The motion of a charged particle in magnetic field
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In case of motion of a charge in a magnetic field, the magnetic force is perpendicular to the velocity of the particle. So no work is done and no change in the magnitude of the velocity is produced (though the direction of momentum may be changed). We shall consider the motion of a charged particle in a uniform magnetic field. First, consider the case of v perpendicular to B.
In case of motion of a charge in a magnetic field, the magnetic force is perpendicular to the velocity of the particle. So no work is done and no change in the magnitude of the velocity is produced (though the direction of momentum may be changed). We shall consider the motion of a charged particle in a uniform magnetic field. First, consider the case of v perpendicular to B.Learn more about Magnetic Force and Magnetic Field.
In case of motion of a charge in a magnetic field, the magnetic force is perpendicular to the velocity of the particle. So no work is done and no change in the magnitude of the velocity is produced (though the direction of momentum may be changed). We shall consider the motion of a charged particle in a uniform magnetic field. First, consider the case of v perpendicular to B.Learn more about Magnetic Force and Magnetic Field.The perpendicular force, q v × B, acts as a centripetal force and produces a circular motion perpendicular to the magnetic field. If velocity has a component along B, this component remains unchanged as the motion along the magnetic field will not be affected by the magnetic field.
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Motion of a charged particle in a uniform magnetic field :
Motion of a charged particle in a uniform magnetic field :
Let us consider a uniform magnetic field of induction B acting along the Z-axis. A particle of charge q and mass m moves in XY plane. At a point P, the velocity of the particle is v (figure).
The magnetic lorentz force on the particle is
F=q(v×B)
Hence F acts along PO perpendicular to the plane containing v and B. Since the force acts perpendicular to its velocity, the force does not do any work. So, the magnitude of the velocity remains constant and only its direction changes. The force F acting towards the point O acts as the centripetal force and makes the particle to move along a circular path.
At points Q and R, the particle experiences force along QO and RO respectively.
Since v and B are at right angles to each other
q(v×B)=Bqvsin90o=Bqv
This magnetic lorentz force provides the necessary centripetal force.
Bqv=rmv2
r=Bqmv ....(1)
It is evident from this equation, that the radius of the circular path is proportional to (i) mass of the particle and (ii) velocity of the particle.From equation (1), rv=mBq
ω=mBq ....(2)
This equation gives the angular frequency of the particle inside the magnetic field.
Period of rotation of the particle,
T=ω2π
T=Bq2πm ....(3)
From equations (2) and (3), it is evident that the angular frequency and period of rotation of the particle in the magnetic field do not depend upon (i) the velocity of the particle and (ii) radius of the circular path.