The thereshold wavelenght for a metal of work function `W_(0)` is `lambda`. The threshold wavelenght for a metal having work function `3W_(0)` will be

Options

A. `lambda`
B. `(lambda)/(2)`
C. `(lambda)/(3)`
D. `(lambda)/(4)`

Explanation

To solve the problem, we need to understand the relationship between the work function of a metal and its threshold wavelength. The threshold wavelength (\( \lambda \)) is related to the work function (\( W_0 \)) by the equation:

\[ W_0 = \frac{hc}{\lambda} \]

where:
- \( h \) is Planck's constant,
- \( c \) is the speed of light,
- \( \lambda \) is the threshold wavelength.

### Step-by-Step Solution:

1. **Identify the given values:**
- The threshold wavelength for a metal with work function \( W_0 \) is \( \lambda \).
- We need to find the threshold wavelength for a metal with work function \( 3W_0 \).

2. **Write the equation for the threshold wavelength:**
- For the first metal, we have:
\[ W_0 = \frac{hc}{\lambda} \]

3. **Set up the equation for the second metal:**
- For the second metal with work function \( 3W_0 \):
\[ 3W_0 = \frac{hc}{\lambda_1} \]
- Here, \( \lambda_1 \) is the threshold wavelength we want to find.

4. **Substitute the value of \( W_0 \) from the first equation into the second equation:**
- From the first equation, we can express \( W_0 \):
\[ W_0 = \frac{hc}{\lambda} \]
- Substitute this into the equation for \( 3W_0 \):
\[ 3 \left(\frac{hc}{\lambda}\right) = \frac{hc}{\lambda_1} \]

5. **Simplify the equation:**
- Cancel \( hc \) from both sides:
\[ \frac{3}{\lambda} = \frac{1}{\lambda_1} \]
- Rearranging gives:
\[ \lambda_1 = \frac{\lambda}{3} \]

6. **Conclusion:**
- The threshold wavelength for the metal with work function \( 3W_0 \) is:
\[ \lambda_1 = \frac{\lambda}{3} \]

### Final Answer:
The threshold wavelength for a metal having work function \( 3W_0 \) will be \( \frac{\lambda}{3} \).

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