Animated Solution for Physics - Dual Nature of Matter and Radiation: The maximum kinetic energy of electrons emitted in the photoelectric effect is linearly dependent on the ......... of the incident radiation.

Imagine a beam of light striking a metal surface. If the light has enough energy, it can knock electrons out of the metal. This phenomenon is known as the photoelectric effect. But how much energy do these ejected electrons carry? Albert Einstein provided the answer with his elegant photoelectric equation:

Here, $K_{\max }$ represents the maximum kinetic energy of the emitted photoelectrons. The term $h u$ is the energy of the incident photon, where $h$ is Planck's constant and $ u$ is the frequency of the light. Finally, $W$ is the work function, which is the minimum energy required to just free an electron from the metal's surface.

Let's look at this equation through the lens of coordinate geometry. Does it remind you of something familiar?

If we compare $K_{\max} = h u - W$ with the standard equation of a straight line, $y=mx+c$, we can see a perfect match.

In this analogy, our dependent variable $y$ is the maximum kinetic energy ($K_{\max }$), and our independent variable $x$ is the frequency ($ u$) of the incident radiation. The slope of this line, $m$, corresponds to Planck's constant ($h$), which is a universal constant. The y-intercept, $c$, corresponds to the negative of the work function ($-W$).

Because the relationship between $K_{\max }$ and $ u$ perfectly fits the equation of a straight line, we can definitively say that the maximum kinetic energy of the emitted electrons is linearly dependent on the frequency of the incident radiation.

If you were to plot a graph of $K_{\max }$ against $ u$, you would get a straight line that intersects the frequency axis at the threshold frequency ($ u_0$) and has a constant slope equal to $h$. Therefore, the missing word in our statement is frequency.