Discuss the motion of a charged particle in a uniform magnetic field with initial velocity perpendicular to the magnetic field.

Explanation

1. When a charged particle having charge q and velocity `vecV` enters a magnetic field `vecB`, it experiences a force,
`vecF=q(vecVxxvecB)`
This force is centripetal or which circulate this particle to uniform circular motion.
2. (i) When `vecVandvecB` both are perpendicular to each other then only it can perform uniform circular motion which is shown in figure.

3. Magnetic force Bqv provides by the centripetal force.
`therefore` For uniform circular motion,
Centripetal force = Magnetic force
`(mv^(2))/r=Bqv`
`thereforer=(mv)/(Bq)" "...(1)`
but substitute p = mv,
`thereforer=p/(Bq)`
which indicate that as momentum increase radius of circle also increase.
4. We know that linear velocity `v=omegar`
`thereforev/omega=(mv)/(Bq)" "(because"Substituting "r=v/omega"in equ. (1)")`
`thereforeomega=(Bq)/m" "...(2)`
where `omega` is angular frequency.
but `omega=2piv` where v is frequency.
`thereforev=omega/(2pi)=(Bq)/(2pim)`
which is also known as cyclotron frequency.
Thus, frequency of charge doesn.t depend on linear velocity or energy which is very useful in construction principle of cyclotron.
5. Time period of it is,
`T=1/v`
`thereforeT=(2pim)/(Bq)" "...(3)`
(ii) If velocity particle makes angle `theta` with magnetic field then two perpendicular components of velocity have to be considered :
(a) parallel or antiparallel to field
(b) perpendicular to field.

6. If there is a component of the velocity parallel to the magnetic field ir will make the particle move along the field and the path of the particle would be a helical one as shown in figure.
7. The distance moved along the magnetic field in one rotation is called pitch (P).
`therefore` Pitch p = `v_(||)T`
(where `v_(||)` is parallel component of velocity)
`thereforep=vcosthetaxx(2pim)/(Bq)`
`thereforep=(2pimvcostheta)/(Bq)orp=(2pimv_(||))/(Bq)`
8. The radius of the circular component of motion is called the radius of the helix.