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Understanding and using theminus sign in Faraday’s lawChris Jones

Physics Department, Hereford Sixth Form College, Folly Lane, Hereford HR1 1LU, UK

E-mail: CLJ@hereford.ac.uk

AbstractThe inclusion of minus signs in physics equations is often a barrier tostudent understanding, and perhaps never more so than in Faraday’s Law ofelectromagnetic induction. This paper carefully explains the origin of theminus sign in Faraday’s law and proposes a simple method by whichstudents can use it to directly predict the direction of induced current in acoil.

IntroductionAt advanced level, the Faraday–Neumann law forthe induced electromotive force (emf) in a rigidcoil of wire is often presented in the form

E = −N��

�t(1)

where E is the induced emf, N the number of turnsof wire in the coil and � the magnetic flux linkingthe coil. The latter is usually presented as

� = BA (2)

where B is the magnetic flux density and A thearea of the coil.

The minus sign in equation (1) inevitablybecomes an issue with students and it is alwaystempting to explain it briefly as being a simpleconsequence of Lenz’s law, safe to ignore incalculations but included in the equation to showthat the induced emf opposes the change of flux.However, exactly how the minus sign shows thisis by no means intuitive and requires some carefulexplanation. To illustrate this, consider the useof the minus sign in another equation, that whichdescribes the dependence of the acceleration a ofa simple harmonic oscillator (frequency f ) on itsdisplacement from equilibrium position, x:

a = −(2πf )2x. (3)

NSPush

Push

SN

(a) Induced emf (or current)

(b) Induced emf (or current)

Figure 1. The problem of associating the minus signwith the direction of induced emf. (a) According toequation (1) E is negative; (b) according toequation (1) E is still negative!

In this equation, both a and x are vector quantities,and the minus sign is used simply to indicate thata points in the opposite direction to x.

The use of the minus sign in Faraday’s law(equation (1)) is clearly different, since none of the

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Understanding and using the minus sign in Faraday’s law

Box 1. Using the equation E = −N∆Φ∆t

to work out the

direction of the induced current around a coil.

Step 1:Point the thumb of your right hand in the direction of the applied magnetic field through the coil,and curl your fingers. The way your fingers point around the coil establishes what we’ll call thepositive (+) sense of the induced emf.

Step 2:Look at the equation at the top of this box. Think of � as representing the number of field linesgoing through the coil (‘flux linkage’). If this is getting larger, then �� is positive and hence Eis negative. If, on the other hand, the number of lines is getting smaller, then �� is negative andhence E is positive.

Example:

NSPush

Step 1Thumb points tothe right so apositive E means

Step 2Flux linkage is increasing. Hence ∆Φis positive and therefore E is negative.

quantities in it are vector quantities. This meansthat we cannot associate the minus sign explicitlywith the direction of induced emf, direction of fluxor direction of current. Figure 1 illustrates thepotential pitfalls of such an association: a minussign for E can mean current induced either wayaround a coil, depending on the circumstances.

Two issues arise from the discussion above.Firstly, what does a negative sign for E actuallymean? Secondly, is there a way in which we canteach students to use the minus sign in order topredict the direction of the induced current? Thisarticle deals with the second point first, since this isof more immediate practical relevance to students.The section on the origin of the minus sign andits link to Lenz’s law includes vector and calculusnotation which may be unfamiliar to students but isincluded to help teachers clarify how an equation

that appears to include only scalar quantities cangive information about the direction of inducedcurrent.

Getting students to use the minus signThe two-step method described in Box 1 enablesstudents to work out the direction of inducedcurrent in a coil directly from equation (1), that iswithout the need for an independent application ofLenz’s law. A variety of practical examples, whichillustrate the use of the method, are described inBox 2.

Discussion of the methodA small amount of practice with this procedureenables students to predict the direction of theinduced emf (and hence the induced current)

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C Jones

Box 2.

A. Withdrawing a bar magnet from a coil.

Pull

NS

Step 2Flux linkage decreasing, so ∆Φ isnegative and therefore E is positive.

Step 1Positive sense.

B. Stretching a loop of conducting rubber.

(Magnetic field into page)

Stretch Stretch

Step 1Positivesense.

Step 2Flux linkage increasing,so ∆Φ is positive andtherefore E is negative.

C. Rolling a conducting bar along conducting rollers.

Roll

A

A

(Magnetic field into page)

Step 2Flux linkage decreasing,so ∆Φ is negative andtherefore E is positive.

Step 1Positivesense.

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Understanding and using the minus sign in Faraday’s law

relatively quickly in all cases involving electro-magnetic induction in coils or single loops.

Once students have become familiar with theuse of the method, it is of course necessary tojustify its use by demonstrating that the methodgives results that are in agreement with experiment.That is, if the minus sign is left out of the equation,the method predicts current the opposite wayaround to that actually observed! This would bea convenient point to introduce Lenz’s law in itsusual form.

I believe that there are advantages in usingthe method discussed in this paper to predictthe direction of the induced current. Firstly,students gain an appreciation that the minus signis of practical importance in equation (1), and isnot something included almost as an afterthoughtfor ‘completeness’. Secondly, by practisingthe method, students become familiar with thedelta notation and the various symbols involvedin Faraday’s equation. Thirdly, the method isfully consistent with the right-handed conventionused in more mathematical treatments; a studentpursuing physics at an undergraduate level wouldnot have to ’unlearn’ the method; indeed practicewith the method could help to clarify the use ofnegative signs in vector algebra. These points arediscussed further in the next section.

The origin of the minus signAlthough the quantities E and � presented inequation (1) are not themselves vector quantities,they are each related to the dot product of twoother vector quantities. Hence they each containinformation about the relative directions of thesetwo quantities.

For the case of the induced emf the two vectorquantities concerned are E, the induced electricfield and dl, an increment of path around a loop C:

E =∮

CE · dl. (4)

Here E · dl is the dot product of the quantities E

and dl (the dot product is positive if E and dl arein the same direction).

Similarly, the magnetic flux across a surfaceS is

� =∫

SB · dS (5)

dSdl

Figure 2. The definitions of dl and dS.

where B is the magnetic flux density and dS is anincrement of area, part of any surface bounded byC; see figure 2.

The directions of E and B follow the usualconventions for electric fields and magnetic fieldsrespectively. The direction of dS may be chosenarbitrarily, either outwards from the surface (asshown in figure 2) or inwards. This arbitrarychoice means that we might as well choose dS tobe in the direction of the magnetic field, i.e. in thedirection of vector quantity B (see for example[1, 2]). This is convenient since the dot productB · dS is then always positive, and hence themagnetic flux linkage � may always be taken aspositive. This justifies the use of � as a positivenumber indicative of the flux linkage (see step 2of the method).

Once a direction has been chosen for thevector quantity dS, it is necessary to use amathematical convention to relate the direction ofdl to dS, and it is this convention that is ultimatelyresponsible for the necessity of the minus sign inequation (1). The convention is simply the oneused in step 1: the right-hand convention. That is,the positive sense for dl is set by the right-handgrip rule with the thumb pointing in the directionof dS (see figure 2).

A positive E is therefore one in which theinduced electric field E (and hence the inducedcurrent) is in the same direction as dl, makingthe dot product E · dl (and hence the right-handside of equation (4)) positive. A negative E is onein which the induced electric field E (and hencethe induced current) is in the opposite directionto dl, making the dot product E · dl (and hencethe right-hand side of equation (4)) negative. Ofcourse, the positive sense for dl must first havebeen determined using the convention describedabove; this is what is involved in step 1 of themethod presented in this paper.

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ConclusionExplaining the minus sign in Faraday’s law bysaying it is a simple consequence of Lenz’s lawis not telling the whole story! It is in fact aconsequence of two things: (i) the decision to usethe right-handed convention, and (ii) the necessitythat the resulting equation predicts a direction forthe induced current which agrees with experiment(i.e. Lenz’s law). In other words, if we had chosento use a left-handed convention, then a minus signwould not be necessary but the equation would stillagree with Lenz’s law!

This article has presented a method by whichAdvanced level students (16–18 years) can use theminus sign in Faraday’s law to predict the directionof an induced current in a coil or loop of wire.The method is simple to use for all students andalso completely consistent with the right-handed

convention, which is useful for students going onto study undergraduate physics.

Received 17 June 2003, in final form 11 July 2003PII: S0031-9120(03)64836-7

References[1] Duffin W J 1990 Electricity and Magnetism

3rd edn (New York: McGraw-Hill) ch 9[2] Lorrain P, Corson D P and Lorrain F 1988

Electromagnetic Fields and Waves 3rd edn(San Francisco: Freeman) ch 23

Chris Jones has taught AS-level andA-level physics at Hereford Sixth FormCollege for five years, after gaining aPhD in semiconductor physics from theUniversity of Exeter.

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