Animated Solution for Physics - Dual Nature of Matter and Radiation: The stopping potential $V_0$ (in volt) as a function of frequency ($\nu$) for a sodium emitter, is shown in the figure. The work function of sodium, from the data plotted in the figure, will be (Take, Planck's constant ($h$) $=6.63\times 10^{-34}\text{ J-s}$, electron charge, $e=1.6\times 10^{-19}\text{ C}$)

Imagine you are conducting a classic photoelectric effect experiment. You shine light of varying frequencies onto a sodium metal plate and carefully measure the stopping potential required to halt the fastest ejected electrons. When you plot this stopping potential $V_0$ against the frequency $ u$ of the incident light, you get a beautiful straight line, exactly as shown in the given graph.

Our goal is to extract the work function of sodium directly from this visual data. To do this, we need to connect the geometry of the graph to the physics of the photoelectric effect.

The cornerstone of this phenomenon is Einstein's photoelectric equation, which states that the maximum kinetic energy of the emitted electrons is the energy of the incident photon minus the work function of the metal:

Here, $e$ is the elementary charge, $V_0$ is the stopping potential, $h$ is Planck's constant, $ u$ is the frequency, and $W$ is the work function. Let's rearrange this equation to isolate $V_0$, making it look like the standard equation of a straight line, $y=mx+c$:

This rearranged form is incredibly revealing. It tells us that the slope of our graph is $\frac{h}{e}$ (a universal constant!), and the y-intercept is $-\frac{W}{e}$. More importantly for our problem, it tells us what happens at the x-intercept.

Look closely at the graph where the line crosses the horizontal axis. At this exact point, the stopping potential $V_0$ is zero. This specific frequency is known as the threshold frequency, denoted by $ u_0$. It is the minimum frequency required to just barely eject an electron with zero kinetic energy.

From the graph, we can read this value directly:

Since the kinetic energy is zero at the threshold frequency, the incident photon's energy is entirely consumed by the work function. Therefore, we can write:

Let's substitute the known values into this elegant relation:

We have our work function, but it's in Joules. The options provided are in electron-volts (eV). To convert Joules to eV, we divide by the charge of a single electron ($1.6\times 10^{-19}\text{ C}$):

This perfectly matches option (b). The beauty of this problem lies in how seamlessly it bridges a visual graph with fundamental quantum principles!