8.0 V power supply and two resistors R and S as shown in
Fig. 5.2.

8.0 V

R
25 kΩ

S
220 kΩ

Fig. 5.2

The resistance of R is 25 kΩ and the resistance of S is 220 kΩ.

The switch can be in either position X or position Y.

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(a) The switch is in position X so that the capacitor is fully charged.

Calculate the energy E stored in the capacitor.

E = ....................................................... J [2]

(b) The switch is now moved to position Y.

(i) Show that the time constant of the discharge circuit is 3.3 s.

[2]

(ii) The fully charged capacitor in (a) stores energy E.

Determine the time t taken for the stored energy to decrease from E to E / 9.

t = ....................................................... s [4]

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5 (a) Define the capacitance of a parallel plate capacitor.

...................................................................................................................................................

...................................................................................................................................................

............................................................................................................................................. [2]

(b) Two capacitors, of capacitances C1 and C2, are connected in parallel to a power supply of
electromotive force (e.m.f.) E, as shown in Fig. 5.1.

C1

C2

Fig. 5.1

Show that the combined capacitance CT of the two capacitors is given by

CT = C1 + C2.

Explain your reasoning. You may draw on Fig. 5.1 if you wish.

[3]

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(c) Two capacitors of capacitances 22 μF and 47 μF, and a resistor of resistance 2.7 MΩ, are
connected into the circuit of Fig. 5.2.

12 V
X

2.7 MΩ
Y

22 μF 47 μF

Fig. 5.2

The battery has an e.m.f. of 12 V.

(i) Show that the combined capacitance of the two capacitors is 15 μF.

[1]

(ii) The two-way switch S is initially at position X, so that the capacitors are fully charged.

Use the information in (c)(i) to calculate the total energy stored in the two capacitors.

total energy = ...................................................... J [2]

(iii) The two-way switch is now moved to position Y.

Determine the time taken for the potential difference (p.d.) across the 22 μF capacitor to
become 6.0 V.

time = ...................................................... s [3]

[Total: 11]
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5 A capacitor of capacitance 470 μF is connected to a battery of electromotive force (e.m.f.) 24 V in

the circuit of Fig. 5.1.

X Y

24 V 470 μF V

P Q
5.6 kΩ 5.6 kΩ

Fig. 5.1

The two-way switch S is initially at position X.

P and Q are identical long straight wires, each with a resistance of 5.6 kΩ. These wires are placed
near to, and parallel to, each other. Wire Q is connected to a voltmeter.

At time t = 0, switch S is moved to position Y so that the capacitor discharges through wire P.

(a) (i) Calculate the charge Q0 on the capacitor at time t = 0.

Q0 = ..................................................... C [2]

(ii) Calculate the current I0 in wire P at time t = 0.

I0 = ...................................................... A [1]

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(iii) Calculate the time constant τ of the discharge circuit.

τ = ...................................................... s [2]

(iv) On Fig. 5.2, sketch a line to show the variation with t of the current I in wire P as the
capacitor discharges.

I0

0
0 t

Fig. 5.2
[2]

(b) (i) Explain why there is an induced e.m.f. across wire Q during the discharge of the
capacitor.

...........................................................................................................................................

...........................................................................................................................................

...........................................................................................................................................

..................................................................................................................................... [3]

(ii) On Fig. 5.3, sketch a line to suggest the variation with t of the voltmeter reading V.

0
0 t
Fig. 5.3
[1]

[Total: 11]

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6 A capacitor C is charged so that the potential difference (p.d.) V across its terminals is 8.0 V.
The capacitor is connected into the circuit of Fig. 6.1.

8.0 V

Fig. 6.1

The switch is initially open. The switch is closed at time t = 0.

(a) Fig. 6.2 shows the variation of V with the charge Q on the plates of capacitor C as the
capacitor discharges.

V/V

0
0 200 400 600
Q / μC

Fig. 6.2

(i) Show that the energy stored in capacitor C at time t = 0 is 1.8 mJ.

[2]

(ii) Determine the capacitance of capacitor C. Give a unit with your answer.

capacitance = ................................. unit ................ [2]

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(b) Fig. 6.3 shows the variation with t of –ln 18.0V V2.
2.0

–ln 18.0V V2
1.0

0
0 2 4 6 8
t/s

Fig. 6.3

(i) Show that, when t is equal to one time constant, the value of –ln 18.0V V2 is equal to 1.0.

[2]

(ii) Determine the time constant τ of the circuit in Fig. 6.1.

τ = ....................................................... s [1]

(iii) Calculate the resistance of resistor R.

resistance = ...................................................... Ω [2]

[Total: 9]

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