Magnetism / Lorentz Force

(1) Introduction

• We have allready studied about thermal effects of current and now in the present chapter we are studied about magnetic effect of current.
  
• Earlier it was believe that there is no connection between electric and magnetic force and both of them are completely different.
  
• But in 1820 Oersted showed that the electric current through a wire deflect the magnetic needle placed near the wire and the direction of deflection of needle is reversed if we reverse the direction of current in the wire.
  
• So, Oersted's experiments establishes that a magnetic field is assoiated with current carrying wire.
  
• Again if we a magnetic needlle near a bar magnet it gets deflectid and rests in some other direction.
  
• This needle experiences the tourque which turn the needle to a definite direction.
  
• Thus, the reagion near the bar magnet or current carrying where magnetic needle experience and suffer deflection is called magnetic field.
  

(2) The Magnetic Field

• We all ready know that a stationery charges gets up a electric field E in the space surrounding it and this electric field exerts a force F=q₀E on the test charge q₀ placed in magnetic field.
  
• Similarly we can describe the intraction of moving charges that, a moving charge excert a magnetic field in the space surrounding it and this magnetic field exert a force on the moving charge.
  
• Like electric field, magntic field is also a vector quantity and is represented by symbol B
  
• Like electric field force which depend on the magnitude of charge and electric field, magnetic force is propotional to the magnitude of charge and the strength of magnetic field.
  
• Apart from its dependence on magnitude of charge and magnetic field strength magnetic force also depends on velocity of the particle.
  
• The magnitude magnetic force increase with increase in speed of charged particle.
  
• Direction of magnetic force depends on direction of magnetc field B and velocity v of the chared particle.
  
• The direction of magnetic force is not alonge the direction of magnetic field but direction of force is always perpendicular to direction of both magnetic field B and velocity v
  
• Test charge of magnitude q₀ is moving with velocity v through a point P in magnetic field B experience a deflecting force F defined by a equation
  F=qv X B
  
• As mentioned earlier this force on charged particle is perpendicular to the plane formed by v and B and its direction is determined right hand thumb rule.
  
  
  
  
  
• When moving charge is positive the direction of force F is the direction of advance of hand screw whose axis is perpendicular to the plane formed by v and B.
  
  
  
  
  
• Direction of force would be opposit to the direction of advance screw for negative charge moving in same direction.
  
• Magnitude of force on charged particle is
  F=q₀vBsinθ
  where θ is the angle between v and B.
  
• If v and B are at right angle to each other i.e. θ=90 then force acting on the particle would be maximum and is given by
  F_max=q₀vB                   ----(3)
  
• When θ=180 or θ=0 i.e. v is parallel or antiparallel to B then froce acting on the particle would be zero.
  
• Again from equation 2 if the velocity of the palticle in the magnetic field is zero i.e., particle is stationery in magnetic field then it does not experience any force.
  
• SI unit of strength of magnetic field is tesla (T). It can be defined as follows
  B=F/qvsinθ
  for F=1N,q=1C and v=1m/s and θ=90
  1T=1NA^-1m^-1
  Thus if a charge of 1C when moving with velocity of 1m/s along the direction perpendicular to the magnetic field experiences a force of 1N then magnitude of field at that point is equal to 1 tesla (1T).
  
• Another SI unit of magnetic field is weber/m² Thus
  1 Wb-m^-2=1T=1NA^-1m^-1
  In CGS system, the magnetic field is expressed in 'gauss'. And 1T= 10⁴ gauss. Dimention formula of magnetic field (B) is [MT^-2A^-1]

(3) Lorentz Force

• We know that force acting on any charge of magnitude q moving with velocity v inside the magnetic field B is given by
  F=q(v X B)
  and this is the magnetic force on charge q due to its motion inside magnetic field.
  
• If both electric field E and magnetic field B are present i.e., when a charged particle moves through a reagion of space where both electric field and magnetic field are present both field exert a force on the particle and the total force on the particle is equal to the vector sum of the electric field and magnetic field force.
  F=qE+q(v X B)                   (4)
  
• This force in equation(4) is known as Lorentz Force.
  
• Where important point to note is that magnetic field is not doing any work on the charged particle as it always act in perpendicular direction to te motion of the charge.
  

(4)Motion of Charged Particle in The Magnetic Field

• As we have mentioned earlier magnetic force F=(vXB) does not do any work on the particle as it is perpendicular to the velocity.
  
• Hence magnetic force does not cause any change in kinetic energy or speed of the particle.
  
• Let us consider there is a uniform magnetic field B perpendicular to the plane of paper and directed in downward direction and is indicated by the symbol C in figure shown below.
  
  
• Now a charge particle +q is projected with a velocity v to the magnetic field at point O with velocity v directed perpendicular to the magnetic field.
  
• Magnetic force acting on the particle is
  F=q(v X B) = qvBsinθ
  Since v is perpendicular to B i.e., angle between v and B is θ=90 Thus charged particle at point O is acted upon by the force of magnitude
  F=qvB
  and the direction of force would be perpendicular to both v and B
  
• Since the force f is perpendicular to the velocity, it would not change the magnitude of the velocity and the peffect of this force is only to change the direction of the velocity.
  
• Thus under the action of the magnetic force of the particle will more along the circle perpendicular to the field.
  
• Therefore the charged particle describe an anticlockwise circular path with constant speed v and here magnetic force work as centripetal force. Thus
  F=qvB=mv²/r
  where radius of the circular path traversed by the particle in the magnetic in field B is given as
  r=mv/qB                   ---(5)
  thus radius of the path is proportional to the momentum mv pof the charged particle.
  
• 2πr is the distance traveled by the particle in one revolution and the period T of the complete revolution is
  T=2 πr /v
  From equation(5)
  r/v=m/qB
  time period T is
  T=2πm/qB                   (6)
  and the frequency of the particle is f=1/T=qB/2πm                   (7)
  
• From equation (6) and (7) we see that both time period and frequency does not dependent on the velocity of the moving charged particle.
  
• Increasing the speed of the charged particle would result in the increace in the radius of the circle. So that time taken to complete one revolution would remains same.
  
• If the moving charged particle exerts the magnetic field in such a that velocity v of particle makes an angle θ with the magnetic field then we can resolve the velocity in two components v_parallel : Compenents of the velocity parallel to field
  v_{perpendicular} :component of velocity perpendicular to magnetic field B
  
• The component v_par would remain unchanged as magnetic force is perpendicular to it.
  
• In the plane perpendicular to the field the particle travels in a helical path. Radius of the circular path of the helex is r=mv_{perpendicular}/qB=mvsinθ/qB                   (8)