Problem 92 - Master Resource Book in JEE Main Physics

Question

Monochromatic light incident on a metal surface emits electrons with kinetic energies from zero to \(2.6 \mathrm{eV}\). What is the least energy of the incident photon, if the tightly bound electron needs \(4.2 \mathrm{eV}\) to remove? (a) \(1.6 \mathrm{~V}\) (b) From \(1.6 \mathrm{eV}\) to \(6.8 \mathrm{eV}\) (c) \(6.8 \mathrm{eV}\) (d) More than \(6.8 \mathrm{eV}\)

Step-by-Step Solution

Answer

The least energy of the incident photon is 6.8 eV, choice (c).

Step 1: Understanding the Energy Requirements

To find the energy of the incident photon, we need to consider the energy needed to remove the electron from the metal (work function, \(W\)) and the maximum kinetic energy (\(KE_{max}\)) of the emitted electrons. The work function \(W\) is given as \(4.2 \mathrm{eV}\). The maximum kinetic energy of the emitted electrons is \(2.6 \mathrm{eV}\).

Step 2: Applying the Photoelectric Equation

The photoelectric equation relates the energy of the incident photon \(E_{photon}\) to the work function \(W\) and the maximum kinetic energy \(KE_{max}\): \[ E_{photon} = W + KE_{max} \]

Step 3: Solving for Photon Energy

Substitute the known values into the equation: \[ E_{photon} = 4.2 \mathrm{eV} + 2.6 \mathrm{eV} = 6.8 \mathrm{eV} \] Thus, the least energy of the incident photon is \(6.8 \mathrm{eV}\).

Step 4: Determining the Correct Answer

With \(E_{photon} = 6.8 \mathrm{eV}\), we identify the correct value among the given options: (c) \(6.8 \mathrm{eV}\).

Key Concepts

Work Function

The work function is a fundamental concept in the photoelectric effect. It represents the minimum energy required to eject an electron from the surface of a metal. Each metal has a characteristic work function, which depends on its atomic structure. In the problem, this energy is valued at \( 4.2 \, \mathrm{eV} \).
Understanding the work function is crucial because it sets the threshold at which electrons can begin to escape the metal. Only when the energy of the incident photon exceeds this threshold will the electrons have enough surplus energy to be emitted with kinetic energy. It’s vital to note that if the photon energy is lesser than the work function, no electrons will be emitted.
In simpler terms, the work function is like a hurdle on a track. Photons need to give electrons enough energy to not only overcome this hurdle but to also allow them to run past it with some leftover speed (kinetic energy).
This characteristic energy is not just a barrier but a determining factor in the amount and speed of emitted electrons during the photoelectric effect.

Kinetic Energy

In the context of the photoelectric effect, the kinetic energy (KE) of emitted electrons is pivotal as it dictates how fast these electrons move after being ejected. After a photon gives its energy to an electron, the part that is left over after overcoming the work function manifests as kinetic energy.
In the given exercise, the maximum kinetic energy of the emitted electrons is \( 2.6 \, \mathrm{eV} \). This value is derived from the energy surplus of the incident photons. It signifies the fastest speed at which an electron is ejected when a photon of sufficient energy hits an electron at the surface of the metal.
The photoelectric equation \[ KE_{max} = E_{photon} - W \] is used to determine this kinetic energy. The importance of this kinetic value lies in its use to calculate the actual energy of the incident photons when combined with the work function.

• Hence, understanding KE is important not only for resolving what happens post-emission but also for calculating initial conditions when experimental data is involved.

Photon Energy

Photon energy is the crux of the photoelectric effect. It refers to the energy carried by a photon of light, which interacts with electrons in a material to potentially release them. Calculating this energy is integral to understanding whether electrons will be liberated from the metal surface.
In our scenario, the energy of the incident photon \( E_{photon} \) must be at least \( 6.8 \, \mathrm{eV} \) to allow the electrons to overcome the work function \( 4.2 \, \mathrm{eV} \) and still possess a kinetic energy \( 2.6 \, \mathrm{eV} \).

By applying the equation:
\[ E_{photon} = W + KE_{max} \], substituting the given values yields \( E_{photon} = 4.2 \, \mathrm{eV} + 2.6 \, \mathrm{eV} = 6.8 \, \mathrm{eV} \).
This calculation proved pivotal in confirming that option (c) \( 6.8 \, \mathrm{eV} \) is the correct least energy needed for the photon.

• Ultimately, photon energy is the driving force that enables the ejection of electrons, making it a vital concept for anyone studying the interactions of light and matter.