Tiêu đề SP-4_Ch-19_Moving Charges and Magnetism.pdf ✅

Chapter Contents Aakash Educational Services Limited - Regd. Office : Aakash Tower, 8, Pusa Road, New Delhi-110005 Ph.011-47623456 Introduction Magnetic Force Motion of Charged Particle in a Magnetic Field Motion in Combined Electric and Magnetic Fields Magnetic Field due to a Current Element : Biot-Savart Law Magnetic Field on the Axis of a Circular Current Loop Ampere's Circuital Law Applications of Ampere's Circuital Law The Solenoid and the Toroid Magnetic force on a current carrying conductor Force between two Parallel Currents, the Ampere Torque on Current Loop, Magnetic Dipole The Moving Coil Galvanometer Introduction In the previous chapters we discussed how a charged object produces an electric field. Similarly a magnet produces a magnetic field-at all points in space around it. About 200 years ago, in 1820 it was realised that both electricity and magnetism are intimately related. During a lecture demonstration, the Danish physicist Hans Christian Oersted noticed that a current in a straight wire caused a noticeable deflection in a nearby magnetic compass needle. Oersted concluded that moving charges or currents produce a magnetic field in the surrounding space. In this chapter, we will see how magnetic field exerts force on moving charged particle, like electrons, protons and current carrying wires. We shall also learn how current produces magnetic fields. We shall see how particles can be accelerated to very high energies in a cyclotron. We shall study how currents and voltages are detected by a galvanometer. Chapter 19 Moving Charges and Magnetism MAGNETIC FORCE Magnetic Field, Lorentz Force Consider a point charge q moving with velocity v is located at position vector r at a given time t. If electric field E and magnetic field B are existing at that point, then force on the electric charge q is given by       electric magnetic   F qE v B F F The force exerted by a magnetic field is called magnetic force and is written as   ( ) F qv B , here q is the charge of the particle, v is its velocity and B is the magnetic field. The direction of magnetic force is perpendicular to both v and B .
2 Moving Charges and Magnetism NEET Aakash Educational Services Limited - Regd. Office : Aakash Tower, 8, Pusa Road, New Delhi-110005 Ph.011-47623456 Example 1 : A charge particle having charge 2 coulomb is thrown with velocity 2 3 i j - -inside a region having E j  2 and magnetic field -5 . k Find the initial Lorentz force acting on the particle. Solution : Lorentz force   ( ) F qE v B   -F j ij k   22 2 3 5     = 2 2 10 15 j ji         -- = 30 16 i j - -Direction of Magnetic Force The direction of magnetic force experienced by a moving charge is the direction of cross-product Bv for a positive charge and opposite for a negative charge. It can be determined by right hand thumb rule or right hand screw rule. When  = 90°, magnetic force is maximum in magnitude Fmax = qvB and direction is obtained by  v B × B v v B ×  B v Fleming’s left hand rule : If we stretch the thumb and first two fingers of left hand in mutually perpendicular directions such that forefinger points along B and middle finger points along v , then the thumb points along F . B v F F B v Properties of Magnetic Force on Charge (i) As F qv B is ,    thus from the property of cross product, force is perpendicular to velocity as well as magnetic field vector. (ii) If the charge is not moving   ||0 v  , the magnetic force is zero.
NEET Moving Charges and Magnetism 3 Aakash Educational Services Limited - Regd. Office : Aakash Tower, 8, Pusa Road, New Delhi-110005 Ph.011-47623456 (iii) If the charged particle is moving either parallel or antiparallel to magnetic field then force on it is zero ( = 0o or  = 180o). (iv) Magnetic Lorentz force cannot change the speed and kinetic energy of a charged particle but changes its velocity and momentum (only direction). (v) As F v  , F v. (power delivered by magnetic Lorentz force) = 0. It can also be noted that work done by it is also always zero. Example 2 : A charged particle of charge 1 mC and mass 2 g is moving with a speed of 5 m/s in a uniform magnetic field of 0.5 tesla. Find the maximum acceleration of the charged particle. Solution : m = 2 × 10–3 kg; v = 5 m/s; B = 0.5 tesla; q = 1 × 10–3 C F qv B      F = qvBsin  F = 10–3 × 5 × 0.5 × sin 90° N  F = 2.5 × 10–3 N F a m  –3 3 2.5 10 N/kg 2 10 a      a = 1.25 m/s2 Example 3 : A charged particle of specific charge (i.e. charge per unit mass) 0.2 C/kg has velocity 2 3 i j - -(m/s) at some instant in a uniform magnetic field 52 .    --i j tesla Find the acceleration of the particle at this instant. Solution : ( ) F q a vB m m     0.2 C/kg q m v ij   (2 3 ) m/s B ij   (5 2 ) T Therefore, a ij ij   0.2(2 3 ) (5 2 )   m/s2 = --0.2[4 15 ] k k  m/s2
4 Moving Charges and Magnetism NEET Aakash Educational Services Limited - Regd. Office : Aakash Tower, 8, Pusa Road, New Delhi-110005 Ph.011-47623456 Example 4 : A charged particle moving in a magnetic field     tesla B ij , has an acceleration of 2i j   --(m/s2) at some instant. Find the value of . Solution : F qv B   ( ) ( ) q a vB m    a B  Thus a B. 0   (2 ).( ) 0 i ji j     ---- 2 –  = 0   = 2 MOTION OF CHARGED PARTICLE IN A MAGNETIC FIELD As the magnetic force on a charged particle is perpendicular to the velocity, it does not do any work on the particle. Hence, the kinetic energy or the speed of the particle doesn’t change due to the magnetic force. When a charged particle q is thrown in magnetic field B with a velocity v then the force acting on the particle is given by F = Bqvsin where  is the angle between the velocity and the magnetic field. Case I :  = 0° or 180° Path followed : Straight line q q  = 0°  = 180° B   sin0 0 F qvB So it will continue to move in a straight line with constant velocity. Case II :  = 90° Path followed : Circular The magnetic lorentz force acts perpendicular to velocity; so it provides the centripetal force causing the charged particle to move in a circular path of radius R with constant speed. 2 mv F qvB R   Radius of circular path  mv R qB  If charged particles enters with same momentum mv P R qB qB   If charged particles enters with some kinetic energy, 2  km R qB Angular velocity ( ) v qB R m   F P q F B q F q r Time period of revolution (T) = 2 2 m qB     Frequency of revolution (f) = 1 2 qB T m  
NEET Moving Charges and Magnetism 5 Aakash Educational Services Limited - Regd. Office : Aakash Tower, 8, Pusa Road, New Delhi-110005 Ph.011-47623456 When a charged particle is projected perpendicular to a magnetic field, (i) Its path is circular in a plane perpendicular to the plane of magnetic field. (ii) The speed and kinetic energy of the particle remains constant. (iii) The velocity and momentum of the particle changes only in direction. (iv) The time period of revolution, angular velocity and frequency of revolution is independent of velocity of the particle and radius of circular path, this fact is used in cyclotron. Case III :  is other than 0°, 180° and 90° Path Followed : Helical A particle of mass m and charge q is projected in x-y plane with speed v making an angle  with magnetic field B (x-axis). The velocity of the particle can be resolved into two components. The component parallel to the field is given by v|| = v cos and the component perpendicular to the magnetic field is v = v sin. B x y v z v v||  The component v|| makes it to move parallel to B (i.e., along x-axis) while, due to v the charge particle experiences a force that makes it to move in a circle. The result is a helical path. (a) Radius of the helix = 2 mv qv B r    mv r qB    mv sin r qB   B y v Pitch  (b) Frequency of revolution ( ) 2 qB f m   (c) Time period of revolution 2 ( ) m T qB   (d) Pitch of the helix : It is the distance moved by the charge along B axis in one complete revolution or one time period. Pitch = v|| T = 2 cos m v qB   Notes on motion of charge in magnetic field (i) 2r O  v q × × × × × × × × × × × × × × × × × × B (ii) O v q, v × × × × × × × × × × × × × × × × × × r   2 B Radius of circular path qB m r v  Radius of circular path v  m r qB Time spent in magnetic field qB m t   Time spent in magnetic field 2  m t qB