A particle of charge ` +q` and mass `m` moving under the influence of a uniform electric field ` E hat(i)` and uniform magnetic field ` B hat(k)` follows a trajectory from ` P to Q` as shown in fig. The velocities at `P and Q` are ` v hat(i)` and ` - 2v hat(j)` . which of the following statement(s) is/are correct ?

Explanation

(a) Work is done by electric force, as magnetic force does no work. Work done by electric force from `P` to `Q`
`W =` force `xx` displacement along direction of-force
`= (qEhat(i)).(2ahat(i)) = 2qEa`
By work-energy theory
`W = Delta K = K_(Q) - K_(P)`
`2qEa = (1)/(2)m[(2v)^(2) - v^(2)] = (3)/(2) mv^(2)`
`E = (3)/(4)(mv^(2))/(qa)`
(b) Rate of work done by electric field
At `P`:
`P_(P) = vec(F).vec(v) = qEhat(i).vhat(i) = qEv = (3mv^(3))/(4a)`
`P_(Q) = 0`, as force and velocity are perpendicular.
( c) `vec(F)_(m)` is always perpendicular to `vec(v)`,
`P = 0`