Non-relativistic protons move reactilinearly in the region of space where there are uniform mutually perpendicular electric and magnetic fields with E and B. The trajectory of the protons lie in the plane X-Y as shown in Fig. 1.107 and forms an angle `phi` with X-axis. Find the pitch of the helical trajectory along which the protons will move after the electric field is switched off.

Explanation

When electric field is switched off, the path followed by the particle will be helical.
Pitch=`v_(||)T....(i)`
where `v_(||)` =velocity of the particle parallel to B, and
T=time of revolution `=((2pim)/(qB))`
Here, `v_(||)=v cos(90-phi)=v sinphi`
Hence, form Eq. (i)
`Pitch=(vsinphi)((2pim)/(qB))....(ii)`
When both fields are present
`qE=qvB sin (90-phi)=qvB cosphi`
or `v=E/(Bcosphi)....(iii)`
Subtituting the value of v from Eq. (iii) in Eq. (ii), we get
`Pitch=(E/(Bcos phi)xxsinphi)(2pim)/(qB)=(2pimE)/(qB^2) tanphi`