42 Springs and Capacitors

Physics Factsheet September 2002

Number 42

Comparison of Capacitors and Springs The purpose of this Factsheet is to explore the comparison between capacitors and springs for synoptic papers of A2 examinations.These examinations require candidates to see similarities in apparently disparate phenomenon and draw deeper comparisons between different types of behaviour.

For those less confident mathematically, it can be considered as the area under the graph of F against x

FF Factsheet 29 explains capacitors in detail and Factsheet 20 looks at simple harmonic motion, which covers the behaviour of springs. Before studying this Factsheet you should be familiar with both of these. Another analogy that you should consider is that between the decay of charge on a capacitor and radioactive decay. Factsheet 36 looks at this comparison. ∆x Δx

Summary of capacitors • A capacitor is a device that stores charge. • The charge stored by a given capacitor is proportional to the voltage applied across it. • Different capacitors store different quantities of charge for the same voltage applied. The constant of proportionality is designated C, the capacitance of a capacitor. • Q = CV

total area = ½ F× x = ½ kx × x F = ½ kx2

xx

(Since F = kx)

Energy stored in a stretched spring = ½ kx2

Summary of springs • When a force is applied to a spring, the extension is proportional to the force applied. • Different springs extend by different amounts for the same force applied. The constant of proportionality is designated k, the spring constant or spring stiffness. • F = kx

Capacitor If V were constant, then energy stored would be V Q (since V is the energy per unit charge). Since V is not constant as the capacitor stores charge then we must consider a tiny change in charge, ∆Q for which V can be considered constant. The sum of all these tiny changes will give the total energy stored – the area under the graph of V against Q

From the summaries given above, similarities begin to appear, but we really need to rearrange the equations to arrive at a valid comparison, because the independent variable in the case of capacitors is V and in springs F.

V

V

So Q = CV x = 1 F k These two equations show similarities in the processes, but the factors that affect the constants are obviously very different.

∆Q

ΔQ

Factors that affect C include the distance between the plates, the area of the plates and the dielectric material. Factors that affect k include the Young’s modulus of the material from which it is made.

QQ

Area = ½ V × Q = ½ V× CV (Since Q = CV) = ½ CV2

Energy stored

Energy stored in a capacitor = ½ CV2

A comparison, which is not evident from the summary above, is the energy stored in capacitors and stretched springs.

Although these appear to be very dissimilar phenomena, the expression for the energy stored is of the same general form in each case because the quantities concerned (Q and C; F and x) are proportional to each other.

Spring When the force, F, applied to the spring is constant, the energy stored will be Fx (force multiplied by extension). In reality as we add weights to a spring the force is not constant. In order to calculate the energy stored we must consider very small changes in extension (∆x) for which F can be considered constant, then the total energy stored would be the sum of these small intervals. Those who are mathematically minded will realize that this sum is the integral of F∆x.

Worked Example: What is the energy stored in a capacitor of capacitance 6.2 microfarads when a voltage of 3V is applied across it? E = ½CV2 = ½ × 6.2 × 10-6 × 3 × 3 = 2.79 × 10-5 J

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Physics Factsheet

Comparison of Capacitors and Springs Questions

Exam Workshop This is a typical weak student’s answer to an exam question. The comments explain what is wrong with the answers and how they can be improved. The examiners’ mark scheme is given below.

1. Sketch graphs to show (a) the variation of charge with applied voltage for a capacitor (b) the variation of extension with applied force for an elastic spring.

(a) Name two devices which store energy: one mechanical device and one electrical. Explain how they each store energy. (4)

2. Using the graphs from question 1, or otherwise, derive the formulae for energy stored by capacitors and springs.

A capacitor stores energy. A stretched spring stores energy.

3. Explain why the mathematical models for springs and capacitors give rise to similar expressions for stored energy.

1/4

The candidate should have realised that this was insufficient for 4 marks. The candidate has only named two devices. No attempt has been made to explain how they store energy. More detail is required.

4. When a force of 8N is applied to a spring, it stretches by 2cm. (a) Calculate the spring constant for this spring. (b) Calculate the energy stored in this spring when a force of 6N is applied to it.

(b) Describe how the energy is stored and give the equation for the energy stored. (4) The energy is stored in the device. E = ½CV2 (capacitor) E = ½kx2 (spring)

5. A capacitor stores energy of 0.00004 J when a voltage of 5 V is applied across it. (a) Calculate the capacitance of the capacitor, giving your answer in microfarads. (b) Calculate the charge stored by the capacitor when a voltage of 6V is applied across it.

2/4

The candidate has scored marks for quoting the equations for energy stored, but has not described how it is stored in each case.

Answers

Examiner’s answers (a) A capacitor stores charge. When a p.d. is applied across the plates of a capacitor, charge is transferred between the plates. The charge stored is proportional to the p.d applied. The electrical energy stored is related to the p.d. and the charge stored.

1, 2 and 3. See text 4. (a) F = kx 8 = k0.02 k = 8/0.02 k = 400 Nm-1

A stretched spring stores elastic strain energy. When a force is applied to the spring it stretches. The extension is proportional to the force applied and energy stored in the stretched spring is related to the force and the extesion. .

(b) We must first find x 6 = 400x x = 0.015m

(b) If the p.d were constant, the energy stored would be Q V, but since V varies we must consider a small change in Q for which V can be considered constant, then the total energy stored is the sum of the small changes i.e. the area under the V/Q graph. E = ½ CV2

E = ½kx2 E = ½ (400) (0.0152) E = 0.045J

If the force were constant, the energy stored would be Fx but since F varies we must consider a small change in x over which F can be considered constant, then the total energy stored is the sum of the small changes i.e. the area under the F/x graph. E = ½ Fx2

5. (a) E = ½CV2 0.00004 = ½ C 52 C = 0.00004/ 12.5 C = 3.2 × 10-6 F = 3.2 µF (b) Q = CV Q = 3.2 × 10-6 × 6 Q = 1.92 × 10-5 C

Acknowledgements: This Physics Factsheet was researched and written by Janice Jones. The Curriculum Press,Unit 305B, The Big Peg,120 Vyse Street, Birmingham, B18 6NF. Physics Factsheets may be copied free of charge by teaching staff or students, provided that their school is a registered subscriber. No part of these Factsheets may be reproduced, stored in a retrieval system, or transmitted, in any other form or by any other means, without the prior permission of the publisher. ISSN 1351-5136

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