Animated Solution for Physics - Dual Nature of Matter and Radiation: According to Einstein's photoelectric equation, the plot of the kinetic energy of the emitted photoelectrons from a metal Versus the frequency of the incident radiation gives a straight line whose slope

The photoelectric effect is one of the most beautiful phenomena in modern physics, serving as the cornerstone for the particle nature of light. When light of a certain frequency strikes a metal surface, it can eject electrons. But how do we understand the relationship between the energy of these electrons and the light that freed them?

Enter Albert Einstein and his elegant photoelectric equation.

Einstein proposed that light consists of packets of energy called photons. The energy of a single photon is given by $E = h u$, where $h$ is Planck's constant and $ u$ is the frequency of the light.

When this photon hits an electron, it transfers its entire energy. However, the electron must pay a "toll fee" to escape the metal's surface. This minimum energy required is called the work function, denoted by $\varphi$.

Whatever energy is left over becomes the maximum kinetic energy ($K{E}_{max}$) of the ejected electron. Mathematically, this is expressed as:

This simple yet profound equation is the key to unlocking our graph.

Imagine plotting this relationship on a graph. We place the frequency of the incident light ($ u$) on the x-axis and the maximum kinetic energy ($K{E}_{max}$) on the y-axis.

To understand the shape of this graph, let's compare Einstein's equation to the standard equation of a straight line in coordinate geometry:

By aligning the variables, we can see a perfect match: - $y$ corresponds to $K{E}_{max}$ - $x$ corresponds to $ u$ - $m$ (the slope) corresponds to $h$ - $c$ (the y-intercept) corresponds to $-\varphi$

This tells us that the graph of $K{E}_{max}$ versus $ u$ is a perfect straight line!

Now, let's focus on the slope of this line. As we derived, the slope $m$ is exactly equal to Planck's constant $h$.

Planck's constant is a fundamental, universal constant of nature. It does not change if you use a different metal, nor does it change if you shine a brighter light (which only increases the intensity or number of photons, not their individual energy).

Therefore, the slope of the $K{E}_{max}$ versus $ u$ graph is the same for all metals and is completely independent of the intensity of the radiation.

If you were to plot this graph for copper, zinc, and sodium on the same axes, you would get a series of perfectly parallel lines. They would have different x-intercepts (threshold frequencies) and different y-intercepts (work functions), but their slopes would be identical, forever bound by the universal value of $h$.

This elegant graphical representation not only confirms Einstein's theory but also provides a direct experimental method to measure Planck's constant!