A particle of specific charge `(q//m)` is projected from the origin of coordinate with initial velocity `[ui-vj]` unifrom electric magnetic fields exist in the region along the +ydirection, of magnitude `E` and `B` The particle will definitely return to the origin once if .

Options

A. `[vb//2piE]` is an integer
B. `(u^(2)+v^(2))^(1//2)[B//piE]` is an integer
C. `[vB//piE]` in an integer
D. `[uB//piE]` is an integer

Answer

C

Explanation

Initial velocity `u_(0)=uhati-vhatj`
`vecE=Ehatj,vecB=Bhatj`
Time period of rotation `t=(2pim)/(qB)implies` velocity along `y=-vhatj`
Whereas acceleration along `y=(qE)/m`
Displacemnet along y aixis will be zero. `impliesy=y_(y)t+1/2a_(y)t^(2)`
`0=-vt+(qE)/(2m)t^(2)impliest=(2mv)/(qE), T=(2pim)/(qB)`
`t/T=((vB)/(piE))impliest=((vB)/(piE))Timplies(vB)/(piE)` Inteter then it will come at origin.