Question

Two electromagnetic radiations have wave numbers in the ratio \(2: 3\). Their energies per quanta will be in the ratio (a) \(3: 2\) (b) \(9: 4\) (c) \(4: 9\) (d) \(2: 3\)

Short answer

The ratio of energies per quanta will be in the ratio (d) 2:3.

Step-by-step solution

Understanding the Relationship Between Wave Number and Energy

First, recall that the energy (E) per quantum of electromagnetic radiation is directly proportional to its frequency (ν), according to Planck's equation, E = hν. The wave number (k) is directly proportional to the frequency, where k = ν/c, and c is the speed of light. Therefore, energy is also directly proportional to the wave number, E ∝ k.

Setting Up the Ratio of Energies Based on Wave Numbers

Given that the ratio of wave numbers is 2:3, and since energy is directly proportional to wave number, the ratio of energies E1/E2 will be the same as the ratio of wave numbers k1/k2. So, if k1/k2 = 2/3, then E1/E2 = 2/3 as well.

Determining the Correct Option

From the direct proportionality, we can conclude that the ratio of energies per quanta will be the same as the ratio of the wave numbers. Therefore, the ratio of energies per quanta will also be 2:3.

Key concepts

Understanding Electromagnetic Radiation

Electromagnetic radiation encompasses a variety of waves, including light waves, radio waves, and X-rays, which all share the common property of traveling at the speed of light in a vacuum. These waves carry energy, which can be quantified in packets known as quanta or photons.

The energy carried by a photon is linked to its electromagnetic properties, which include frequency and wavelength. High-frequency electromagnetic radiation, like gamma rays, carries more energy per photon than low-frequency waves, such as radio waves.

Exploring the electromagnetic spectrum further, different types of radiation exhibit distinct behaviors and characteristics which can be understood through their interaction with matter. For example, ultraviolet light can cause chemical reactions, while infrared radiation is mainly associated with heat.

Deciphering Wave Number

The wave number is a measure of the spatial frequency of a wave, representing the number of wave cycles in a unit distance. Symbolically denoted by 'k,' it is inversely related to the wavelength \( \lambda \) as \( k = \frac{1}{\lambda} \) and has the units of reciprocal meters \( m^{-1} \) in the International System of Units (SI).

Notice how the wave number can provide insights into the energy per quantum of radiation; a higher wave number indicates smaller wavelengths and, consequently, higher energy according to Planck's equation.

Understanding the relationship between wave number and other properties of the wave can greatly enhance our comprehension of complex phenomena such as diffraction patterns, resonance and the interaction of matter with electromagnetic waves.

Exploring Planck's Equation

Planck's equation \( E = hu \) describes the quantized nature of the energy carried by electromagnetic waves. Here, 'E' represents the energy per quantum, 'h' denotes Planck's constant (\(6.62607015 \times 10^{-34} J\cdot s\)), and 'ν' is the frequency of the wave.

This equation is a cornerstone of quantum physics and elucidates the direct proportionality between energy and frequency, indicating that each photon of light carries a discrete amount of energy that increases with the light's frequency. Planck's work leads us to understand blackbody radiation, photoelectric effect, as well as chemical and biological processes that depend on the absorption and emission of light at specific frequencies.

Incorporating Planck's equation in our study of electromagnetic radiation not only allows for an understanding of energy quantization but also bridges the gap between classical and quantum physics, providing a coherent interpretation of light's dual wave-particle nature.