An electron is accelerated from rest through a potential difference V. This electron experiences a force F in a uniform magnetic field. On increasing the potential difference to V', the force experienced by the electron in the same magnetic field becomes 2F. Then, the ratio `(V'//V)` is equal to

Options

A. `1//4`
B. `2//1`
C. `1//2`
D. `1//4`

Explanation

To solve the problem, we will follow these steps:

1. **Understanding the relationship between potential difference and kinetic energy**:
When an electron is accelerated through a potential difference \( V \), the work done on the electron is equal to its kinetic energy. This can be expressed as:
\[ eV = \frac{1}{2} mv^2 \]
where \( e \) is the charge of the electron, \( m \) is its mass, and \( v \) is its velocity after being accelerated through the potential difference \( V \).

2. **Finding the velocity of the electron**:
Rearranging the equation gives us:
\[ v = \sqrt{\frac{2eV}{m}} \]

3. **Magnetic force on the electron**:
The magnetic force \( F \) experienced by a charged particle moving in a magnetic field \( B \) is given by:
\[ F = e v B \]
Substituting the expression for \( v \) from step 2 into this equation, we get:
\[ F = e B \sqrt{\frac{2eV}{m}} \]

4. **Analyzing the change in potential difference**:
When the potential difference is increased to \( V' \), the new velocity \( v' \) can be expressed as:
\[ v' = \sqrt{\frac{2eV'}{m}} \]
The new force \( F' \) experienced by the electron is:
\[ F' = e v' B = e B \sqrt{\frac{2eV'}{m}} \]

5. **Setting up the ratio of forces**:
According to the problem, when the potential difference is increased to \( V' \), the force becomes \( 2F \):
\[ F' = 2F \]
Substituting the expressions for \( F \) and \( F' \):
\[ e B \sqrt{\frac{2eV'}{m}} = 2 \left( e B \sqrt{\frac{2eV}{m}} \right) \]

6. **Simplifying the equation**:
We can cancel \( e B \) from both sides (assuming \( e \) and \( B \) are non-zero):
\[ \sqrt{\frac{2eV'}{m}} = 2 \sqrt{\frac{2eV}{m}} \]
Squaring both sides gives:
\[ \frac{2eV'}{m} = 4 \cdot \frac{2eV}{m} \]
This simplifies to:
\[ V' = 4V \]

7. **Finding the ratio \( \frac{V'}{V} \)**:
Therefore, the ratio of the new potential difference to the original potential difference is:
\[ \frac{V'}{V} = \frac{4V}{V} = 4 \]

Thus, the final answer is:
\[ \frac{V'}{V} = 4 \]