The energies `E_1 and E_2` of two radiations are 25 eV and 50 eV respectively. The relation between their wavelengths, i.e., `lambda_(1) and lambda_(2)` will be

Options

A. `lambda_(1)=2lambda_(2)`
B. `lambda_(1)=4lambda_(2)`
C. `lambda_(1)=1/2`
D. `lambda_(1)=lambda_(2)`

Answer

To solve the problem of finding the relationship between the wavelengths \( \lambda_1 \) and \( \lambda_2 \) of two radiations with energies \( E_1 = 25 \, \text{eV} \) and \( E_2 = 50 \, \text{eV} \), we can use the relationship between energy and wavelength as described by Planck's equation.

### Step-by-Step Solution:

1. **Write the Energy-Wavelength Relationship**:
According to Planck's theory, the energy \( E \) of a photon is related to its wavelength \( \lambda \) by the equation:
\[ E = \frac{hc}{\lambda} \]
where \( h \) is Planck's constant and \( c \) is the speed of light.

2. **Set Up the Equations for Each Energy**:
For the first radiation with energy \( E_1 = 25 \, \text{eV} \):
\[ E_1 = \frac{hc}{\lambda_1} \implies 25 = \frac{hc}{\lambda_1} \quad \text{(Equation 1)} \]
For the second radiation with energy \( E_2 = 50 \, \text{eV} \):
\[ E_2 = \frac{hc}{\lambda_2} \implies 50 = \frac{hc}{\lambda_2} \quad \text{(Equation 2)} \]

3. **Express \( \lambda_1 \) and \( \lambda_2 \)**:
Rearranging Equation 1 gives:
\[ \lambda_1 = \frac{hc}{25} \]
Rearranging Equation 2 gives:
\[ \lambda_2 = \frac{hc}{50} \]

4. **Find the Relationship Between \( \lambda_1 \) and \( \lambda_2 \)**:
Now, we can express the ratio of \( \lambda_1 \) to \( \lambda_2 \):
\[ \frac{\lambda_1}{\lambda_2} = \frac{\frac{hc}{25}}{\frac{hc}{50}} = \frac{50}{25} = 2 \]
Therefore, we can conclude:
\[ \lambda_1 = 2 \lambda_2 \]

5. **Final Result**:
The relationship between the wavelengths is:
\[ \lambda_1 = 2 \lambda_2 \]

### Conclusion:
The correct option is that \( \lambda_1 \) is equal to twice \( \lambda_2 \).