Magnetic Field Charged Particle
When a charged particle moves through a magnetic field, the ‘magnetic’ force exerted by the field on the particle is,
Fm = q v × B
where v is the instantaneous velocity of the particle. The direction of Fm is perpendicular to both v and B. If v is parallel to B.Fm = 0, since v × B = 0.
Since the force is always perpendicular to direction of motion, it does not work on the particle:
= 0
As vectors d r and (d r)/dt are in the same direction. Hence, kinetic energy of the particle remains constant:
K = ½ mv2 = ½ m v . v
The speed of the particle, v = | v |, therefore does not change; only its direction changes. The equation of motion of the particle is,
We consider the motion under uniform constant magnetic field, B = B k; the Z-axis is defined along the direction of magnetic field.
The velocity v of the particle can be decomposed into components, parallel and perpendicular to B. That is, we write
v = v|| + v⊥
where v is along Z-axis and v⊥ lies in X-Y plane. Hence, we have
m d/dt (v|| + v⊥) = q (v|| + v⊥) × B
= q v⊥ × B
Since v|| × B = 0. The force vector v⊥ × B lies in X-Y plane (i.e. in plane perpendicular to B), and has no component parallel to B. Hence, we get
m d/dt v|| = 0
and, m d/dt v⊥ = q (v⊥ × B)
that is, the parallel component of velocity of the particle is not affected; particle continues to move along B (i.e. Z-axis) with constant velocity v|| = vz k. The perpendicular component interacts with magnetic field. In terms of X-Y components, we write
v⊥ = vx i + vy j
these are coupled equations.
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