Problem 34 - Master Resource Book in JEE Main Physics

Question

Light of wavelength 4000 A incident on a sodium surface for which the threshold wavelength of photoelectrons is \(5420 \mathrm{~A}\). The work function of sodium is (a) \(0.57 \mathrm{eV}\) (b) \(1.14 \mathrm{eV}\) (c) \(2.29 \mathrm{eV}\) (d) \(4.58 \mathrm{eV}\)

Step-by-Step Solution

Answer

The work function of sodium is (b) 1.14 eV.

Step 1: Understanding Work Function and Threshold Wavelength

The work function \( \phi \) of a material is the minimum energy required to eject an electron from its surface. This is related to the threshold wavelength \( \lambda_{0} \) by the formula: \[ \phi = \frac{hc}{\lambda_{0}} \] where \( h \) is Planck's constant \( (6.626 \times 10^{-34} \text{ Js}) \) and \( c \) is the speed of light \( (3 \times 10^{8} \text{ m/s}) \). Convert \( \lambda_{0} \) from angstroms to meters for calculation.

Step 2: Convert Threshold Wavelength to Meters

The threshold wavelength given for sodium is \( 5420 \text{ A} \). Convert this to meters:\[ 5420 \text{ A} = 5420 \times 10^{-10} \text{ m} \] Result: **5420 \( \times 10^{-10} \text{ m} \)**.

Step 3: Calculate the Work Function

Plug the converted wavelength into the work function equation:\[ \phi = \frac{(6.626 \times 10^{-34} \text{ Js})(3 \times 10^{8} \text{ m/s})}{5420 \times 10^{-10} \text{ m}} \]Calculate this to find \( \phi \) in Joules first.

Step 4: Convert Work Function from Joules to Electronvolts

Use the conversion factor \( 1 \, \text{eV} = 1.602 \times 10^{-19} \, \text{J} \) to convert \( \phi \) from Joules to electronvolts:\[ \phi_{(\text{eV})} = \frac{\phi_{(\text{J})}}{1.602 \times 10^{-19} \text{ J/eV}} \]

Step 5: Match Calculated Work Function to Given Options

Calculate \( \phi \) using the numbers from previous steps and convert it into electronvolts. Compare the calculated value with the provided options to identify the correct one.

Key Concepts

Work Function

The work function, denoted as \( \phi \), is a fundamental concept in the photoelectric effect. It represents the smallest amount of energy required to liberate an electron from the surface of a metal. This is crucial in understanding how and why different materials react to light in various ways.

• The work function is material-specific, meaning every element or compound has its own unique value.
• You can calculate the work function using the formula: \( \phi = \frac{hc}{\lambda_{0}} \).
• Here, \( h \) is Planck's constant and \( c \) is the speed of light in a vacuum.

By using this formula with the correct values, you can derive the energy required, making it possible to predict how a substance will behave when exposed to light.

Threshold Wavelength

Threshold wavelength, symbolized as \( \lambda_{0} \), is another pivotal concept. It is the longest wavelength of light capable of causing electron ejection from a material. In simpler terms, if the wavelength of light is longer than this threshold, the light doesn't have enough energy.

• Every material's threshold wavelength is different, unique to its atomic structure.
• This wavelength is directly linked to the work function by the relation \( \phi = \frac{hc}{\lambda_{0}} \).
• A shorter wavelength corresponds to higher energy photons, pivotal in photoelectric emission.

Considering these facts allows us to understand why certain frequencies or colors of light can cause electron emission while others cannot.

Planck's Constant

Planck's constant \( h \) is a foundational pillar of quantum mechanics. It links the energy of a photon to its frequency, playing a vital role in the equations governing the photoelectric effect.

• The value of Planck's constant is approximately \( 6.626 \times 10^{-34} \text{ Js} \).
• It appears in the formula \( E = hf \), demonstrating that energy is proportional to frequency.
• In the context of the photoelectric effect, it is used in \( \phi = \frac{hc}{\lambda_{0}} \), connecting light's energy and material properties.

This constant helps us understand the quantized nature of energy, an essential concept for comprehending modern physics.

Speed of Light

The speed of light \( c \) is one of the most fundamental constants in physics. It represents the velocity at which light travels in a vacuum, and is integral to many physical calculations, including those involving electromagnetic waves.

• The established value is \( 3 \times 10^{8} \text{ m/s} \).
• It is used in formulas to relate energy, frequency, and wavelength, such as \( \phi = \frac{hc}{\lambda_{0}} \).
• Understanding \( c \) is crucial for interpreting how light and other electromagnetic waves interact with matter.

By incorporating the speed of light into various equations, we can explain and predict a wide array of physical phenomena, from optical effects to particle physics.