Question

A cyclotron with dees of radius \(50\text{ cm}\) and magnetic field \(1.5\text{ T}\) is used to accelerate protons and alpha particles separately. The ratio of the maximum kinetic energies acquired by proton and alpha particle is:

• A. \(1:4\)
• B. \(1:1\)
• C. \(1:2\)
• D. \(1:8\)

Hint:

In cyclotron problems, use \(K_{\max}\propto q^2/m\). For alpha particles, both charge and mass increase, leading to cancellation.

Answer 1

The Correct Option is B

Concept: Maximum kinetic energy in a cyclotron is \[ K=\frac{q^2B^2R^2}{2m} \] Thus, \[ K\propto\frac{q^2}{m} \]

Step 1: For proton. \[ q=e,\qquad m=m_p \] Therefore, \[ K_p\propto\frac{e^2}{m_p} \]

Step 2: For alpha particle. \[ q=2e,\qquad m=4m_p \] Hence, \[ K_\alpha \propto \frac{(2e)^2}{4m_p} = \frac{4e^2}{4m_p} = \frac{e^2}{m_p} \]

Step 3: Compare the energies. \[ K_p=K_\alpha \] Therefore, \[ K_p:K_\alpha = 1:1 \] \[ \boxed{1:1} \]