A charge particle moves along circular path in a uniform magnetic field in a cyclotron. The kinetic energy of the charge particle increases to 4 times its initial value. What will be the ratio of new radius to the original radius of circular path of the charge particle :

Solution

Physics sir · 4 steps

Write the radius in terms of kinetic energy: r goes as the square root of K, so 4 times K means 2 times r.

1. Radius from momentum. In a uniform field the magnetic force supplies the centripetal force: qvB = mv²/r. This gives r = mv/qB = p/qB.
2. Bring in kinetic energy. Kinetic energy and momentum are linked by K = p²/2m, so p = √2mK.

   r = √2mK/qB ⇒ r ∝ √K

3. Compare the radii. The charge, the mass and B stay the same, so r₂/r₁ = √(K₂/K₁) = √4 = 2.
4. So. The new radius is twice the original, a ratio of 2 : 1, which is option (c).

The idea to keep. For a charge in a magnetic field, r = p/qB = √2mK/qB. Radius scales with momentum directly and with kinetic energy only as its square root.

Answer from the NTA final answer key. Solution by PrepWiser.