16

7 (a) State what is meant by the capacitance of a parallel plate capacitor.

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

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

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

(b) A capacitor of capacitance C is connected into the circuit shown in Fig. 7.1.

A B

sensitive
+ ammeter
V A
–
C

Fig. 7.1

When the two-way switch is in position A, the capacitor is charged so that the potential
difference across it is V.
The switch moves to position B and the capacitor fully discharges through the sensitive
ammeter.

The switch moves repeatedly between A and B so that the capacitor charges and then
discharges with frequency f.

(i) Show that the average current I in the ammeter is given by the expression

I = fCV.

[2]

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(ii) For a potential difference V of 150 V and a frequency f of 60 Hz, the average current in
the ammeter is 4.8 μA.

Calculate the capacitance, in pF, of the capacitor.

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

(c) A second capacitor, having the same capacitance as the capacitor in (b), is connected into
the circuit of Fig. 7.1. The two capacitors are connected in series.

State and explain the new reading on the ammeter.

new reading = ......................................................... μA

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

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

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

[Total: 9]

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6 (a) Two flat metal plates are held a small distance apart by means of insulating pads, as shown
in Fig. 6.1.

metal plate

insulating
metal plate pad

Fig. 6.1

Explain how the plates could act as a capacitor.

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

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

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

(b) The arrangement in Fig. 6.1 has capacitance C.

The arrangement is connected into the circuit of Fig. 6.2.

A B

sensitive
ammeter
V A
C

Fig. 6.2

When the two-way switch is moved to position A, the capacitor is charged so that the potential
difference across it is V. When the switch moves to position B, the capacitor fully discharges
through the sensitive ammeter.

The switch moves repeatedly between A and B so that the capacitor charges and then
discharges with frequency f.

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(i) Show that the average current I in the ammeter is given by

I = CVf.

[2]

(ii) For a potential difference V of 180 V and a frequency f of switching of 50 Hz, the average
current I in the ammeter is 2.5 μA.

Calculate the capacitance, in pF, of the parallel plates.

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

(c) A second capacitor is connected into the circuit of Fig. 6.2.

The two capacitors are connected in parallel.

State and explain the change, if any, in the average current in the ammeter.

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

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

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

[Total: 8]

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6 (a) A capacitor consists of two parallel metal plates, separated by air, at a variable distance x
apart, as shown in Fig. 6.1. The capacitance C is inversely proportional to x.

x
metal plates

Fig. 6.1

The capacitor is charged by a supply so that there is a potential difference (p.d.) V between
the plates.

State expressions, in terms of C and V, for the charge Q on one of the plates and for the
energy E stored in the capacitor.

Q = ............................................. E = ............................................. [1]

(b) The charged capacitor in (a) is now disconnected from the supply. The plates of the capacitor
are initially separated by distance L. They are then moved closer together by a distance D, as
shown in Fig. 6.2.

D new position

original position

Fig. 6.2

State expressions, in terms of C, V, L and D, for:

(i) the new capacitance CN

CN = ......................................................... [1]

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(ii) the new charge QN on one of the plates

QN = ......................................................... [1]

(iii) the new p.d. VN between the plates.

VN = ......................................................... [1]

(c) Explain whether reducing the separation of the plates in (b) results in an increase or decrease
in the energy stored in the capacitor.

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

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

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

[Total: 5]

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6 (a) Define electric potential.

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

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

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

(b) An isolated conducting sphere in a vacuum has radius r and is initially uncharged. It is then
charged by friction so that it carries a final charge Q. This charge can be considered to be
acting at the centre of the sphere.

By considering the electric potential at its surface, show that the capacitance C of the sphere
is given by

C = 4πε0r

where ε0 is the permittivity of free space.

[2]

(c) The dome of an electrostatic generator is a spherical conductor of radius 13 cm. It is initially
charged so that the electric potential at the surface is 4.5 kV.

A smaller isolated sphere of radius 5.2 cm, initially uncharged, is brought near to the dome.
Sparking causes a current between the two spheres until they reach the same potential.
Assume that any charge on a sphere may be considered to act as a point charge at its centre.

Calculate the charge that is transferred between the two spheres.

charge = ..................................................... C [3]

[Total: 7]

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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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 16

(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]

(c) A second identical capacitor is connected in parallel with the first capacitor.

State and explain the change, if any, to the time constant of the discharge circuit.

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

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

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

[Total: 10]

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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 of capacitance C and a resistor of resistance R are connected as shown in Fig. 6.1.

Fig. 6.1

Initially, the capacitor is charged and the switch is open.

The switch is closed at time t = 0.

Fig. 6.2 and Fig. 6.3 show, respectively, the variations with t of the charge Q on the capacitor and
the potential difference (p.d.) V across the resistor.

1.0 10

Q / mC V/V

0.5 5

0 0
0 5 10 15 0 5 10 15
t/s t/s

Fig. 6.2 Fig. 6.3

(a) Explain the shape of the line in Fig. 6.3 representing the variation of V with t.

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

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

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

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

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

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(b) Use Fig. 6.2 to show that the time constant of the circuit in Fig. 6.1 is 5.5 s.

[3]

(c) Use Fig. 6.2, Fig. 6.3 and the information in (b) to determine:

(i) capacitance C, in μF

C = .................................................... μF [2]

(ii) resistance R, in kΩ.

R = ................................................... kΩ [2]

[Total: 10]

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5 A capacitor, a battery of electromotive force (e.m.f.) 12 V, a resistor R and a two-way switch are
connected in the circuit shown in Fig. 5.1.

T
S

12 V

Fig. 5.1

The switch is initially in position S. When the capacitor is fully charged, the switch is moved to
position T so that the capacitor discharges. At time t after the switch is moved the charge on the
capacitor is Q.

The variation with t of ln (Q / μC) is shown in Fig. 5.2.

ln (Q /μC)
2

0
0 1 2 3 4 5
t/s

Fig. 5.2

(a) Show that the capacitance of the capacitor is 1.5 μF.

[3]

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(b) Determine the resistance of R.

resistance = ..................................................... Ω [3]

(c) Calculate the energy stored in the capacitor at time t = 0.

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

(d) A second identical resistor is now connected in parallel with R.

The switch is initially in position S. When the capacitor is fully charged, the switch is moved to
position T so that the capacitor discharges. At time t after the switch is moved the charge on
the capacitor is Q.

On Fig. 5.2, sketch a line to show the variation of ln (Q / μC) with t between time t = 0 and
time t = 5.0 s. [2]

[Total: 10]

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5 Two capacitors A and B are connected into the circuit shown in Fig. 5.1.

X
A

Y
B

Fig. 5.1

Capacitor A has capacitance C and capacitor B has capacitance 3C.

The electromotive force (e.m.f.) of the cell is V.
The two-way switch S is initially at position X, and capacitor B is initially uncharged.

(a) State, in terms of V and C, expressions for:

(i) the initial charge QA on the plates of capacitor A

QA = ......................................................... [1]

(ii) the initial energy EA stored in capacitor A.

EA = .......................................................... [1]

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

(i) State and explain what happens to the charge that was initially on the plates of capacitor A.

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

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

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

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(ii) Show that the final potential difference (p.d.) VB across capacitor B is given by
V
VB = .
4
Explain your reasoning.

[3]

(iii) Determine an expression, in terms of V and C, for the decrease ΔE in the total energy
that is stored in the capacitors as a result of the change of the position of the switch.

ΔE = ......................................................... [2]

[Total: 9]

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 14

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 8.0V V.
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 8.0V V 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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