Suppose a nucleus initally at rest undergoes `alpha` decay according to equation
`._(92)^(235)X rarr Y +alpha`
At `t=0`, the emitted `alpha`-partilces enter a region of space where a uniform magnetic field `vec(B)=B_(0) hat j` and elcertis field `vec(E)=E_(0) hati` exist. The `alpha`-prticles enters in the region with velocity `vec(V)=v_(0)hat j` from `x=0`. At time`t=sqrt3xx10^(6) (m_(0))/(q_(0)E_0)s`, the particle was observed to have speed twice the initial velocity `v_(0)`. Then, find (a) the velocity `v_(0)` of the `alpha`-particles,
(b) the initial velocity `v_(0)` of the `alpha`-particle, (c ) the binding energy per nucleon of the `alpha`-particle.
`["Given that" m(Y)=221.03 u,m(alpha)=4.003 u,m(n)=1.09u,m(P)=1.008u]`.

Answer

`10^(7) ms^(-1)`

Explanation

Charge on `alpha`-particle,`q_(alpha)=4 xx 1.6 xx10^(-19)C` and `1u=931 MeV//c^(2)`. Force acting on `alpha`- particle,
`vecF_(B)=q(vecvxxvecB)+q_(alpha)E_(0)hat i`
Hence, the particle will move in a circular path in `y_z`Plane . In the same time, it will move x-direction. Hence, resulting path will be a helical spiaral with increasing pitch. Thus,velocity of the circulating `alpha`-particles after time `t`.
`vecv_(0)=v_(0) cos theta hat j-v_(0) sin thetahat k`
Due to electric field present, the velocity of the `alpha`-particle will incerease along x-axis . Hence, velocity of `alpha` particle along x-axis,
`vec(v_x)=(q_(0) E_(0))/(m_(0))t hati`

Hence,velocity of `alpha`-particle,
`vecv=((q_(0)E_(0)t)/(m_(0)))hat i+v_(0) cos theta hat j-v_(0) sin theta hat k` (i)
`theta =omega t=(2 pit)/(T)`
where `T= (2pim)/(q_(alpha) B)`
Hence ,
`vecv=((q_(0)B_(0))/(m_(0)))t hat i+v_(0) cos ((2pi t)/(T)) hat j-v_(0) sin ((2pi)/(T)) hat k` (ii)`
Give, `|vecv|=2v_(0), at t=sqrt3xx10^(7) (m_(0))/(q_(0)E_(0))s`
`4v_(0)^(2)=(sqrt2xx10^(7))^(2)+v_(0)^(2)`
`because v_(0) =10^(7) m s^(-1)`
When an `alpha`-particle is emitted with velocity `v_(0)` from a stationery nucleus` X`, decay product (nucleus `Y`) recoils. Hence,
`m_(Y)V=m_(alpha)v_(0)`
`rArr V=(m_(alphav))/(m_(Y))=1.811xx10^(5) m s^(-1)`.