Animated Solution for Physics - Dual Nature of Matter and Radiation: In a photoelectric effect experiment, the graph of stopping potential $V$ versus reciprocal of wavelength $(1/\lambda )$ obtained is shown in the figure. As the intensity of incident radiation is increased,

Imagine you are in a physics lab, setting up a classic photoelectric effect experiment. You shine light of various wavelengths onto a metal plate and carefully measure the stopping potential required to halt the fastest emitted electrons. When you plot this stopping potential $V$ on the y-axis against the reciprocal of the wavelength $\frac{1}{\lambda }$ on the x-axis, a beautiful straight line emerges.

This line doesn't start from the origin; it intersects the x-axis at a specific point and makes an angle $\theta$ with it. The question challenges us to predict what happens to this exact graph if we simply turn up the brightness—that is, increase the intensity of the incident light.

To decode the secrets hidden within this straight line, we must call upon Einstein's legendary photoelectric equation. This equation is the ultimate bridge between the quantum world of photons and the macroscopic world of voltages. It states that the energy of an incoming photon is split into two parts: overcoming the metal's work function ${\varphi }_0$ and giving kinetic energy to the ejected electron.

We know that the energy of a photon can be written in terms of its wavelength as $E=\frac{hc}{\lambda }$. Furthermore, the maximum kinetic energy is directly related to the stopping potential by $K_{\text{max}}=eV$.

Substituting these into our master equation gives us the raw setup:

Now, let's perform a little algebraic gymnastics. Our goal is to make this equation look like the graph, which means isolating the stopping potential $V$ on one side.

First, we move the work function to the other side:

Next, we divide the entire equation by the elementary charge $e$:

Take a step back and look at this beautiful result. It perfectly mirrors the standard equation of a straight line, $y=mx+c$.

Here, our y-variable is $V$, and our x-variable is $\frac{1}{\lambda }$. By direct comparison, we can extract the physical meaning of the graph's geometric features: - The slope $m$ is exactly $\frac{hc}{e}$. - The y-intercept $c$ is exactly $-\frac{{\varphi }_0}{e}$.

Now for the grand finale. The question asks how increasing the intensity affects this graph.

In the quantum picture, increasing the intensity of monochromatic light simply means firing more photons per second, not firing more energetic photons. Let's look at our slope and intercept. The slope $\frac{hc}{e}$ is composed entirely of universal constants (Planck's constant, the speed of light, and the charge of an electron). The y-intercept $-\frac{{\varphi }_0}{e}$ depends only on the work function, which is an intrinsic property of the specific metal being used.

Since neither the universal constants nor the metal's work function care about how many photons are hitting the surface, both the slope and the intercept remain absolutely constant.

Therefore, the straight line does not shift, it does not steepen, and it does not flatten. The graph does not change at all!