Question

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.

Answer 1

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`