Animated Solution for Physics - Dual Nature of Matter and Radiation: The anode voltage of a photocells kept fixed. The wavelength $\lambda$ of the light falling on the cathode is gradually changed. The plate current $I$ of photocell varies as follows

The photoelectric effect is a fascinating phenomenon where light shining on a metal surface causes the emission of electrons. However, this emission is not guaranteed just because light is present. It strictly depends on the energy of the incoming light particles, known as photons.

According to Planck's quantum theory, the energy of a single photon is given by the equation $E=\frac{hc}{\lambda }$, where $h$ is Planck's constant, $c$ is the speed of light, and $\lambda$ is the wavelength of the incident light. This equation reveals a crucial inverse relationship: as the wavelength $\lambda$ increases, the energy $E$ of the photon decreases.

For an electron to be ejected from the metal surface, the incoming photon must possess a minimum amount of energy. This minimum energy is called the work function ($\varphi$) of the metal.

If the photon's energy is less than the work function ($E<\varphi$), no electrons will be emitted, regardless of how intense the light is or how long it shines. We can express this condition in terms of wavelength. There exists a maximum wavelength, called the threshold wavelength (${\lambda }_0$), beyond which no photoelectric emission occurs. Mathematically, this is written as $\lambda \le {\lambda }_0$.

In our specific problem, the anode voltage is kept fixed, and we are gradually increasing the wavelength $\lambda$ of the incident light. Let's trace what happens to the plate current (photocurrent) $I$.

Initially, when $\lambda$ is well below ${\lambda }_0$, the photons have plenty of energy to eject electrons. Assuming the intensity (number of photons per second) is relatively constant, a steady stream of photoelectrons is emitted, resulting in a constant or slowly varying positive photocurrent $I$.

As we continue to increase $\lambda$, the energy of each individual photon drops. Eventually, we reach the critical point where $\lambda ={\lambda }_0$. At this exact wavelength, the photon energy is just barely enough to overcome the work function, leaving the ejected electrons with zero kinetic energy.

If we increase the wavelength even slightly beyond this point ($\lambda >{\lambda }_0$), the photon energy becomes strictly less than the work function. The emission of photoelectrons abruptly ceases. Consequently, the photocurrent $I$ drops to exactly zero and remains zero for all higher wavelengths.

Looking at the given options, we need a graph that starts at a positive value and eventually drops to zero at a specific wavelength, remaining zero thereafter. Graph (d) perfectly captures this physical reality, showing a curve that smoothly descends to the $\lambda$-axis and stops, representing the cutoff at the threshold wavelength.