Animated Solution for Physics - Dual Nature of Matter and Radiation: Radiation of wavelength $\lambda$ is incident on a photocell. The fastest emitted electron has speed $v$. If the wavelength is changed to $\frac{3\lambda }{4}$, the speed of the fastest emitted electron will be

The photoelectric effect is one of the most beautiful demonstrations of the quantum nature of light. When a photon strikes a metal surface, it doesn't just gently push an electron; it delivers its entire energy payload in one explosive event.

According to Einstein's photoelectric equation, this energy ($E=\frac{hc}{\lambda }$) is spent in two ways. First, a fixed 'tax' must be paid to free the electron from the metal's atomic lattice. This tax is the work function ($\varphi$). Whatever energy remains is converted into the kinetic energy of the escaping electron.

For the fastest emitted electron, which escapes without any internal collisions, the equation is:

In our problem, we are given two scenarios. In the first scenario, light of wavelength $\lambda$ ejects electrons with a maximum speed $v$.

In the second scenario, the wavelength is reduced to $\frac{3\lambda }{4}$. Because wavelength is inversely proportional to energy, a shorter wavelength means a more energetic photon. Let's call the new maximum speed $v^{\prime }$. We can write the equation for this new state:

Which simplifies to:

Now, we need to find a relationship between $v^{\prime }$ and $v$. The bridge between our two equations is the term $\frac{hc}{\lambda }$. We can substitute the entire expression from the first equation directly into the second equation:

Let's expand the left side to see how the energy distributes:

To isolate the new kinetic energy, we subtract $\varphi$ from both sides:

This final equation is incredibly revealing. If the work function $\varphi$ were zero, the new kinetic energy would be exactly $\frac{4}{3}$ times the old kinetic energy. However, because the work function is a positive constant, the new kinetic energy is $\frac{4}{3}$ times the old kinetic energy plus an extra positive term ($\frac{1}{3}\varphi$).

Let's divide the entire equation by $\frac{1}{2}m$ to look at the velocities:

Since $\varphi$ and $m$ are strictly positive physical quantities, the term $\frac{2\varphi }{3m}$ is strictly greater than zero. Therefore, we can establish a strict inequality:

Taking the square root of both sides, we arrive at our final, elegant conclusion:

This tells us that when you increase the incident energy by a certain factor, the kinetic energy of the electrons increases by more than that factor. Why? Because the 'tax' (work function) remained constant, allowing a larger fraction of the new energy to be converted purely into kinetic energy.