The multiplication of bananas by umbrellas - …

1 Electromagnetic phenomena 24 The multiplication of bananasby umbrellasby Laithwaite'Adult people are very lothe to show their ignorance totheir fellow men, which is a pity, because they then goaround feeling inferior when, were they to declare theirignorance to a companion or small group, the odds are'that nobody would know anyway!Small children are notso commonly afflicted this they become teenagerstheybegin to have pride and to pretend, in the main, thatthey understand, certainly the fashionable words of the alas, it is with older they get themore reluctant they become to say"I don't know."Juste-Fler World War II, there was a delightful film called" what sailors are," in which three junior naval officersin port went out drinking and had rather too way back to the ship they found an old perambulatorchassis on a bombed site and used it to wheel along themost inebriated of the they saw a pawn-broker's sign-three brass balls-and climbed up a tele-graph pole and took it down as a trophy (as drunks will).

2 When they reached the ship they were slightly soberedbut decided that it would be fun to sneak aboard a neigh-bouring ship belonging to a foreign power and mount thebrass balls and pram chassis on a turntable which lookedas if it should carry a radar inthe early hours of the morning and even managed to paintit grey to match the turntable. They returned to their shipwithout of the officers in question was the junior signalsofficer and in the morning his immediate superior saw theerection on the foreign ship and got his binoculars on it."What's that?"he snapped to the junior with the discovery of their misdeeds he thought^uickly and said with confidence, "It's a type 291.

3 Sir!"`A type 291?" queried his elder."Yes sir, the verylatest in radar detectors." The senior man frowned butwas not going to show his ignorance by asking furtherquestions, so his face suddenly lit up in recognition-" Ofcourse." he said, "the 291."When the Captain spottedit. he sent for the senior signals officer and the conversa-tion was repeated almost word for word except that thesignals officer said it with even more assurance:"Youknow sir, the jolly old type 291." "Yes of course "-andthe higher in rank it went the less likely was anyone tosay "What's a type 291?" In the end the news reachedthe foreign powers on the ship itself whowereeven lesslikely to admit that something had got itself put therewithout -their knowing, so they set their spies to work andthe junior signals officer was tracked down as probably theonly man who knew, so they kidnapped him and all hisprotests that it was three brass balls and a pram chassiswere in vain!

4 So we teach our students about vector quantities-things that have both magnitude and direction, such asvelocity, force. B and H." We tell our students how toadd such things-" parallelogram of vectors" and all that. ImperialCollege,London822 Fig. 1. The " parallelogram "( "in this case) ofvectors is obviousifthe vectors are distancesIf you can add distances AB and BC and get AC, as isobvious from Fig. 1, you can travel in these directions atcertain speeds so you can add velocities in the same ,whynotd2x/dt2?Whynotindeed?Multiply the diagram by mass and you are adding forces(M d2x/dt2).All this is fine, but the electrical student thenruns up against it.

5 To cause forces due to electromagnetismyou need to have magnetic flux density B not in the sameline as an electric current which then cuts it, and experi-ment shows that the force thereby produced is proportionalboth to current and to B and to thesineof the singlebetween find out which way the force acts youhold up your left hand (or was it your right?) in the mannershown in Fig. 2, and repeat the magic words:"thuMb-motion. Forefinger-Field. seCond finger-Current"!Were not weallreared on this rule? But to how manyof us was it openly admitted. let alone stressed, that wewere multiplying essentiallydifferentthings together andgetting a third thing which was not related to the firsttwo, except in dimensions were differentin our particular case flux density was a fiction anyway-and the rule made as much sense as saying that when youmultiply bananas by umbrellas you get grandfather clocks!

6 (except that in science at least one of the things involvedis a fairy castle and never so earthy as a banana ).Some of our physics teachers might have us believethat the process we call" vector multiplication "is simple-even obvious."You multiply a length by a length andget an area (Fig. 3a).An area is represented by a vectornormal to its plane."Stuff and nonsense!In the firstplace you cannot tell whether the so-called" vector "areais to be drawn upwards or downwards, but suppose weattempt to go into the third x lengthx length = volume (Fig. 3b). as we all letus do it by vectors. AB XBC = OP (so some of us aretold). If that be so, then OP X BD = O.

7 For the two all volumes are zero!Let's face do not now look like true vectors,yetwhen werefer to distances walked along the ground they must beadded like velocities and forces. whicharetrue Revicw20;27 December1974 Fig. 2. The"sacred"righthandruleElectricalReview 20127 December1974(a)(b) (a)Multiplicationofdistanceby distanceappears to beconsistentwiththerules of vector multiplication , but (b) a triplemultiplication of distances does notThe solutiontothis apparent paradox lies in a truthwhich I was never told as a pupil or as a thisgame of multiplying bananas by umbrellas , youarenorallowed to multiply bananas by bananas , nor umbrellas tells us we can onlyaddbananas to cannot multiply them is a ludicrous whycan we multiply bananas by umbrellas , and therefore fluxdensity by electric current?

8 Why did no one tell me early'n my career that the vector products"velocityxvelocity"and"force x force" are meaningless? and what is quantities like force, acceleration,current and fluxcannot be taken in pairs at random and multiplied vectori-ally or the answer may be as meaningless as tryingtomultiply beef steak by physical the latterproduct might even be said to make more sense as if wewrote:Beef steakxphysical exercise = athleticsI have no doubt that if this equation were to be "putacross"in the classroom with as much conviction as areFleming's Left and Right Hand Rules the children wouldhave no difficulty in swallowing they could notdigest it, and I suggest that we have never digested vectormultiplication, for if we had, we would not continue toregard gyroscopes as" very difficult to understand," whichis the reaction of most engineers when askedwhat theyknow about these curious more learned and those preferring the mathematicalapproach to their engineering will probably reply.

9 " Theiraction can all be understoodby the Conservation ofMomentum," with the same degree of conviction as "TheType 291."For there is more to gyroscopes than merevector multiplication and the extension of -the linear applica-tion of Newton's Laws which gives the equation:Force = massxacceleration (1)to the rotary case of:Torque = Moment of inertia x angularacceleration(2)for equation (1) has effectively been multipliedthroughoutby length, which may or may not remain a constant as amass revolves, as for example, if it consisted of a deform-able we venture into the uncharted seas of the us look at the first occasion on which thewool waswell and truly pulled over our eyes on the subject of appliedmathematics, as taught at" A "level.

10 "Circular Motion,"I believe, is the heading. Centrifugal force (or should it be"centripetal"?)is about to be derived, and I have heardmany pedants argue on which name shouldhe you prefer, it is the thing that keeps the stringin tension when you whirl a chestnut around on the endof refresh your memories, the"proof"goeslike this:Consider a wheel (Fig. 4a) to be rotating atconstantangular velocity at. Instantaneously, the point A is travel-ling at velocity v, (and v, is a vector ).At an instant dtlater, the point A is displaced by angle dh), and its velocity,although of the same magnitude, is now find thechangein velocity, subtract v,from v, (vectorially, ofcourse) and this has been done on Fig.