Assertion (A) : If the potential difference applied to an electron is made `4` time , the de Broglie wavelength associated is halved
Reason (R ) : On making potential difference `4` times , velocity is doubled and hence `lambda` is halved

Options

A. If both (A) and (R ) correct and (R ) is the correct explanation for (A)
B. If both (A) and (R ) correct and (R ) is the correct explanation for (A)
C. If (A) is correct but (R ) is incorrect
D. If (A) is incorrect but (R ) is correct

Explanation

To solve the question, we need to analyze the assertion and the reason provided, and determine their validity based on the principles of de Broglie wavelength and kinetic energy of electrons.

### Step-by-Step Solution:

1. **Understanding the de Broglie Wavelength**:
The de Broglie wavelength (λ) of a particle is given by the formula:
\[ \lambda = \frac{h}{mv} \]
where \(h\) is Planck's constant, \(m\) is the mass of the particle, and \(v\) is its velocity.

2. **Relating Potential Difference to Kinetic Energy**:
When an electron is accelerated through a potential difference (V), it gains kinetic energy (KE) equal to the work done on it by the electric field:
\[ KE = eV \]
where \(e\) is the charge of the electron.

3. **Expressing Velocity in Terms of Potential Difference**:
The kinetic energy can also be expressed in terms of velocity:
\[ KE = \frac{1}{2}mv^2 \]
Setting these equal gives:
\[ eV = \frac{1}{2}mv^2 \]
Rearranging this gives:
\[ v = \sqrt{\frac{2eV}{m}} \]

4. **Effect of Increasing Potential Difference**:
If the potential difference is increased to 4 times its original value (let's say from V to 4V), the new velocity \(v'\) becomes:
\[ v' = \sqrt{\frac{2e(4V)}{m}} = \sqrt{4} \cdot \sqrt{\frac{2eV}{m}} = 2v \]
This shows that the velocity doubles when the potential difference is quadrupled.

5. **Calculating the New Wavelength**:
Substituting the new velocity into the de Broglie wavelength formula:
\[ \lambda' = \frac{h}{mv'} = \frac{h}{m(2v)} = \frac{1}{2} \cdot \frac{h}{mv} = \frac{1}{2} \lambda \]
Thus, the new wavelength is half of the original wavelength.

6. **Conclusion**:
Since both the assertion and reason are correct and the reason correctly explains the assertion, we conclude that:
- Assertion (A) is correct: If the potential difference applied to an electron is made 4 times, the de Broglie wavelength associated is halved.
- Reason (R) is correct: On making the potential difference 4 times, velocity is doubled and hence λ is halved.

### Final Answer:
The correct option is **Option A** – Both Assertion and Reason are correct and reason is the correct explanation of the assertion.