Motion of a charged particle in uniform magnetic field
In this topic Motion of a charged particle in uniform magnetic field we will discuss about how a charged particle moves in uniform magnetic field, when it is projected with some velocity at an angle with magnetic field.
Before to know about this topic students must know about the force acting on a charge particle when it moves in a uniform magnetic field. To know about this topic force on a charge particle in uniform magnetic field click here.
Motion of a charged particle in uniform magnetic field –
Suppose a charged particle of mass ‘m’ and charge ‘q’ is projected with velocity ‘v’ at angle ‘ϴ’ with the magnetic field ‘B’ as shown in figure (a) . Here v has two components (v1= vcosϴ ) along x -axis and ( v2= v sinϴ) along y-axis.
Due to v1 it moves along x-axis and due to v2 a force F acts which is given as F = qv2 B = qvBsinϴ. Since this force F acts perpendicular to the velocity and magnitude of v2 not changes hence it moves on a circular path. As shown in fig(b).
let r is the radius of the circular path , due to these two components of v1 and v2 particle follows a helical path as shown in fig(a) .
As we know when body moves on a circular path the centripetal forc F = mv22/r = mv2sin2ϴ/r ;
Which is provided by magnetic force F= qvBsinϴ
So we can write mv2sin2ϴ/r = qvBsinϴ;
So, r = mvsinϴ/qB ……………….(1)
or vsinϴ=qBr/m …………………….(2)
angular velocity ω= vsinϴ/r = Bqr/mr = Bq/m ……………..(3)
here frequency f=ω/2∏ = Bq/2∏m ( here frequency is independent of velocity)………….(4)
and time period T = 1/f = 2∏m/Bq ……………………………(5).
The pitch of the helix = vcosϴ x T = v cosϴ 2∏m/Bq .
Special cases –
Case (i) -If ϴ=00 i.e. v1=v and v2=0
So there will be no force and particle will move in the direction of B .
Case (ii) – if ϴ=900 then v1=0 and v2=v i.e. body will move only in the circular path
And radius of the circular path r= mv/qB or r= v/(q/m)B .