Animated Solution for Physics - Dual Nature of Matter and Radiation: The surface a metal is illuminated with the light of 400 nm. The kinetic energy of the ejected photoelectrons was found to be 1.68 eV. The work function of the metal is ($hc=1240\text{ eV-nm}$)

Finding the Work Function

Imagine you are observing a microscopic world where a metal surface is being bombarded by light. This isn't just any light; it's a stream of tiny energy packets called photons. In our specific problem, the wavelength of this incident light is given as $\lambda =400\text{ nm}$.

When one of these energetic photons strikes the metal surface, it transfers its energy to an electron. If the energy is sufficient, the electron breaks free and flies away! The problem tells us that the maximum kinetic energy of these escaping electrons is $K_{\max }=1.68\text{ eV}$. Our mission is to find the work function ($W_0$) of the metal, which is the minimum energy required just to pull an electron out of the surface.

To solve this, we rely on one of the most elegant equations in modern physics: Einstein's Photoelectric Equation. It beautifully states the conservation of energy in this quantum event:

Here, $E$ is the total energy brought in by the incident photon. This energy is split into two parts: a toll fee paid to the metal to escape ($W_0$), and whatever is left over becomes the kinetic energy of the electron ($K_{\max }$).

Before we can find the work function, we need to know exactly how much energy our $400\text{ nm}$ photon carries. The energy of a photon is inversely proportional to its wavelength, given by the formula:

The problem conveniently provides the value of the product $hc$ as $1240\text{ eV-nm}$. This is a fantastic shortcut because it allows us to calculate the energy directly in electron volts without messing with joules or powers of ten.

Let's substitute our values:

Notice how the nanometer units perfectly cancel out. Dividing $1240$ by $400$ gives us exactly $3.1\text{ eV}$. This is the total energy delivered by each individual photon.

Now that we have the photon's energy, we can bring back our master equation. We know the photon brings in $3.1\text{ eV}$, and the electron leaves with a maximum kinetic energy of $1.68\text{ eV}$.

To find the work function, $W_0$, we simply subtract the kinetic energy from the total photon energy:

And there we have it! The work function of this specific metal is $1.42\text{ eV}$. This means that any photon hitting this metal must have at least $1.42\text{ eV}$ of energy to eject an electron. If we used light with a longer wavelength (and thus lower energy), we might not see any electrons ejected at all!