Two paricles have idential charges. If they are accelerated throgh identical potential differences, then the ratio of their deBrogile wavelength would be

Options

A. `lambda_(1) : lambda_(2) = 1 : 1`
B. `lambda_(1) : lambda_(2) = m_(2) : m_(1)`
C. `lambda_(1) : lambda_(2) = sqrt(m_(2)) : sqrt(m_(1))`
D. `lambda_(1) : lambda_(2) = sqrt(m_(1)) : sqrt(m_(2))`

Explanation

To solve the problem of finding the ratio of the de Broglie wavelengths of two particles with identical charges that are accelerated through identical potential differences, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Kinetic Energy**:
The kinetic energy (KE) gained by a charged particle when it is accelerated through a potential difference (ΔV) is given by the formula:
\[ KE = q \cdot \Delta V \]
where \( q \) is the charge of the particle.

2. **Relating Kinetic Energy to Momentum**:
The kinetic energy can also be expressed in terms of momentum (p) and mass (m):
\[ KE = \frac{p^2}{2m} \]
Equating the two expressions for kinetic energy gives:
\[ q \cdot \Delta V = \frac{p^2}{2m} \]
Rearranging this, we find:
\[ p = \sqrt{2m \cdot q \cdot \Delta V} \]

3. **Finding the de Broglie Wavelength**:
The de Broglie wavelength (λ) is given by:
\[ \lambda = \frac{h}{p} \]
where \( h \) is Planck's constant. Substituting the expression for momentum into this equation gives:
\[ \lambda = \frac{h}{\sqrt{2m \cdot q \cdot \Delta V}} \]

4. **Calculating the Ratio of Wavelengths**:
For two particles (1 and 2) with identical charges and accelerated through the same potential difference, we can write:
\[ \lambda_1 = \frac{h}{\sqrt{2m_1 \cdot q \cdot \Delta V}} \quad \text{and} \quad \lambda_2 = \frac{h}{\sqrt{2m_2 \cdot q \cdot \Delta V}} \]
To find the ratio of their wavelengths:
\[ \frac{\lambda_1}{\lambda_2} = \frac{\sqrt{2m_2 \cdot q \cdot \Delta V}}{\sqrt{2m_1 \cdot q \cdot \Delta V}} = \frac{\sqrt{m_2}}{\sqrt{m_1}} = \sqrt{\frac{m_2}{m_1}} \]

5. **Conclusion**:
Thus, the ratio of the de Broglie wavelengths of the two particles is:
\[ \frac{\lambda_1}{\lambda_2} = \sqrt{\frac{m_2}{m_1}} \]
Therefore, the answer is that the ratio of their de Broglie wavelengths depends on the square root of the ratio of their masses.