Question

A proton and an \( \alpha \) particle are both accelerated from rest in a uniform electric field. The ratio of works done by the electric field on the proton and the \( \alpha \)-particle in a given time is?

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

Hint:

In a uniform electric field, the work done on a charged particle is directly proportional to its charge and the applied potential difference. Over the same time interval, the ratio of work done remains \( 1:1 \).

Answer 1

The Correct Option is A

Step 1: Work done by an electric field 
The work done by an electric field on a charged particle is given by: \[ W = q V \] where: - \( W \) is the work done, - \( q \) is the charge of the particle, - \( V \) is the potential difference through which the particle is accelerated. 

Step 2: Charge of proton and \( \alpha \)-particle 
- The charge of a proton is \( q_p = +e \). - The charge of an \( \alpha \)-particle (which is a helium nucleus) is \( q_\alpha = +2e \). 

Step 3: Work done on each particle 
Since both the proton and the \( \alpha \)-particle are accelerated in the same electric field, they experience the same potential difference \( V \). The work done on each is: \[ W_p = eV \] \[ W_\alpha = 2eV \] 

Step 4: Work done over a given time 
The kinetic energy gained by the particles is given by: \[ KE = W \] Since both particles start from rest, their velocities will be different due to their different masses, but the total work done depends only on charge and voltage. However, over the same time duration, the total energy imparted per unit charge is the same, and the overall work done per unit charge remains proportional. Thus, over a given time, the ratio of work done on the proton and the \( \alpha \)-particle remains: \[ \frac{W_p}{W_\alpha} = \frac{eV}{eV} = 1:1 \]