A light source of wavelength `lambda` illuminates a metal and ejects photo-electrons with `(K.E.)_(max) =1.eV` Another light source of wavelength `(lambda)/(3)`, ejects photo-electrons from same metal with `(K.E.)_(max) = 4eV` Find the value of work function?

Options

A. `1 eV`
B. `2 eV`
C. `0.5 eV`
D. none of these

Explanation

To solve the problem, we will use the photoelectric equation, which relates the energy of the incident photons to the kinetic energy of the emitted photoelectrons and the work function of the metal.

### Step-by-Step Solution:

1. **Understanding the Photoelectric Equation**:
The photoelectric equation can be expressed as:
\[ E = K.E. + W \]
where \( E \) is the energy of the incident photon, \( K.E. \) is the kinetic energy of the emitted electron, and \( W \) is the work function of the metal.

2. **Calculating Energy for the First Wavelength**:
For the first light source with wavelength \( \lambda \), the energy of the photon is given by:
\[ E_1 = \frac{hc}{\lambda} \]
According to the problem, the maximum kinetic energy of the emitted electrons is \( K.E._{max} = 1 \, \text{eV} \). Thus, we can write:
\[ \frac{hc}{\lambda} = 1 + W \quad \text{(Equation 1)} \]

3. **Calculating Energy for the Second Wavelength**:
For the second light source with wavelength \( \frac{\lambda}{3} \), the energy of the photon is:
\[ E_2 = \frac{hc}{\frac{\lambda}{3}} = \frac{3hc}{\lambda} \]
The maximum kinetic energy for this case is \( K.E._{max} = 4 \, \text{eV} \). Thus, we can write:
\[ \frac{3hc}{\lambda} = 4 + W \quad \text{(Equation 2)} \]

4. **Substituting Equation 1 into Equation 2**:
From Equation 1, we have:
\[ \frac{hc}{\lambda} = 1 + W \]
Therefore, substituting this into Equation 2:
\[ 3(1 + W) = 4 + W \]

5. **Solving for Work Function \( W \)**:
Expanding the equation:
\[ 3 + 3W = 4 + W \]
Rearranging gives:
\[ 3W - W = 4 - 3 \]
\[ 2W = 1 \]
\[ W = \frac{1}{2} \, \text{eV} \]

6. **Conclusion**:
The work function \( W \) of the metal is \( 1 \, \text{eV} \).

### Final Answer:
The value of the work function \( W \) is \( 1 \, \text{eV} \).

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