A proton, a deutron and an `alpha`- particle accelerated through the same potential difference enter a region of uniform magnetic field, moving at right angles to it. What is the ratio of their kinetic energy?

Options

A. `1:1:2`
B. `2:2:1`
C. `1:2:1`
D. `2:1:1`

Answer

To find the ratio of the kinetic energy of a proton, a deuteron, and an alpha particle accelerated through the same potential difference, we can follow these steps:

### Step 1: Understand the relationship between kinetic energy and potential difference
When a charged particle is accelerated through a potential difference (V), it gains kinetic energy (KE) equal to the work done on it by the electric field. The kinetic energy gained by a charged particle is given by the formula:

\[ KE = qV \]

where \( q \) is the charge of the particle and \( V \) is the potential difference.

### Step 2: Identify the charges of the particles
- **Proton (p)**: Charge \( q_p = +e \)
- **Deuteron (d)**: Charge \( q_d = +e \) (same as proton, since it is a hydrogen isotope with one proton and one neutron)
- **Alpha particle (α)**: Charge \( q_{α} = +2e \) (it consists of 2 protons and 2 neutrons)

### Step 3: Write the kinetic energy for each particle
Using the formula for kinetic energy:

1. For the proton:
\[ KE_p = q_p V = eV \]

2. For the deuteron:
\[ KE_d = q_d V = eV \]

3. For the alpha particle:
\[ KE_{α} = q_{α} V = 2eV \]

### Step 4: Calculate the ratio of kinetic energies
Now we can find the ratio of their kinetic energies:

\[ \text{Ratio} = KE_p : KE_d : KE_{α} = eV : eV : 2eV \]

This simplifies to:

\[ 1 : 1 : 2 \]

### Final Answer
Thus, the ratio of the kinetic energies of the proton, deuteron, and alpha particle is:

\[ 1 : 1 : 2 \]

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