Question

Non-relativistic protons move rectinearly in the region of space where there are unifrom mutually perpendicular electric and magnetic fields with `E = 4.0 kV//m` and `B = 50 mT`. The trajectory of the protons lies in the plates `xz` (Fig) and forms an angle `varphi = 30^(@)` with the `x` axis. Find teh pithc of the helical trajectory along which the protons will move after the electric field is swiched off.

Answer 1

When the electric field is swichted off, the path followed by the particle will be helical. And pitch, `Delta l = v_(||) T`, (where `v_(||)` is the velocity of the particle, parallel to `vec(B)`, and `T`, the time period of revolution.)
`= v cos (90 - varphi) T = v sin varphi T`
`= v sin varphi (2pi m)/(aB)` (as `T = (2pi)/(qB)`) ...(1)
Now, when both the fields were present , `qE = qvB sin (90 - varphi)`, as no net force was effective on the system.
or, `v = (E)/(B cos varphi)` ....(2)
From (1) and (2), `Delta l =v (E)/(B) (2pi m)/(qB) tan varphi = 6 cm`.