Animated Solution for Physics - Dual Nature of Matter and Radiation: The following figure shows few data points in a photoelectric effect experiment for a certain metal. The minimum energy for ejection of electron from its surface is (Take, Planck's constant, $h=6.62\times 10^{-34}$ J-s)

Imagine you are conducting the classic photoelectric effect experiment. You shine light of various frequencies onto a metal surface and measure the stopping potential required to halt the fastest ejected electrons. The graph provided in the problem is a direct visual representation of this experiment, plotting stopping potential $V_{\text{Stop}}$ against the incident frequency $f$.

The question asks for the minimum energy required to eject an electron. In the language of physics, this minimum energy is known as the work function (${\varphi }_0$) of the metal.

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

At the threshold frequency ($f_0$), the incident photons have just enough energy to overcome the work function, meaning the ejected electrons have zero kinetic energy. Consequently, the stopping potential $V_{\text{Stop}}$ is exactly zero.

Looking closely at the graph, the line intersects the x-axis at point B. The coordinates of point B are $(5.5,0)$. This x-intercept is our golden ticket—it tells us that the threshold frequency $f_0$ is exactly $5.5\times 10^{14}\text{ Hz}$.

Now, we simply substitute this value into our work function equation along with Planck's constant ($h=6.62\times 10^{-34}\text{ J}\cdot \text{s}$):

I know this number looks a bit messy, but let's take a breath. The options provided are in electron-volts (eV), not Joules. To convert our energy into eV, we must divide by the elementary charge ($e=1.6\times 10^{-19}\text{ C}$):

To make the division easier, let's adjust the decimal point in the numerator:

Calculating this gives us approximately $2.27\text{ eV}$. This perfectly matches option (c). The beauty of this problem lies in how a simple geometric feature—the x-intercept—directly unlocks the quantum properties of the metal!