Question

An alpha particle (mass 6.4 × 10⁻²⁷ kg and charge 3.2 × 10⁻¹⁹ C) having 8.0 MeV energy, enters a region of a uniform magnetic field of 0.5 T. If the field is directed perpendicular to the velocity of the particle, find the radius of the circular path described by the particle. Mention the condition under which the particle in this region (i) describes a helical path, and (ii) goes straight undeviated.

Answer 1

Given: Mass of the alpha particle, m = 6.4 × 10⁻²⁷ kg

Charge of the alpha particle, q = 3.2 × 10⁻¹⁹ C

Energy of the alpha particle, E = 8.0 MeV

Magnetic field strength, B = 0.5 T

Formula: Magnetic force provides centripetal force:

`qvB = (mv^2)/r`

`r = (mv)/(qB)`

Convert energy to joules

E = 8.0 MeV

= 8 × 10⁶ × 1.6 × 10⁻¹⁹ 

E = 1.28 × 10⁻¹² J

The total energy is related to the kinetic energy K.E. (since the particle starts from rest):

K.E. = `1/2 mv^2`

E = `1/2 mv^2`

v = `sqrt((2E)/m)`

= `sqrt((2 xx 1.28 xx 10^-12)/(6.4 xx 10^-27))`

= `sqrt((2.56 xx 10^-12 xx 10^27)/6.4)`

= `sqrt((2.56 xx 10^15)/6.4)`

= `sqrt((0.4 xx 10^15)`

= `sqrt(4 xx 10^14)`

= 2 × 10⁷ m/s

The radius r of the circular path of a charged particle moving in a magnetic field is given by the formula:

r = `(mv)/(qB)`

= `((6.4 xx 10^-27) xx (2 xx 10^7))/((3.2 xx 10^-19) xx (0.5))`

= `(12.8 xx 10^-20)/(1.6 xx 10^-19)`

= `(1.28 xx 10^-19)/(1.6 xx 10^-19)`

= 0.8 m

(i) Helical path: For the particle to describe a helical path, the velocity must have both parallel and perpendicular components relative to the magnetic field. That means:

v_⊥ ≠ 0, v_|| ≠ 0

In this case, the particle moves along a spiral trajectory because the magnetic field affects only the perpendicular component of its velocity.

(ii) Straight undeviated path: If the velocity is parallel to the magnetic field `(vecv || vecB)`, then there is no magnetic force acting on the particle because the magnetic force is proportional to the perpendicular component of the velocity relative to the magnetic field. In this case:

`vecv || vecB` ⇒ No magnetic force, and the particle moves in a straight line.