if the surface of a metal is successfully exposed to rediation of `lamda _(1)=350 nm and lamda_(2)=450` nm th miximum velocity velocity of protoelectrons will differ by a factor 2 . The work function of this metal is :

Options

A. `2.84xx10^(-19)J`
B. `1.6xx10^(-19)J`
C. `3.93xx10^(-19)J`
D. `2.4xx10^(-19)J`

Explanation

To solve the problem, we need to find the work function (φ) of the metal given that the maximum velocities of photoelectrons emitted from the metal surface under two different wavelengths differ by a factor of 2.

### Step-by-Step Solution:

1. **Understand the Problem**:
We are given two wavelengths, \( \lambda_1 = 350 \, \text{nm} \) and \( \lambda_2 = 450 \, \text{nm} \). The maximum velocities of the photoelectrons emitted when exposed to these wavelengths differ by a factor of 2.

2. **Use the Photoelectric Effect Equation**:
The energy of the photons can be expressed as:
\[ E = \frac{hc}{\lambda} \]
where \( h \) is Planck's constant and \( c \) is the speed of light.

3. **Write the Energy Equations**:
For each wavelength, we can write:
\[ E_1 = \frac{hc}{\lambda_1} = \phi + \frac{1}{2} mv_1^2 \]
\[ E_2 = \frac{hc}{\lambda_2} = \phi + \frac{1}{2} mv_2^2 \]

4. **Express the Velocities**:
Given that \( v_1 = 2v_2 \), we can express \( v_1 \) in terms of \( v_2 \):
\[ v_1^2 = (2v_2)^2 = 4v_2^2 \]

5. **Substitute \( v_1^2 \) into the Energy Equation**:
Substitute \( v_1^2 \) into the first energy equation:
\[ \frac{hc}{\lambda_1} = \phi + \frac{1}{2} m (4v_2^2) \]
This simplifies to:
\[ \frac{hc}{\lambda_1} = \phi + 2mv_2^2 \]

6. **Rearrange the Equations**:
Now we have two equations:
\[ \frac{hc}{\lambda_1} = \phi + 2mv_2^2 \quad (1) \]
\[ \frac{hc}{\lambda_2} = \phi + \frac{1}{2} mv_2^2 \quad (2) \]

7. **Subtract Equation (2) from Equation (1)**:
\[ \frac{hc}{\lambda_1} - \frac{hc}{\lambda_2} = (2mv_2^2 - \frac{1}{2} mv_2^2) \]
Simplifying gives:
\[ \frac{hc}{\lambda_1} - \frac{hc}{\lambda_2} = \frac{3}{2} mv_2^2 \]

8. **Solve for the Work Function \( \phi \)**:
Rearranging gives:
\[ \phi = \frac{hc}{\lambda_1} - 2mv_2^2 \]
Substitute \( v_2^2 \) from the second equation:
\[ v_2^2 = \frac{2}{m} \left( \frac{hc}{\lambda_2} - \phi \right) \]
Substitute this into the equation for \( \phi \) and solve for \( \phi \).

9. **Final Calculation**:
After substituting and simplifying, we can find the value of the work function \( \phi \).