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MOVING CHARGES AND MAGNETISM, , 4) INTRODUCTION, , In olden days. electricity and magnetism were treated as separate subjects. Electricity, dealt with interactions of charged bodies while magnetism dealt with interactions of magnets, and compass needles. However, in 1820, it was realised that they were intimately related., Experiments of Danish scientist Hans Christian Oersted showed that a compass needle is, deflected by passing electric current through a wire near the needle. From these experiments,, Oersted concluded that electric current through a conductor produces magnetic field in the, surrounding space. Ampere supported this observation by saying that electric charges in, motion produce magnetic fields. Michael Faraday showed that moving magnets or changing, magnetic fields generate electricity. Maxwell unified the laws of electricity and magnetism, and developed a new field called electromagnetism. Most of the phenomenon occuring, around us can be described under electromagnetism., , 42 OERSTED'S EXPERIMENTS, , Consider a magnetic needle NS which can rotate freely about a vertical axis in, horizontal plane. A conducting wire AB is held over the magnetic needle NS, paralle! t, it. When current flows from A to B, the N-pole of the magnetic needle gets deflected toward, west [Fig 4.1 (a)]. If the direction of the current is from B to A, the N-pole of the magne!, needle gets deflected towards east (Fig 4.1(b)]. That means, if the direction of current 15, reversed, the direction of the deflection of the magnetic needle is also reversed. The deflectiin, increases on increasing the current in the wire or bringing the magnetic need!z closer to, the wire. Since the magnetic needle can be deflected only by the interaction of anoth1, magnetic field, it is concluded that the current in the wire produces a magnetic field in, the surrounding space. The direction of deflection of magnetic needle due to the current, in the wire is given by Ampere's swimming tule., , , , , , Fig 4.1, , , , , , , , Ampere's swimming rule: Imagine a man swimming along the wire in the direction of, current with his fce always turned towards the needle such that current enters at his, feet and leaves w is head. Then, the north pole (N-pole) of the magnetic needle will, be deflected towards his left hand., , 4.3.5 MAGNETIC FORCE, , 4.3.1 Concept of magnetic field:, , In electrostatics we have studied that a static charge is the source of electric field., , Scanned with CamScanner

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fro UNIT - IV @, , >, A static charge produces an electric field E at every point in the space arount it, in, Ao sti Any, , 2 > od, charge q placed any point interacts with this electric field and experiences a force F =qE, Just as static charges produce an electric field, conductors carrying currents or Moving, , charges produce magnetic field, denoted as Bin the space around them. The Magnetic, soon as the current is switched off or the charges stop moving, [t Means, , , , field disappears, that a moving charge is a source of both electric field E and magnetic field B ,, >, The magnetic field B_ is a vector field. It has several Properties identical to electric, field, It is defined at cach pojnt in space, It is found to obey the principle of superposition,, , That is, the magnetic field B ata point due several sources is the vector sum of Magnetic, , a >> o> SO, field of each individual source. The magnetic field B atapointis B = B\+B,+B,+, , where B B, B, are the magnetic fields at a point due to each individual source, 19 Bay: Bay ssercisseemes ; ;, , at that point. Just as electric field is represented by field lines, magnetic field also is represented, , by field lines. Magnetic field lines have characteristics similar to that of electric field lines,, , but the difference is that magnetic field lines are closed paths without a starting point or, , ending point. If the magnetic field is emerging out of the plane of the paper it is depicted, , by a dot (¢). If the Magnetic field is going into the plane of the paper, it is depicted by, a cross (x),, , 4.3.2 Magnetic force on a moving charge, Lorentz force:, , . tee . 5 . . > . ., Consider a positive charge q moving with velocity y in the presence of magnetic, , , , >, field B. It is experimentally found that the magnetic force on the charge q is given by, > YX, F =q( vx B) stteesssnecsseee (4,1), , =, The magnetic force F on the moving charge has the, (i) — The magnetic force F depends on the, , charge q, velocity y and the magnetic ., , >, field B. Force on a Negative charge, is opposite to that on positive charge,, , following features,, , , , . > oe + (ii) Since the expression force F includes 7 SR wi, => a Bc oe eo or ay can t ie, a vector product of y and B, the Fig 4.2, , , , direction of the force is always Perpendicular to both velocity and magnetic field., , The direction of the force is given by tight hand rule or the right handed screw, rule for vector product or Cross product., , Scanned with CamScanner

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Cal, , ae ae |, MOVING CHARGES AND MAGNETISM |/197, , iii) The for 2 : «, a It is satan aaa if velocity and magnetic field are parallel to each other., velocity and magnetic field are perpendicular to each other., , iv) The magni i i, (iv) i ee of the force is zero if the charge is not moving (i.e. v = 0). Only, moving charge experiences force in a magnetic field., , The magnitude of the force on the charge q moving with velocity v in the magnetic, , field Bis Fa la(v«B)| = QVBSINO scscssssssssscsserseen (4.2)., , where © is the angle between the directions of y and R- We have F = B when, r _ >, q = +1, v = 1 and @ = 90°. Hence, magnetic field B at a point is defined as the, , force acting on a unit positive charge moving with unit velocity perpendicular to, the direction of magnetic field., . . >, The SI unit of magnetic field B is tesla (T). The magnetic field at a point, , is one tesla if a positive charge of one coulomb moving at right angle to the magnetic, , field. with a velocity of one metre per second experiences a force of one newton. That, , is B=1T ifq=+10C, v= 1 ms", 0 = 90° and F=1N., Another unit used for magnetic field is gauss. 1 gauss (G) = 10+ tesla., , , , Since Berane? dimensional formula of B is, , [oar, , [BI far] LT| oe, , , , If the positive charge q is moving with velocity y in the presence of both electric, , > >, field — and magnetic field B. the, , F=qEtd(vxB), , > >>, or F =q [e+ vxB (4.3)., , This force is called Lorentz force. The Lorentz force only in magnetic field is, , Paced), , 4.3.3 Magnetic force on a current-carrying conductor:, , force on the charge due to both of them is, , , , i ine rod (or a straight wire : os, Consider a conducting rod ( tralg ) of length / and uniform cross-sectional, , area A in an external magnetic field p . Let the number density of the mobile charge, carriers (electrons) be n. Then, the total number of mobile charee carriers in the rod is, 2g ‘, , Scanned with CamScanner

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198)| UNIT- IV @, , nAl. If q is the charge on each charge carrier, total charge is equal to nAlg,, , a steady current flows in this conducting rod, each mobile ange carrier has an de en, Crap,, , , , drift velocity In Mi presence of an external magnetic field B » the force on thes, , > charge carriers is F =nAlq Cy, x B). But nqvy= j is the current densiy,, , va, , >eat j ix B), cl :, The magnitude of current density is Jer: Therefore, I = jA. But current I is g not, , a vector. Hence, write, , where a is a vector of magnitude / and the vector sign is transfered from 7 to 7., >, The direction of / is same as the direction of + (or the direction of current). In this, >, , equation, B is not the magnetic field produced by the current carrying conductor (rod, , or wire) but it is the external magnetic field., , > > >, The direction of the force F is perpendicular to both / and B. It is given by, , the right hand rule used for cross product. The magnitude of the force is, , F=1/B sin@ -- .. (4.5) ', , , , >, , >, where @ is the angle between the directions of / and B. The direction of, _y, , | is same as that of current. The force is zero when the conductor is parallel or, , antiparallel to magnetic field. (i.e., 0 = 0° or 180°). It is maximum when the conductor, is perpendicular to the magnetic field (i.c., 8 = 90°). This maximum value of the force, is F=17B., , Eq. (4.4) holds good for a straight rod or straight wire. If the rod or wire has, an arbitrary shape, it can be considered to be made up of a large number of linear, , a : ~~ 5 ap gives, strips of length d/. The vector sum of the force = B. for each linear strip 8, , > >, the resultant force r on the conductor. That is Fe yl di xB, , In most of the cases the summation can be converted to integral., i+ MOTION IN A MAGNETIC FIELD, , 2. ‘peaae on, 6 gas yee a ree, When a charged particle is moving in a magnetic field, the magnetlc fo!, , Scanned with CamScanner

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ot MOVING CHARGES AND MAGNETISM [99, , ihe charged particle is perpendicular to its velocity. Therefore, no work is done by, the magnetic. force on the charged particle and there is no change in the magnitude of, yelocity: However, the direction of velocity (and momentum) may be changed., , <>, , Consider a pails of mass m and charge q moving with velocity v in a uniform, . ~ =| -» \, magnetic field B- The mingnetic force on the charged particle is F=q v~B |. When, ee, 4 ‘ . . ., vis perpendicular to B, this force acts as centripetal force and produces circular motion, , perpendicular to the magnetic field. Therefore, the charged particle will describe a circle, , =>, if vy and B are perpendicular to each other [Fig 4.3]. In this case, the magnitude of, , the magnetic force on the charged particle is F=q v B (‘." 6 = 90°). The centripetal force, , ~ A . 5 % . mv?, on the charged particle of mass m moving in a circular path of radius r is ——., r, , 2, ; MMe =iN. . 2., a =qvB or r Fr a aca (4.6), , The larger the momentum p , larger is the radius and bigger is the circular path., The stronger the magnetic field, smaller is the radues and smaller is the circular path. If @ is, , y_4B _ 9B, r, , , , the angular frequency, then velocity v = or. So, @=2av= =——and frequency V=, m, , 2am, , Therefore, the frequency v of revolution is independent of velocity or kinetic energy of, the particle., , , , , , , , >. >, If vy is not perpendicular to B, then poh ae,, —> = —>., the velocity has a component Vp along B Bk we OPT, > > V7 \e ff \ 3, and a componennt v, perpendicular to B. x fs ps AY, > /, The component Vp remains unchanged since ; [ > « © & Reed, the motion along the magnetic field is unaffected Ge #. 4, by the Magnetic field, The component of \ ae, 5. ee > : =, velocity v,, Perpendicularto B keeps changing “ 7, its direert... = : ‘ x x ., oe in a plane perpendicular to the Fig 4.3, netic field. Thus, the charged particle has, , , , . > , Cir . . ‘ ., Cular motion in a plane perpendicular to B and also moves along the direction B., S ‘ < : ae . ., , ma : result, it has helical motion as shown in Fig 4.4. The distance moved along the, Snetic field in one rotation is called pitch P., , Scanned with CamScanner