Light of wavelength `lamda` strikes a photoelectric surface and electrons are ejected with kinetic energy K. If K is to be increased to exactly twice its original value, the wavelength must be changed to `lamda'` such that

Options

A. `lamda'lt(lamda)/(2)`
B. `lamda,gt(lamda)/(2)`
C. `lamdagtlamda'gt(lamda)/(2)`
D. `lamda'=(lamda)/(2)`

Explanation

To solve the problem, we will use the principles of the photoelectric effect. The kinetic energy of the ejected electrons is related to the energy of the incident photons and the work function of the material.

### Step-by-Step Solution:

1. **Understanding the Photoelectric Effect**:
The energy of the incident photons is given by the equation:
\[ E = \frac{hc}{\lambda} \]
where \( h \) is Planck's constant, \( c \) is the speed of light, and \( \lambda \) is the wavelength of the incident light.

2. **Kinetic Energy Relation**:
The kinetic energy \( K \) of the ejected electrons can be expressed as:
\[ K = E - \phi \]
where \( \phi \) is the work function of the material. Substituting for \( E \):
\[ K = \frac{hc}{\lambda} - \phi \]

3. **Doubling the Kinetic Energy**:
If we want to increase the kinetic energy to \( 2K \), we have:
\[ 2K = \frac{hc}{\lambda'} - \phi \]
where \( \lambda' \) is the new wavelength.

4. **Setting Up the Equations**:
We have two equations:
- From the original condition:
\[ K = \frac{hc}{\lambda} - \phi \quad \text{(1)} \]
- From the new condition:
\[ 2K = \frac{hc}{\lambda'} - \phi \quad \text{(2)} \]

5. **Substituting Equation (1) into Equation (2)**:
We can express \( K \) from equation (1) and substitute it into equation (2):
\[ 2\left(\frac{hc}{\lambda} - \phi\right) = \frac{hc}{\lambda'} - \phi \]
Simplifying this gives:
\[ \frac{2hc}{\lambda} - 2\phi = \frac{hc}{\lambda'} - \phi \]

6. **Rearranging the Equation**:
Rearranging gives:
\[ \frac{hc}{\lambda'} = \frac{2hc}{\lambda} - \phi + 2\phi \]
\[ \frac{hc}{\lambda'} = \frac{2hc}{\lambda} + \phi \]

7. **Finding the Ratio of Wavelengths**:
Now, we can express the ratio of the wavelengths:
\[ \frac{\lambda'}{\lambda} = \frac{hc}{\phi + \frac{2hc}{\lambda}} \]
This indicates that \( \lambda' \) is related to \( \lambda \) and the work function \( \phi \).

8. **Analyzing the Conditions**:
Since we know that \( K \) is being doubled, we can conclude that:
- \( \lambda' < \lambda \) (because increasing energy corresponds to decreasing wavelength).
- Also, since \( K \) is doubled, \( \lambda' \) must be less than \( \frac{\lambda}{2} \) to achieve this increase in kinetic energy.

### Conclusion:
From the analysis, we conclude that:
- \( \lambda' < \lambda \)
- \( \lambda' > \frac{\lambda}{2} \)

Thus, the correct option is:
- **Option C**: \( \lambda \) is greater than \( \lambda' \) and \( \lambda' \) is greater than \( \frac{\lambda}{2} \).