Animated Solution for Physics - Dual Nature of Matter and Radiation: Two identical photocathodes receive the light of frequencies $f_1$ and $f_2$, respectively. If the velocities of the photoelectrons coming out are $v_1$ and $v_2$ respectively, then

Imagine you are standing in a quantum laboratory. In front of you are two identical pieces of metal—photocathodes. Because they are forged from the exact same material, they share a fundamental property: their work function, $\varphi$. This work function is like a toll booth; it demands a specific amount of energy before it lets an electron escape the metal's surface.

When we shine light on these metals, we are essentially bombarding them with packets of energy called photons. The energy of each photon is directly proportional to the frequency of the light, given by the equation $E=hf$, where $h$ is Planck's constant.

In our problem, we have two distinct scenarios playing out on these identical metals.

In the first scenario, light of frequency $f_1$ strikes the first photocathode. The photons deliver an energy of $h{f}_1$. The metal takes its toll, $\varphi$, and the remaining energy is transferred to the ejected electron as kinetic energy. According to Einstein's photoelectric equation, the maximum kinetic energy of this electron, moving with velocity $v_1$, is:

In the second scenario, light of frequency $f_2$ strikes the second photocathode. Similarly, the photons deliver an energy of $h{f}_2$. The metal again takes the exact same toll, $\varphi$, because it is identical to the first. The ejected electron, now moving with velocity $v_2$, has a maximum kinetic energy of:

We now have a system of two equations. The key to solving physics problems often lies in looking at the destination before starting the journey. If we glance at the options provided in the question, we notice a glaring absence: the work function $\varphi$ is nowhere to be found.

This is our biggest clue. We must eliminate $\varphi$ from our mathematical model. Since $\varphi$ appears as a subtracted term in both equations, the most elegant way to banish it is to subtract equation (ii) from equation (i).

Let's perform the subtraction:

Notice how beautifully the physics aligns with the math. The $-\varphi$ and the $-(-\varphi )$ perfectly cancel each other out, leaving us with a pure relationship between kinetic energies and photon energies:

We are almost there. The final step is mere algebraic rearrangement to isolate the velocity terms on one side, matching the structure of the given options.

We multiply both sides by $2$ and divide by the mass of the electron, $m$:

This perfectly matches option (a).

A Word of Caution

This problem was a straightforward application of Einstein's photoelectric equation, but it carried a subtle trap. The entire solution hinged on the word "identical".

If the problem had stated that the photocathodes were made of different materials, their work functions would be ${\varphi }_1$ and ${\varphi }_2$. Subtracting the equations would have left a messy $({\varphi }_2-{\varphi }_1)$ term, and the elegant cancellation would have failed. Always read the physical constraints of a problem carefully before diving into the algebra!