When a metallic surface is illuminated by a light of frequency `8xx10^(14)`Hz, photoelectron of maximum energy 0.5 eV is emitted. When the same surface is illuminated by light of frequency `12xx10^(14)` Hz, photoelectron of maximum energy 2 eV is emitted. The work function is

Options

A. 0.5 eV
B. 2.85 eV
C. 2.5 eV
D. 3.5 eV

Explanation

To find the work function of the metallic surface, we will use the Einstein's photoelectric equation, which relates the maximum kinetic energy of the emitted photoelectrons to the energy of the incident photons and the work function of the material.

### Step-by-Step Solution:

1. **Understand the Einstein's Photoelectric Equation**:
The equation is given by:
\[ E_{\text{max}} = h \nu - W \]
where:
- \( E_{\text{max}} \) = maximum kinetic energy of the emitted photoelectrons
- \( h \) = Planck's constant (\(6.626 \times 10^{-34} \, \text{Js}\))
- \( \nu \) = frequency of the incident light
- \( W \) = work function of the metallic surface

2. **Set Up Equations for Each Case**:
For the first case (frequency \( \nu_1 = 8 \times 10^{14} \, \text{Hz} \)):
\[ E_{\text{max1}} = h \nu_1 - W \]
Given \( E_{\text{max1}} = 0.5 \, \text{eV} \):
\[ 0.5 = h (8 \times 10^{14}) - W \quad \text{(Equation 1)} \]

For the second case (frequency \( \nu_2 = 12 \times 10^{14} \, \text{Hz} \)):
\[ E_{\text{max2}} = h \nu_2 - W \]
Given \( E_{\text{max2}} = 2 \, \text{eV} \):
\[ 2 = h (12 \times 10^{14}) - W \quad \text{(Equation 2)} \]

3. **Express Work Function from Both Equations**:
From Equation 1:
\[ W = h (8 \times 10^{14}) - 0.5 \]
From Equation 2:
\[ W = h (12 \times 10^{14}) - 2 \]

4. **Set the Two Expressions for Work Function Equal**:
\[ h (8 \times 10^{14}) - 0.5 = h (12 \times 10^{14}) - 2 \]

5. **Rearranging the Equation**:
\[ h (12 \times 10^{14}) - h (8 \times 10^{14}) = 2 - 0.5 \]
\[ h (4 \times 10^{14}) = 1.5 \]

6. **Solve for Planck's Constant**:
\[ h = \frac{1.5}{4 \times 10^{14}} = \frac{1.5}{4} \times 10^{-14} \, \text{eV.s} \]

7. **Substituting Back to Find Work Function**:
Substitute \( h \) back into either expression for \( W \). Using Equation 1:
\[ W = h (8 \times 10^{14}) - 0.5 \]
Calculate \( W \):
\[ W = \left(\frac{1.5}{4} \times 10^{-14}\right)(8 \times 10^{14}) - 0.5 \]
\[ W = 3 - 0.5 = 2.5 \, \text{eV} \]

### Final Answer:
The work function \( W \) of the metallic surface is \( 2.5 \, \text{eV} \).