Animated Solution for Physics - Dual Nature of Matter and Radiation: The maximum kinetic energy of photoelectrons emitted from a surface when photons of energy 6 eV fall on it is 4 eV. The stopping potential in volt is

Mastering the Stopping Potential

Have you ever been handed a puzzle where half the pieces belong to a completely different picture? That’s exactly what the examiners have done in this classic photoelectric effect problem. It’s a test of not just your mathematical ability, but your conceptual confidence. Let’s break down the physics and see why this question is a beautiful trap for the unwary student.

The Photoelectric Dance

Imagine the microscopic world of a metal surface. Photons, acting as tiny packets of light energy, are bombarding this surface. In our specific scenario, each incoming photon carries an energy of $6\text{ eV}$. When a photon strikes an electron, it transfers its energy. If this energy is sufficient to overcome the metal's binding energy (the work function), the electron breaks free and flies away.

Because electrons are bound with varying strengths depending on their depth within the metal, they emerge with a range of kinetic energies. The problem tells us that the absolute fastest of these escapees—the elite sprinters of the electron world—have a maximum kinetic energy ($K_{\max }$) of $4\text{ eV}$.

Now, imagine we want to stop this flow of electrons. We set up a target plate (the anode) and apply a negative voltage to it. This creates an electric field that pushes back against the incoming electrons. It’s like trying to roll a marble up a steep hill.

To completely stop the photoelectric current, we don't just need to stop the slow electrons; we must stop the absolute fastest ones. The minimum reverse voltage required to halt even the most energetic electron is called the stopping potential, denoted as $V_0$.

The physics here is a simple conservation of energy: the work done by the electric field must exactly equal the maximum kinetic energy of the electron. Mathematically, this is expressed as: where $e$ is the elementary charge of an electron.

Here is where the magic happens. We know that $K_{\max }=4\text{ eV}$. Let's plug this directly into our master equation:

Look closely at the units. The energy is given in electron-volts (eV). An electron-volt is literally the energy an electron gains (or loses) when moving through a potential difference of one volt. Because we have the elementary charge $e$ on both sides of the conceptual equation, it elegantly cancels out.

Dividing both sides by $e$, we are left with:

The stopping potential is exactly $4\text{ V}$.

Did you notice what we didn't use? The $6\text{ eV}$ energy of the incident photons! This is the hallmark of a brilliantly designed question. Many students will instinctively try to plug every given number into Einstein's photoelectric equation ($E=\Phi +K_{\max }$) and end up calculating the work function ($\Phi =6\text{ eV}-4\text{ eV}=2\text{ eV}$) before realizing it doesn't help them find the stopping potential any faster.

The examiner provided the $6\text{ eV}$ to see if you truly understood the direct, intimate relationship between maximum kinetic energy and stopping potential. If you know that $K_{\max }$ in eV numerically equals $V_0$ in volts, you can bypass the extra data and arrive at the answer in seconds.

This problem is a powerful reminder: in physics, clarity of concept is your greatest weapon against the illusion of complexity.