As a charged particle 'q' moving with a velocity `vec(v)` enters a uniform magnetic field `vec(B)`, it experience a force `vec(F) = q(vec(v) xx vec(B)). For theta = 0^(@) or 180^(@), theta` being the angle between `vec(v) and vec(B)`, force experienced is zero and the particle passes undeflected. For `theta = 90^(@)`, the particle moves along a circular arc and the magnetic force (qvB) provides the necessary centripetal force `(mv^(2)//r)`. For other values of `theta (theta !=0^(@), 180^(@), 90^(@))`, the charged particle moves along a helical path which is the resultant motion of simultaneous circular and translational motions.
Suppose a particle that carries a charge of magnitude q and has a mass `4 xx 10^(-15)` kg is moving in a region containing a uniform magnetic field `vec(B) = -0.4 hat(k) T`. At some instant, velocity of the particle is `vec(v) = (8 hat(i) - 6 hat(j) 4 hat(k)) xx 10^(6) m s^(-1)` and force acting on it has a magnitude 1.6 N
Angular frequency of rotation of particle, also called the `cyclotron frequency' is

Options

A. `8 xx 10^(5) rad s^(-1)`
B. `12.5 xx 10^(4) rad s^(-1)`
C. `6.2 xx 10^(6) rad s^(-1)`
D. `4 xx 10^(7) rad s^(-1)`

Answer

D

Explanation

Given `|qvec(v) xx vec(B)|=1.6`
`implies |q(8hat(i)-6hat(j)+4hat(k)) xx (-0.4hat(k)) xx 10^(6)|=1.6`
`implies 4 xx 10^(6) xx q = 1.6 implies q=0.4 xx 10^(-6)C`
`omega=(qB)/(m) = (0.4 xx 10^(6) xx 0.4)/(4 xx 10^(-15)) = 4 xx 10^(-7) rad//s`.