Two particles X and Y having equal charges are being accelerated through the same potential difference. Thereafter they enter normally in a region of uniform magnetic field and describes circular paths of radii R₁ and R₂ respectively. The mass ratio of X and Y is :

Solution

Physics sir · 4 steps

Same charge, same voltage means same kinetic energy, so the radius depends only on the square root of mass.

1. Equal kinetic energy. A charge q accelerated through a potential difference V gains kinetic energy qV. Both particles have the same q and V, so both have the same kinetic energy K = qV.
2. Radius in the field. In the magnetic field, qvB supplies the centripetal force, so r = mv/qB = p/qB. The momentum is p = √2mK = √2mqV.

   r = √2mqV/qB = 1/B√(2mV/q) ⇒ r ∝ √m

3. Take the ratio. With q, V and B the same for both, R₁/R₂ = √(m_X/m_Y). Squaring both sides gives the mass ratio.
4. So. m_X/m_Y = (R₁/R₂)², which is option (d).

The idea to keep. After acceleration through a potential difference, write r = √2mqV/qB. For equal charges, the radius grows as √m, not as m.

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