Question

A particle of mass m and positive charge q, moving with a uniform velocity v, enters a magnetic field B, as shown in the figure. (a) Find the radius of the circular arc it describes in the magnetic field. (b) Find the angle subtended by the arc at the centre. (c) How long does the particle stay inside the magnetic field? (d) Solve the three parts of the above problem if the charge q on the particle is negative.
Figure

Answer 1

Explanation:

Particle mass = m

On a particle, the positive charge = q

Magnetic field = B

Uniform velocity = v

(a) The circular arc radius explained in the magnetic field by the particle is

We understand that,

r = mvqB

(b) The subtended angle by the centre arc

To arc ABC, the line MAB is tangent, so the angle defined by the charged particle,

∠MAO = 90°

Now, ∠NAC = 90°

OAC = OCA = θ

Then, AOC =

180°− (θ + θ) = π − 2θ

(c) The time taken by the particle to be in the magnetic field

Inside the magnetic field, the distance enclosed by the particle,

l = rθ .

Using r=mvqB

(d) Assume that the particle’s charge q is negative, then

(i) Circular arc radius, r = mvqB.

(ii) Within the magnetic field remains the arc centre. Consequently, the subtended angle = π + 2θ

(iii) Likewise, inside the magnetic field, to cover the path, the time taken by the particle =  mqB(π+2θ).