In a photo-emissive cell, with exciting wavelength `lambda_1`, the maximum kinetic energy of electron is K. If the exciting wavelength is changed to `lambda_2`, the maximum kinetic energy of electron is 2K then

Options

A. `lambda_1=2lambda_2`
B. `lambda_1 gt 2lambda_2`
C. `lambda_1 lt 2lambda_2`
D. `lambda_1=lambda_2/2`

Answer

To solve the problem, we will use the photoelectric effect equations that relate the maximum kinetic energy of emitted electrons to the wavelength of the incident light and the work function of the material.

### Step-by-Step Solution:

1. **Understanding the Photoelectric Effect**:
The maximum kinetic energy (K.E.) of the emitted electrons can be expressed using the equation:
\[ K.E. = E_{\text{photon}} - \phi_0 \]
where \(E_{\text{photon}} = \frac{hc}{\lambda}\) is the energy of the incoming photon, \(h\) is Planck's constant, \(c\) is the speed of light, \(\lambda\) is the wavelength of the incident light, and \(\phi_0\) is the work function of the material.

2. **First Case with Wavelength \(\lambda_1\)**:
For the first wavelength \(\lambda_1\), the maximum kinetic energy of the emitted electrons is \(K\):
\[ K = \frac{hc}{\lambda_1} - \phi_0 \tag{1} \]

3. **Second Case with Wavelength \(\lambda_2\)**:
For the second wavelength \(\lambda_2\), the maximum kinetic energy is \(2K\):
\[ 2K = \frac{hc}{\lambda_2} - \phi_0 \tag{2} \]

4. **Rearranging Equation (1)**:
From equation (1), we can express \(\phi_0\):
\[ \phi_0 = \frac{hc}{\lambda_1} - K \tag{3} \]

5. **Rearranging Equation (2)**:
From equation (2), we can express \(K\):
\[ K = \frac{hc}{\lambda_2} - \phi_0 \tag{4} \]

6. **Substituting Equation (3) into Equation (4)**:
Substitute \(\phi_0\) from equation (3) into equation (4):
\[ K = \frac{hc}{\lambda_2} - \left(\frac{hc}{\lambda_1} - K\right) \]
Simplifying this gives:
\[ K = \frac{hc}{\lambda_2} - \frac{hc}{\lambda_1} + K \]
This implies:
\[ 0 = \frac{hc}{\lambda_2} - \frac{hc}{\lambda_1} \]
Rearranging gives:
\[ \frac{hc}{\lambda_1} = \frac{hc}{\lambda_2} \]

7. **Finding the Relationship Between \(\lambda_1\) and \(\lambda_2\)**:
From the equations derived, we can find the relationship between \(\lambda_1\) and \(\lambda_2\):
\[ K = \frac{hc}{\lambda_2} - \left(\frac{hc}{\lambda_1} - K\right) \]
Rearranging gives:
\[ \frac{hc}{\lambda_1} = 2K + \phi_0 \]
Therefore:
\[ \frac{hc}{\lambda_1} > \frac{hc}{2\lambda_2} \]
This implies:
\[ \lambda_1 < 2\lambda_2 \]

### Conclusion:
Thus, the relationship between the wavelengths is:
\[ \lambda_1 < 2\lambda_2 \]