A particle of charge q and mass m is projected from the origin with velocity 
= v 0 
in a non-uniform magnetic field 
= – B 0 x 
. Here v 0 and B 0 are positive constants of proper dimensions. Find the maximum positive x-coordinate of the particle during its motion.
Text Solution
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Sol . Magnetic force on the particle at origin is along positive y-direction, so its trajectory will be concave up-wards. As the magnetic field is non-uniform, trajectory is not circular.
Magnetic force is always perpendicular to velocity. So its magnitude remains constant. Let at point P(x, y) its velocity vector make as angle θ with positive x-axis. Then the magnetic force will 
be at an angle θ with the positive y-direction. So,
a y = 
cos θ
or 
= 
[F B = Bqv 0 sin 90º]
or 
. 
= 
(v 0 cos θ )
where 
= v x = v 0 cos θ
Thus 
= 
x
At maximum x-displacement, velocity is along positive y-direction
or 
= 
or v 0 = 
or x max = 
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