Problem 13 - Master Resource Book in JEE Main Physics

Question

A \(100 \mathrm{~W}\) sodium lamp radiates energy uniformly in all directions. The lamp is located at the centre of a large sphere that absorbs all the sodium light which is incident on it. The wavelength of the sodium light is \(589 \mathrm{~nm} .\) (i) What is the energy per photon associated with the sodium light? (ii) At what rate are the photons delivered to the sphere? (a) (i) \(4.6 \mathrm{eV}\) (ii) \(1.6 \times 10^{24}\) photon/s (b) (i) \(3.4 \mathrm{eV}\) (ii) \(4.5 \times 10^{24}\) photon/s (c) (i) \(2.1 \mathrm{eV}\) (ii) \(3 \times 10^{20}\) photon/s (d) (i) \(1.1 \mathrm{eV}\) (ii) \(2 \times 10^{24}\) photon/s

Step-by-Step Solution

Answer

The correct option is (c): (i) 2.1 eV, (ii) 3 × 10^{20} photon/s.

Step 1: Calculate Energy of One Photon

The energy of a photon can be calculated using the formula:\[ E = \frac{hc}{\lambda} \]where:- \( h = 6.626 \times 10^{-34} \text{ J·s} \) is Planck's constant,- \( c = 3 \times 10^8 \text{ m/s} \) is the speed of light,- \( \lambda = 589 \times 10^{-9} \text{ m} \) is the wavelength of the sodium light.Substitute the values into the equation to find the energy per photon:\[ E = \frac{6.626 \times 10^{-34} \times 3 \times 10^8}{589 \times 10^{-9}} \approx 3.37 \times 10^{-19} \text{ J} \]Convert this energy from joules to electron volts (eV), knowing that \( 1 \text{ eV} = 1.602 \times 10^{-19} \text{ J} \):\[ E = \frac{3.37 \times 10^{-19}}{1.602 \times 10^{-19}} \approx 2.1 \text{ eV} \]

Step 2: Calculate Photon Delivery Rate

The power of the lamp \( P = 100 \text{ W} \) indicates it emits \( 100 \text{ joules per second} \).The number of photons \( n \) delivered per second can be found using:\[ n = \frac{P}{E} \]where \( E \) is the energy per photon, calculated previously as \( 3.37 \times 10^{-19} \text{ J} \):\[ n = \frac{100}{3.37 \times 10^{-19}} \approx 2.97 \times 10^{20} \text{ photons/s} \]

Step 3: Choose the Correct Option

From our calculations, we found:- Energy per photon is approximately \(2.1 \text{ eV}\).- Photon delivery rate is approximately \(3 \times 10^{20} \text{ photons/s}\).Comparing these values with the given options, option (c) matches our results.

Key Concepts

Planck's Constant

In the realm of quantum mechanics, understanding the fundamental constants can unveil the nature of particles and light. Planck's Constant ( \(h = 6.626 \times 10^{-34} \, \text{J}\cdot\text{s}\)) is one of those essential constants. This quantity signified the smallest action that can be observed, playing a crucial role in linking a photon's energy with its frequency.

• The equation \( E = h u \) demonstrates how energy \(E\) is proportional to frequency \(u\).
• Alternately, with the speed of light \(c\) in play, it relates with wavelength \(\lambda\) as \( E = \frac{hc}{\lambda} \).

These formulas illustrate that Planck's Constant is central in the energy calculation of photons. It's the bridge connecting observable light phenomena with the quantum world. Despite its minuscule size, impacting units as slight as joules times seconds, it is pivotal in quantum physics.

Wavelength of Light

Light as we observe it might seem consistent, but its properties exhibit diversity that unveils a deeper structure. The wavelength of visible light represents this variety, often measured in nanometers (nm).

• Wavelength indicates the distance between consecutive peaks of a light wave.
• In the case of the sodium lamp, the key wavelength is \(589 \, \text{nm}\), a signature of its emission.

Wavelength influences the energy a photon holds. Using the equation \(E = \frac{hc}{\lambda}\), this relationship is evident. As seen, the sodium light's wavelength is employed to compute photon energy. The shorter the wavelength, the higher the energy per photon. Hence, even concepts like color perception relies upon specific light wavelengths.

Photon Delivery Rate

Breaking down the vast energy emission of a lamp into particle form gives us the photon delivery rate. This rate forms a cornerstone in understanding how much light energy is distributed over time.

• The power of the lamp, provided as 100 watts, translates to an energy distribution of 100 joules per second.
• To determine the photon rate, the equation \(n = \frac{P}{E}\) unravels how many photons are emitted every second.

Using our calculated energy of each photon, we get a delivery rate of approximately \(2.97 \times 10^{20} \text{ photons/s}\), which shows how active any given lamp can be. Such measurements help us perceive not only intensity but also the efficiency of light sources in illuminating spaces.