A charged particle of specific charge (charge/mass) `alpha` released from origin at time `t=0` with velocity `vec v = v_0 (hat i + hat j)` in uniform magnetic field `vec B = B_0 hat i.` Coordinates of the particle at time `t= pi//(B_0 alpha)` are

Options

A. `((v_(0))/(2B_(0)alpha), (sqrt2v_(0))/(alpha B_(0)), (-v_(0))/(B_(0)alpha))`
B. `((-v_(0))/(2B_(0)alpha), 0,0)`
C. `(0, (2v_(0))/(B_(0)alpha), (v_(0)pi)/(2B_(0)alpha))`
D. `((v_(0)pi)/(B_(0)alpha), 0, (-2v_(0))/(B_(0)alpha))`

Answer

D

Explanation

Specific charge, `alpha= (q)/(m)`, path of the particle will be a helix of time period,
`T= (2pi m)/(B_(0)q) = (2pi)/(B_(0)alpha)`
The given time, `t= (pi)/(B_(0)alpha)= (T)/(2)`
`:.` Coordinates of particle at time t= T/2 would be `(v_(x)T//2, 0,-2r)`
Here, `r= (mv_(0))/(B_(0)q)= (v_(0))/(B_(0)alpha)`

`:.` The coordinates are `((v_(0)pi)/(B_(0)alpha), 0,(-2v_(0))/(B_(0)alpha))`