Animated Solution for Physics - Dual Nature of Matter and Radiation: In a photoelectric experiment, the wavelength of the light incident on a metal is changed from $300\text{ nm}$ to $400\text{ nm}$. The decrease in the stopping potential is close to $\left(\frac{hc}{e}=1240\text{ nm V}\right)$

Imagine you are driving on a highway and you encounter a toll booth. To pass through, you need to pay a specific fee. If you hand the toll collector a large bill, you get some change back. The photoelectric effect works in a remarkably similar way!

When a photon of light strikes a photosensitive metal surface, it acts like a car arriving at the toll booth. The metal demands a specific "toll fee" called the work function ($\varphi$) to let an electron escape. If the photon has more energy than this work function, the electron escapes with the leftover energy as its kinetic energy.

To measure this maximum kinetic energy, we apply a reverse voltage—a "stopping potential" ($V_0$)—just strong enough to halt even the fastest electrons.

Einstein beautifully captured this energy conservation in his famous photoelectric equation:

Since the energy of a photon is given by $E=\frac{hc}{\lambda }$ and the maximum kinetic energy is $K_{\max }=e{V}_0$, we can rewrite the equation as:

In our problem, we are dealing with two different scenarios on the same metal surface. Let's write the equation for both cases.

For the first wavelength (${\lambda }_1=300\text{ nm}$):

For the second wavelength (${\lambda }_2=400\text{ nm}$):

We are asked to find the decrease in the stopping potential, which is $\Delta V=V_{01}-V_{02}$.

Notice that the work function $\varphi$ is a property of the metal itself. Because we are using the same metal in both cases, $\varphi$ is a constant. If we subtract the second equation from the first, the work function elegantly cancels out!

Now, we can isolate our target, $\Delta V$, by dividing the entire equation by the elementary charge $e$ and factoring out $hc$:

This is where the magic happens. The problem generously provides the value of $\frac{hc}{e}$ as $1240\text{ nm V}$. This saves us from plugging in the microscopic values of Planck's constant, the speed of light, and the charge of an electron individually.

Let's substitute our known values into the isolated equation:

To solve the fraction, we find a common denominator for $300$ and $400$, which is $1200$.

When we divide $31$ by $30$, we get approximately $1.033\text{ V}$. Looking at our multiple-choice options, the closest value is $1.0\text{ V}$.

By understanding the physical reality behind the equations, we turned a potentially messy calculation into a smooth, logical derivation!