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A charged particle is introduced at the origin x=0, y=0, z=0 with a given initial...

A charged particle is introduced at the origin x=0, y=0, z=0 with a given initial velocity v . A uniform electric field E  and a uniform magnetic field B exist everywhere. The velocity v , electric field E  and a uniform magnetic field B are given in the columns below
Column 1 Column 2 Column 3
1. Electron with  v=2E0B0x^  (i) E=E0z^ (P)  B=-B0x^
2. Electron with  v=2E0B0y^ (ii)  E=-E0y^ (Q)  B=B0x^
3.Proton with v =0 (iii)  E=-E0x^ (R)  B=-B0y^
4. Proton with  v=2E0B0x^ (iv)  E=E0x^ (S)  B=-B0z^
In which case will the particle describe a helical path with axis along positive z direction?
(2)  (ii)  (R)
(4)   (i)   (S)
(3) (ii)  (R)
(4)  (ii)   (R)
Solution:

For path to be helix with axis along +ve z-direction, particle should experience a centripetal acceleration in x-y plane.

For the given set of options only option (4(i(S) satisfy the condition. Path is helical with increasing pitch.