Problem 44 - Master Resource Book in JEE Main Physics

Question

Light of wavelength \(\lambda\) strikes a photo sensitive surface and electrons are ejected with kinetic energy \(E\). If the \(\mathrm{KE}\) is to be increased to \(2 E\), the wavelength must be changed to \(\lambda^{\prime}\) where (a) \(\lambda^{\prime}=\frac{\lambda}{2}\) (b) \(\lambda^{\prime}=2 \lambda\) (c) \(\frac{\lambda}{2}<\lambda^{\prime}<\lambda\) (d) \(\lambda^{\prime}>\lambda\)

Step-by-Step Solution

Answer

The wavelength must change to \(\lambda' = \frac{\lambda}{2}\) (option a).

Step 1: Understanding the Photoelectric Effect

The photoelectric effect describes the ejection of electrons from a material when light of certain frequency (or wavelength) strikes its surface. The energy of the incident light depends inversely on its wavelength, described by the formula \( E = \frac{hc}{\lambda} - \phi \), where \( E \) is the kinetic energy of the ejected electrons, \( h \) is Planck's constant, \( c \) is the speed of light, \( \lambda \) is the light's wavelength, and \( \phi \) is the work function of the material.

Step 2: Relating Kinetic Energy to Wavelength

The equation \( E = \frac{hc}{\lambda} - \phi \) tells us that increasing the kinetic energy \( E \) requires an increase in the energy term \( \frac{hc}{\lambda} \). Since \( \frac{hc}{\lambda} \) is inversely proportional to \( \lambda \), decreasing the wavelength \( \lambda \) will increase the kinetic energy.

Step 3: Finding the New Wavelength for Doubling KE

To double the kinetic energy from \( E \) to \( 2E \), the energy of the light which is \( \frac{hc}{\lambda} \) needs to double as well. This means \( \frac{hc}{\lambda'} \) should equal \( 2 \times \frac{hc}{\lambda} \). Solving \( \frac{hc}{\lambda'} = 2 \times \frac{hc}{\lambda} \) leads to \( \lambda' = \frac{\lambda}{2} \).

Step 4: Verifying the Solution

Checking the choices given in the problem, option (a) \( \lambda' = \frac{\lambda}{2} \) matches our result, indicating this is the correct choice to increase the kinetic energy to \( 2E \).

Key Concepts

Kinetic Energy

Kinetic energy refers to the energy an object has due to its motion. In the context of the photoelectric effect, it is the energy imparted to electrons when they are ejected from a material's surface after being hit by light. The relationship between kinetic energy (\( E \)) and the photoelectric effect is described by the equation:

• \( E = \frac{hc}{\lambda} - \phi \)

Here, \( h \) represents Planck's constant, \( c \) the speed of light, \( \lambda \) the wavelength of the light, and \( \phi \) the work function of the material. This equation highlights that the kinetic energy of ejected electrons is directly influenced by the energy of the photons, which in turn depends on the wavelength of light. Consequently, the shorter the wavelength (and thus higher energy), the greater the kinetic energy of the ejected electrons.

Wavelength

Wavelength is a fundamental concept in physics that describes the distance between consecutive peaks of a wave. It is inversely related to energy when considering electromagnetic waves, such as light. In the photoelectric effect,\( \lambda \) determines the energy carried by the photons. Since energy has an inverse relationship with wavelength, as the wavelength of light decreases, the energy of the photons increases:

• \( E = \frac{hc}{\lambda} \)

This relationship implies that reducing the wavelength results in more energetic photons, which can provide more energy to electrons and thus increase their kinetic energy. This principle is used in applications like increasing the kinetic energy of photoelectrons to achieve specific technological results.

Work Function

The work function \( \phi \) of a material is the minimum amount of energy required to eject an electron from its surface. It acts as a threshold that photons must overcome in the photoelectric effect. This intrinsic property of a material depends on the binding energy of electrons and varies across different materials. The equation for the kinetic energy in the photoelectric effect,\( E = \frac{hc}{\lambda} - \phi \), demonstrates that any energy of the incident photons beyond the work function is converted into the kinetic energy of the ejected electrons. This means higher work functions make it harder for electrons to be dislodged unless impacted by light of a shorter wavelength or with greater intensity.

Planck's Constant

Planck's constant \( h \) is a fundamental quantity in quantum mechanics that describes the fixed, quantized nature of electromagnetic radiation energy. It plays a crucial role in understanding the relationship between energy and frequency (or wavelength) of waves. In the photoelectric effect equation:\[ E = \frac{hc}{\lambda} - \phi \]Planck's constant appears as part of the term \( \frac{hc}{\lambda} \), showcasing that energy is determined by frequency or inverse wavelength. By using energy units \( \text{Joule-seconds} \), \( h \) allows scientists to quantify how light interacts with matter at the atomic level, making it a cornerstone constant in modern physics for theoretical development and empirical assessments.

Speed of Light

The speed of light \( c \) is a critical constant in physics representing the speed at which electromagnetic waves, including light, propagate through a vacuum. Known to be approximately 299,792,458 m/s, this universal constant not only ties into relativity but also into quantum mechanics, especially the photoelectric effect, where it is a factor in determining energy levels:

• \( E = \frac{hc}{\lambda} - \phi \)

Because the speed of light remains constant regardless of the observer's state of motion, it ensures that energy computations using the formula maintain coherence across various scenarios. Light's invariant speed is essential for the synchronization of signals, understanding energy transfer at atomic and subatomic levels, and reinforcing the principle that light is the fastest transmitting entity known.