Light of wavelength `lambda` falls on metal having work functions `hc//lambda_(0)` . Photoelectric effect will take place only if :

Options

A. `lamdagelamda_(0)`
B. `lamdalelamda_(0)`
C. `lamdage2lamda_(0)`
D. `lamda=4lamda_(0)`

Explanation

To solve the problem, we need to analyze the conditions under which the photoelectric effect occurs when light of wavelength \( \lambda \) falls on a metal with a work function given by \( \frac{hc}{\lambda_0} \).

### Step-by-Step Solution:

1. **Understanding the Photoelectric Effect**:
The photoelectric effect occurs when light falls on a metal surface and ejects electrons from it. For this to happen, the energy of the incoming photons must be greater than or equal to the work function of the metal.

2. **Using Einstein's Photoelectric Equation**:
According to Einstein's photoelectric equation:
\[ E = h\nu = \text{Work Function} + \text{Kinetic Energy} \]
where \( E \) is the energy of the incoming photon, \( h \) is Planck's constant, \( \nu \) is the frequency of the light, and the work function is the minimum energy required to eject an electron.

3. **Relating Frequency to Wavelength**:
The frequency \( \nu \) can be expressed in terms of wavelength \( \lambda \) as:
\[ \nu = \frac{c}{\lambda} \]
where \( c \) is the speed of light. Therefore, the energy of the photon can be rewritten as:
\[ E = h\nu = \frac{hc}{\lambda} \]

4. **Substituting the Work Function**:
The work function \( \phi \) is given as:
\[ \phi = \frac{hc}{\lambda_0} \]
Thus, the condition for the photoelectric effect to occur can be written as:
\[ \frac{hc}{\lambda} \geq \frac{hc}{\lambda_0} \]

5. **Simplifying the Inequality**:
Dividing both sides by \( hc \) (which is positive), we get:
\[ \frac{1}{\lambda} \geq \frac{1}{\lambda_0} \]
This can be rearranged to:
\[ \lambda_0 \geq \lambda \]

6. **Final Result**:
Therefore, the condition for the photoelectric effect to take place is:
\[ \lambda \leq \lambda_0 \]

### Conclusion:
The photoelectric effect will take place only if the wavelength \( \lambda \) of the incident light is less than or equal to the threshold wavelength \( \lambda_0 \).