Animated Solution for Physics - Dual Nature of Matter and Radiation: Sodium and copper have work functions $2.3\text{ eV}$ and $4.5\text{ eV}$, respectively. Then, the ratio of the wavelengths is nearest to

Imagine a metal surface as a fortress holding onto its electrons. To break an electron free, you must pay a specific energy toll. In the world of the photoelectric effect, this toll is called the work function ($\varphi$). Every metal has its own unique toll. In our problem, Sodium (Na) demands a modest $2.3\text{ eV}$, while Copper (Cu) is much stricter, requiring $4.5\text{ eV}$.

When the problem asks for the "ratio of the wavelengths" associated with these work functions, it is implicitly referring to the threshold wavelength (${\lambda }_0$). This is the absolute maximum wavelength (and thus, the minimum energy) a photon can have to just barely pay the toll and eject an electron.

Planck-Einstein Relation

To connect the energy toll to the wavelength of light, we use the fundamental Planck-Einstein relation:

At the exact threshold of emission, the energy of the incident photon exactly equals the work function of the metal. Therefore, we can write:

Rearranging this to solve for the threshold wavelength gives us:

This equation reveals a beautiful and critical physical truth: Threshold wavelength is inversely proportional to the work function (${\lambda }_0\propto \frac{1}{\varphi }$). A higher energy toll requires a more energetic photon, which means a shorter wavelength.

Armed with this inverse relationship, we can easily compare the two metals without ever needing to plug in the messy constants $h$ and $c$. We want the ratio of the wavelength of Sodium to Copper:

The $hc$ terms gracefully cancel out, leaving us with the inverse ratio of their work functions:

Now, we simply substitute the given values into our streamlined equation. Notice that we don't even need to convert electron-volts (eV) to Joules, because the units will perfectly cancel out in the ratio!

In the context of competitive exams like JEE, we look for the nearest integer ratio. The value $1.956$ is incredibly close to $2$. Therefore, we can confidently conclude that the ratio is approximately $2:1$.