The KE of the photoelectrons is E when the incident wavelength is `(lamda)/(2)`. The KE becomes 2E when the incident wavelength is `(lamda)/(3)`. The work function of the metal is

Options

A. `(hc)/lamda`
B. `(2hc)/(lamda)`
C. `(3hc)/(lamda)`
D. `(hc)/(3lamda)`

Explanation

To find the work function of the metal based on the given kinetic energies of photoelectrons at different wavelengths, we can follow these steps:

### Step 1: Understand the Photoelectric Effect Equation
The kinetic energy (KE) of the emitted photoelectrons can be expressed using the equation:
\[ KE = E - W \]
where:
- \( KE \) is the kinetic energy of the photoelectrons,
- \( E \) is the energy of the incident photons,
- \( W \) is the work function of the metal.

The energy of the incident photons can be expressed in terms of wavelength (\( \lambda \)):
\[ E = \frac{hc}{\lambda} \]
where:
- \( h \) is Planck's constant,
- \( c \) is the speed of light.

### Step 2: Set Up the Equations for the Two Given Conditions
1. When the incident wavelength is \( \frac{\lambda}{2} \), the kinetic energy is \( E \):
\[ E = \frac{hc}{\frac{\lambda}{2}} - W \]
Simplifying this gives:
\[ E = \frac{2hc}{\lambda} - W \quad \text{(Equation 1)} \]

2. When the incident wavelength is \( \frac{\lambda}{3} \), the kinetic energy becomes \( 2E \):
\[ 2E = \frac{hc}{\frac{\lambda}{3}} - W \]
Simplifying this gives:
\[ 2E = \frac{3hc}{\lambda} - W \quad \text{(Equation 2)} \]

### Step 3: Solve the Equations Simultaneously
Now we have two equations:
1. \( E = \frac{2hc}{\lambda} - W \)
2. \( 2E = \frac{3hc}{\lambda} - W \)

From Equation 1, we can express \( W \):
\[ W = \frac{2hc}{\lambda} - E \quad \text{(1)} \]

Substituting this expression for \( W \) into Equation 2:
\[ 2E = \frac{3hc}{\lambda} - \left(\frac{2hc}{\lambda} - E\right) \]
This simplifies to:
\[ 2E = \frac{3hc}{\lambda} - \frac{2hc}{\lambda} + E \]
\[ 2E - E = \frac{hc}{\lambda} \]
\[ E = \frac{hc}{\lambda} \]

### Step 4: Substitute Back to Find Work Function
Now substitute \( E \) back into Equation (1):
\[ W = \frac{2hc}{\lambda} - \frac{hc}{\lambda} \]
\[ W = \frac{hc}{\lambda} \]

### Conclusion
The work function of the metal is:
\[ W = \frac{hc}{\lambda} \]