Question

A slightly divergent beam of charged particles accelerated by a Potential difference V propogates from a point A along the axis of a solenoid. The beam is brought into focus at a distance l from the point A at two successive values of magnetic induction `B_1 and B_2.` Find the specific charge `q//m` of the particles.

Answer 1

Let us first calculate the velocity of the particles from the
energy equation,
`1/2mv^2=Vq implies v=sqrt((2Vq)/m)`
Since the charged particles are slightly divergent, they will follow
a helical path. Let `theta` be the small angle made by a particle with
`B cos theta~=1`
`:.` P(pitch of the particle)=`v_(||)xxT`
`=v cos thetaxx(2pim)/(qB)=(2pivm)/(qB)`
Particles are focussed if l contains integral number of pitches.
`l=np implies p=l//n= l, l//2, l//3,....`
`:.` for two consecutive focussing (as B increases, p decrease)
`l=(2pimv)/(qB_1) and l/2=(2pimv)/(qB_2)`
or `B_1=(2pimv)/(ql) and B_2=(4pimv)/(ql)`,
or `B_2-B_1=(2pimv)/(ql) or B_2-B_1=(2pim)/(ql)sqrt((2Vq)/m),`
or `q/m=(8pi^2V)/(l^2(B_2-B_1)^2)`