For photo - electric effect with incident photon wavelength `lambda` the stopping is `V_(0)` identify the correct variation(s) of `V_(0)` with `lambda and 1 // lambda`

Options

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B.
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D.

Explanation

To solve the problem regarding the variation of stopping potential \( V_0 \) with the wavelength \( \lambda \) and \( \frac{1}{\lambda} \) in the context of the photoelectric effect, we can follow these steps:

### Step 1: Understand the Photoelectric Effect Equation
The photoelectric effect is described by the equation:
\[ E = h \nu = \frac{hc}{\lambda} \]
where:
- \( E \) is the energy of the incident photon,
- \( h \) is Planck's constant,
- \( \nu \) is the frequency of the photon,
- \( c \) is the speed of light,
- \( \lambda \) is the wavelength of the incident photon.

The stopping potential \( V_0 \) is related to the kinetic energy of the emitted electrons and can be expressed as:
\[ eV_0 = E - \phi \]
where:
- \( e \) is the charge of the electron,
- \( \phi \) is the work function of the material.

### Step 2: Relate Stopping Potential to Wavelength
From the above equations, we can substitute \( E \):
\[ eV_0 = \frac{hc}{\lambda} - \phi \]
Rearranging gives:
\[ V_0 = \frac{hc}{e\lambda} - \frac{\phi}{e} \]

### Step 3: Analyze the Equation
In the equation \( V_0 = \frac{hc}{e\lambda} - \frac{\phi}{e} \):
- The term \( \frac{hc}{e\lambda} \) indicates that \( V_0 \) is inversely proportional to \( \lambda \).
- The term \( -\frac{\phi}{e} \) is a constant offset.

### Step 4: Variation with \( \lambda \)
As \( \lambda \) increases, the term \( \frac{hc}{e\lambda} \) decreases, leading to a decrease in \( V_0 \). Therefore, the variation of \( V_0 \) with \( \lambda \) is:
- **Inverse Relationship**: \( V_0 \) decreases as \( \lambda \) increases.

### Step 5: Variation with \( \frac{1}{\lambda} \)
If we express the equation in terms of \( \frac{1}{\lambda} \):
\[ V_0 = \frac{hc}{e} \cdot \frac{1}{\lambda} - \frac{\phi}{e} \]
Here, \( V_0 \) is directly proportional to \( \frac{1}{\lambda} \):
- **Linear Relationship**: \( V_0 \) increases as \( \frac{1}{\lambda} \) increases.

### Conclusion
- The variation of stopping potential \( V_0 \) with \( \lambda \) is inversely proportional.
- The variation of stopping potential \( V_0 \) with \( \frac{1}{\lambda} \) is directly proportional.