Variation of Stopping Potential with Wavelength and Inverse Wavelength

According to Einstein's photoelectric equation, the maximum kinetic energy of the emitted photoelectrons is given by:

$e V_0 = h\nu - \phi = \frac{hc}{\lambda} - \phi$

where:

• $V_0$ is the stopping potential,
• $e$ is the magnitude of charge on an electron,
• $h$ is Planck's constant,
• $c$ is the speed of light in vacuum,
• $\lambda$ is the wavelength of the incident photon,
• $\phi$ is the work function of the metal.

Photoelectric emission takes place only when the incident wavelength $\lambda$ is less than or equal to the threshold wavelength $\lambda_0$ (where $\lambda_0 = \frac{hc}{\phi}$).

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1. Variation of $V_0$ with $1/\lambda$:

Rearranging the equation in terms of $\frac{1}{\lambda}$:

$V_0 = \left( \frac{hc}{e} \right) \frac{1}{\lambda} - \frac{\phi}{e} \quad \text{for } \frac{1}{\lambda} \ge \frac{1}{\lambda_0}$

• For $\frac{1}{\lambda} < \frac{1}{\lambda_0}$, no photoelectrons are emitted, so $V_0 = 0$.
• For $\frac{1}{\lambda} \ge \frac{1}{\lambda_0}$, $V_0$ increases linearly with $\frac{1}{\lambda}$ with a positive slope $\left(\frac{hc}{e}\right)$ and an intercept on the horizontal axis at $\frac{1}{\lambda} = \frac{1}{\lambda_0}$.

This linear behavior with a non-zero threshold intercept is correctly represented in graph (C).

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2. Variation of $V_0$ with $\lambda$:

Rearranging the equation in terms of $\lambda$:

$V_0 = \frac{hc}{e\lambda} - \frac{\phi}{e} \quad \text{for } 0 < \lambda \le \lambda_0$

• For $\lambda > \lambda_0$, no photoelectrons are emitted, so $V_0 = 0$.
• For $0 < \lambda \le \lambda_0$, $V_0$ decreases non-linearly as $\lambda$ increases.
• Taking the derivatives with respect to $\lambda$: $\frac{dV_0}{d\lambda} = -\frac{hc}{e\lambda^2} < 0$ $\frac{d^2V_0}{d\lambda^2} = \frac{2hc}{e\lambda^3} > 0$

Since $\frac{d^2V_0}{d\lambda^2} > 0$, the curve is convex downwards, decreasing continuously to zero as $\lambda \to \lambda_0$, and remaining zero for $\lambda > \lambda_0$.

This non-linear, concave-upward decaying curve is correctly represented in graph (A).

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Thus, the correct option(s) are (A) and (C).