Animated Solution for Physics - Dual Nature of Matter and Radiation: The work functions of silver and sodium are $4.6$ and $2.3\text{ eV}$, respectively. The ratio of the slope of the stopping potential versus frequency plot for silver to that of sodium is

Imagine shining a beam of light onto a piece of metal and watching electrons pop out like tiny sparks. This is the photoelectric effect, a phenomenon that completely revolutionized our understanding of light and matter.

Albert Einstein brilliantly explained this by proposing that light is made of tiny packets of energy called photons. When a photon hits an electron, it transfers its energy. If this energy is enough to overcome the metal's grip—known as the work function ($W$)—the electron breaks free!

Einstein's photoelectric equation beautifully captures this energy exchange:

Here, $K_{\max }$ is the maximum kinetic energy of the escaping electron, $h$ is Planck's constant, and $f$ is the frequency of the incident light.

But how do we measure this kinetic energy? We apply a reverse voltage to stop the electrons. The minimum voltage required to stop even the fastest electrons is called the stopping potential ($V_0$). The energy required to stop them is $e{V}_0$, where $e$ is the elementary charge.

Substituting this into our equation, we get:

Let's rearrange this equation to solve for the stopping potential $V_0$:

Now, look closely at this equation. Does it remind you of something from coordinate geometry? Yes, it perfectly matches the equation of a straight line:

If we plot the stopping potential $V_0$ on the y-axis and the frequency $f$ on the x-axis, we get a straight line!

By comparing the two equations, we can easily identify the slope ($m$) of our graph:

This is where the magic happens! Notice that the slope depends only on Planck's constant ($h$) and the elementary charge ($e$). Both of these are fundamental, universal constants.

What does this mean? It means that the slope is completely independent of the metal's work function ($W$). Whether you are using silver with a work function of $4.6\text{ eV}$, sodium with $2.3\text{ eV}$, or any other metal in the universe, the slope of the $V_0$ versus $f$ graph will always be exactly the same!

The problem asks for the ratio of the slope for silver to that of sodium. Since the slopes are identical, the ratio is simply:

The different work functions only affect the y-intercept ($-\frac{W}{e}$) and the x-intercept (the threshold frequency), shifting the parallel lines horizontally and vertically. But their steepness—their slope—remains forever constant.