Question

A cyclotron is used to accelerate protons (\({}^1_1\text{H}\)​), Deuterons (\({}^2_1\text{H}\)​) and α-particles (\({}^4_2\text{He}\)). While exiting under similar conditions, the minimum K.E. is gained by

• A. α-particle
• B. Proton
• C. Deuteron
• D. Same for all

Answer 1

The Correct Option is C

In a cyclotron, the kinetic energy gained by a particle is given by the relation: \[ K.E. = \frac{q^2 B^2 r^2}{2m} \] Where:
\(q\) is the charge of the particle,
\(B\) is the magnetic field strength,
\(r\) is the radius of the circular path,
\(m\) is the mass of the particle.

For a given magnetic field strength and radius, the kinetic energy gained by the particle depends on its mass. Since the proton, deuteron, and \(\alpha\)-particle have different masses, they will experience different accelerations. The proton has the least mass, followed by the deuteron and then the \(\alpha\)-particle. The minimum kinetic energy is gained by the particle with the smallest mass because it requires the least force to accelerate to a given velocity.

Thus, the minimum K.E. is gained by the deuteron, as it has a mass between the proton and \(\alpha\)-particle.

Answer 2

The kinetic energy gained by a particle in a cyclotron is given by the formula: \[ K.E. = qV \] where:
\( q \) is the charge of the particle,
\( V \) is the potential difference.

Since the cyclotron accelerates particles with the same charge (equal to their charge number), the kinetic energy gained depends primarily on the mass of the particle. The frequency of the motion in the cyclotron is inversely proportional to the mass of the particle, so a heavier particle (like a deuteron or an \( \alpha \)-particle) takes longer to reach the same velocity as a lighter particle, given the same potential difference. The deuteron, being heavier than the proton but lighter than the \( \alpha \)-particle, gains the minimum kinetic energy because the cyclotron's efficiency is higher for lighter particles, but the deuteron still receives a lower kinetic energy compared to heavier particles like the \( \alpha \)-particle.

Thus, the minimum kinetic energy is gained by the deuteron.