In a cyclotron, if a deutron can gain a maximum energy of 40 MeV, then a proton can gain a maximum energy of ____.

80 MeV

The correct answer is Option (4) → 80 MeV

In a cyclotron, the maximum kinetic energy gained by a particle is proportional to its charge-to-mass ratio ($q/m$) assuming same cyclotron parameters.

Given: Maximum energy for deuteron, $E_d = 40\ \text{MeV}$

Charge of deuteron $q_d = +e$, mass $m_d \approx 2 m_p$

Charge of proton $q_p = +e$, mass $m_p$

Energy ratio:

$\frac{E_p}{E_d} = \frac{q_p/m_p}{q_d/m_d} = \frac{(e/m_p)}{(e/2 m_p)} = 2$

∴ $E_p = 2 \cdot E_d = 2 \cdot 40\ \text{MeV} = 80\ \text{MeV}$

∴ Maximum energy a proton can gain = 80 MeV