Understanding Stopping Potential in Photoelectric Effect

The question asks what property the stopping potential for a fast-moving photo-electron is independent of. Let's break down the concept of stopping potential using the principles of the photoelectric effect.

Photoelectric Effect and Einstein's Equation

When light shines on a metal surface, it can eject electrons. This phenomenon is called the photoelectric effect. The minimum energy required to eject an electron is called the work function ($\phi$) of the metal. According to Einstein's photoelectric equation, the maximum kinetic energy ($K_{max}$) of the ejected photo-electrons is given by:

$K_{max} = hf - \phi$

Where:

• $h$ is Planck's constant
• $f$ is the frequency of the incident photon
• $\phi$ is the work function of the metal

Defining Stopping Potential

The stopping potential ($V_s$) is the minimum negative potential applied to the collecting plate that stops even the most energetic photo-electrons from reaching it. At this potential, the work done by the electric field on the electron equals its maximum kinetic energy:

$eV_s = K_{max}$

Where $e$ is the elementary charge.

Combining the two equations, we get:

$eV_s = hf - \phi$

Or,

$V_s = \frac{hf}{e} - \frac{\phi}{e}$

Analysis of Dependencies

From the equation $V_s = \frac{hf}{e} - \frac{\phi}{e}$, we can see how stopping potential depends on various factors:

Stopping Potential Dependence on Frequency and Wavelength

Since frequency ($f$) is directly proportional to the energy of the incident photon ($hf$), the stopping potential ($V_s$) increases as the frequency increases. Wavelength ($\lambda$) is inversely related to frequency ($f = c/\lambda$, where $c$ is the speed of light). Therefore, stopping potential also depends on the wavelength of the incident photon; a shorter wavelength (higher frequency) results in a higher stopping potential.

Stopping Potential Dependence on Type of Metals

The work function ($\phi$) is a property specific to the material of the metal surface. Different metals have different work functions. As the equation shows, $V_s$ is directly related to $\phi$ (specifically, $V_s$ decreases as $\phi$ increases, assuming $hf > \phi$). Thus, the stopping potential is dependent on the type of metal.

Stopping Potential and Intensity of Incident Photon

The intensity of the incident light is related to the *number* of photons striking the metal surface per unit area per unit time. Increasing the intensity increases the *number* of photo-electrons emitted, but it does not change the energy of individual photons ($hf$) or the work function ($\phi$) of the metal. Since the stopping potential ($V_s$) depends only on the maximum kinetic energy ($K_{max}$), which in turn depends on photon energy and work function, it is independent of the light's intensity.

Conclusion

Based on the analysis of Einstein's photoelectric equation, the stopping potential ($V_s$) depends on the frequency (or wavelength) of the incident photon and the type of metal (through its work function). However, it does not depend on the intensity of the incident light.