Question

Light of wavelength, \(\lambda\), strikes a metal surface with intensity \(X\) and the metal emits \(Y\) electrons per second of maximum kinetic energy \(Z\). What will happen to \(Y\) and \(Z\) if \(X\) is halved? (a) \(Y\) will be halved and \(Z\) will be doubled (b) \(Y\) will be doubled and \(Z\) will be halved (c) \(Y\) will be halved and \(Z\) will remain the same (d) \(Y\) will remain same and \(Z\) will be halved

Short answer

Option (c) Y will be halved and Z will remain the same.

Step-by-step solution

Understanding the Photoelectric Effect

The question pertains to the Photoelectric Effect, which occurs when light of sufficient frequency strikes the surface of a metal and causes electrons to be emitted. According to Einstein's theory of the photoelectric effect, the kinetic energy of the emitted electrons is proportional to the frequency of the incident light but is independent of its intensity. The number of emitted electrons per second, however, is directly proportional to the intensity of the light.

Effect of Reducing Intensity on Electron Emission (Y)

Since the number of emitted electrons (Y) depends on the light intensity, reducing the intensity of the incoming light, X, by half, will also reduce the number of electrons emitted per second (Y) by half.

Effect of Reducing Intensity on Maximum Kinetic Energy (Z)

The maximum kinetic energy (Z) of the emitted electrons is not affected by changes in light intensity. It only depends on the frequency (or wavelength) of the light. Since the wavelength, \(\lambda\), is constant and the light intensity (X) has no bearing on the kinetic energy, the maximum kinetic energy (Z) will remain the same.

Key concepts

Kinetic Energy of Emitted Electrons

When discussing the photoelectric effect, the kinetic energy of the emitted electrons is a key focus. It's essential to understand that the kinetic energy, denoted as K.E., of an electron that's ejected from a metal surface is the energy it has due to its motion. This energy is the result of the electron absorbing a quantum of energy from the incident light.
According to the laws of physics, the kinetic energy of an electron after being emitted can be calculated using the equation
\[\begin{equation} K.E. = hu - \phi, \end{equation}\]
where h is Planck's constant, u is the frequency of the incident light, and \phi is the work function of the metal - the minimum energy needed to remove an electron from the metal surface.
If the frequency of light is kept constant, and only the intensity varies, the kinetic energy of the emitted electrons will not change. This is because intensity, which relates to the number of photons striking the surface per unit time, does not affect the energy per photon (
which is associated with frequency). Instead, the intensity affects the rate at which electrons are emitted, a factor we will explore further in the next section.

Intensity of Incident Light

The intensity of incident light, labeled here as X, plays a crucial role in the photoelectric effect, specifically in the number of electrons (
Y) ejected. Intensity is a measure of the power per unit area carried by the light and essentially determines how many photons are hitting the metal's surface per second. Unlike frequency, intensity does not impact the energy each photon carries; rather, it influences the likelihood of electron emissions occurring.
\[\begin{equation}Intensity \propto Number \: \: of \: photons \end{equation}\]
In our example, halving the intensity of the incoming light would also halve the number of electrons emitted per second. Higher intensity means more electrons can absorb photons and gain sufficient energy to overcome the metal's work function. Lowering the intensity reduces the rate at which these interactions occur, ultimately leading to fewer electrons being emitted.

Einstein's Theory of Photoelectric Effect

Einstein's groundbreaking theory of the photoelectric effect provided a quantum explanation for this observed phenomenon, revolutionizing our understanding of light and matter interactions. He proposed that light consists of particles called photons, each with an energy quantum proportional to its frequency
\[\begin{equation}E = hu, \end{equation}\]
where E is the energy of each photon, h is Planck's constant, and u is the frequency. When these photons interact with electrons in the metal, an electron can be ejected if the photon's energy exceeds the work function of the metal.
Einstein's theory implies that only the photons with at least the minimum energy threshold, as dictated by the work function, can result in photoemission. It explains why changing the intensity of light does not affect the kinetic energy of the emitted electrons but changing the frequency does. This theory earned Einstein the Nobel Prize in Physics and paved the way for the development of quantum mechanics.