Question

Compare the energies of two radiations \(E_{1}\) with wavelength \(800 \mathrm{~nm}\) and \(E_{2}\) with wavelength \(400 \mathrm{~nm}\) (a) \(E_{1}=2 E_{2}\) (b) \(E_{1}=E_{2}\) (c) \(E_{2}=2 E_{1}\) (d) \(E_{2}=-\frac{1}{2} E_{1}\)

Short answer

Correct option is (c) E_{2} = 2 E_{1}.

Step-by-step solution

Understand the Relationship between Energy and Wavelength

The energy (E) of radiation is inversely proportional to its wavelength (λ). This relationship is given by the equation: E = h * c / λ, where h is Planck's constant (6.626 x 10^{-34} J*s), and c is the speed of light in a vacuum (3 x 10^8 m/s).

Calculate the Energy of Radiation E1

Use the energy-wavelength relationship to calculate E1. Substitute the given wavelength λ1 = 800 nm (which is 800 x 10^{-9} meters) into the equation to find E1.

Calculate the Energy of Radiation E2

Similarly, calculate E2 using the given wavelength λ2 = 400 nm (which is 400 x 10^{-9} meters).

Compare the Energies E1 and E2

Once both energies are calculated, compare E1 and E2 to determine the correct relationship between them.

Determine the Correct Option Based on Calculations

Through comparison, identify which of the given options correctly describes the relationship between E1 and E2.

Key concepts

Planck's constant

Planck’s constant, denoted as 'h', is a fundamental constant in physics that has a significant role in quantum mechanics. Its value is approximately \(6.626 \times 10^{-34} \) joule-seconds (J·s). Max Planck first introduced this constant in 1900 while studying blackbody radiation. It holds the key to understanding the discrete nature of energy in quantum systems.

Planck’s constant is the proportionality factor that relates the energy of a photon to its frequency, using the equation \( E = h \times u \), where E is the energy, h is Planck’s constant, and \( u \) (nu) is the frequency of the photon. In essence, the constant shows that energy comes in discrete packets called quanta, rather than being continuous. This quantization principle is crucial in the photoelectric effect and has been confirmed by an abundance of experimental evidence.

Speed of light

The speed of light in a vacuum, symbolized as 'c', is another essential universal physical constant. It is approximately \(3 \times 10^{8}\) meters per second (m/s). This extraordinary speed is the ultimate velocity at which all energy, matter, and information in the universe can travel.

Often in the realm of physics, c is not just a speed - it represents a fundamental barrier for the transfer of information. According to Einstein's theory of relativity, nothing can exceed this speed. The speed of light is critical in the energy-wavelength relationship because it lets us connect the energy of photons with their wavelengths using the formula \(E = \frac{hc}{\lambda}\). This equation unifies Planck’s constant, the speed of light, and the concept of wavelength, bridging the gap between the microscopic quantum world and the large-scale world we observe.

Radiation energy calculation

The calculation of radiation energy refers to determining the amount of energy that is carried by a photon of electromagnetic radiation. The energy of a photon, as seen from the formula \(E = \frac{hc}{\lambda}\), can be calculated by multiplying Planck’s constant (h) by the speed of light (c) and dividing the product by the wavelength (λ) of the photon.

As we understand from the step by step solution, different wavelengths will correspond to different energies due to the inverse relationship between energy and wavelength. This means that shorter wavelengths will have higher energy and longer wavelengths will have lower energy, which is why ultraviolet light is more energetic than infrared light. Thus, by knowing the wavelength of radiation, we can use the energy calculation formula to find out the energy of photons, which is critical in fields ranging from astrophysics to quantum computing.