Question

The energy ratio of a photon of wavelength \(3000 \AA\) and \(6000 \AA\) is: (a) \(1: 1\) (b) \(2: 1\) (c) \(1: 2\) (d) \(1: 4\)

Short answer

The energy ratio is (b) 2: 1.

Step-by-step solution

Understand the Photon Energy Formula

The energy of a photon is given by the formula \( E = \frac{hc}{\lambda} \), where \( h \) is Planck's constant \( 6.626 \times 10^{-34} \text{Js} \), \( c \) is the speed of light \( 3 \times 10^8 \text{m/s} \), and \( \lambda \) is the wavelength of the light.

Convert Wavelength Units

The given wavelengths are in angstroms, where \( 1 \text{Å} = 10^{-10} \text{m} \). Thus, convert the given wavelengths:\[ 3000 Å = 3000 \times 10^{-10} \text{m} \]\[ 6000 Å = 6000 \times 10^{-10} \text{m} \]

Calculate Energy for 3000 Å

Using the formula for photon energy,\[ E_1 = \frac{6.626 \times 10^{-34} \times 3 \times 10^8}{3000 \times 10^{-10}} \]Simplifying, we find:\[ E_1 = \frac{6.626 \times 3}{3000} \times 10^{-16} \text{J}\]

Calculate Energy for 6000 Å

Similarly, calculate for the photon with \( 6000 Å \):\[ E_2 = \frac{6.626 \times 10^{-34} \times 3 \times 10^8}{6000 \times 10^{-10}} \]Simplifying this gives:\[ E_2 = \frac{6.626 \times 3}{6000} \times 10^{-16} \text{J}\]

Determine the Energy Ratio

Now compute the ratio \( \frac{E_1}{E_2} \):\[ \frac{E_1}{E_2} = \frac{\frac{6.626 \times 3}{3000}}{\frac{6.626 \times 3}{6000}} \]This simplifies to:\[ \frac{E_1}{E_2} = \frac{6000}{3000} = 2 \]

Interpret Result

The calculated energy ratio is \( 2:1 \), which corresponds to option (b).

Key concepts

Planck's Constant

Planck's constant, denoted as \( h \), is a fundamental constant in quantum mechanics. It describes the scales of quantum effects and is crucial in calculating the energy of photons. This constant has the value \( 6.626 \, \times \, 10^{-34} \) joule seconds (Js). Planck's constant is a pivotal component in the formula \( E = \frac{hc}{\lambda} \), where it relates the energy of a photon to the frequency or wavelength of its electromagnetic radiation.

The impact of Planck's constant reflects how energy and frequency are intertwined. Higher frequency (which means shorter wavelength) results in higher energy, all dictated by this tiny but immensely significant number in the formula. Without Planck's constant, it would not be possible to link the macroscopic properties of light to their microscopic quantum phenomena.

Wavelength Conversion

Conversion of wavelength units is an important task when working with photon energies, especially since wavelengths are often given in angstroms (Å) in various scientific contexts. One angstrom is equal to \( 10^{-10} \) meters. Therefore, when handling calculations involving Planck’s formula \( E = \frac{hc}{\lambda} \), it is crucial to ensure that all units are consistent.

To convert, multiply the given wavelength in angstroms by \( 10^{-10} \). For example, \( 3000 \) Å becomes \( 3000 \times 10^{-10} \) meters. This step is essential to avoid errors and ensures the calculations conform to the standard metric units needed for the constants we use.

Energy Ratio Calculation

Calculating the energy ratio of photons involves comparing their energies when they have different wavelengths. The energy \( E \) of a photon is inversely related to its wavelength \( \lambda \) by the equation \( E = \frac{hc}{\lambda} \). Consequently, a shorter wavelength means higher energy and vice versa.

To determine the ratio between two photon energies with wavelengths of \( 3000 \) Å and \( 6000 \) Å using our converted values, you evaluate \( E \) for each wavelength:

• \( E_1 = \frac{hc}{3000 \times 10^{-10}} \)
• \( E_2 = \frac{hc}{6000 \times 10^{-10}} \)

When the same constants \( hc \) are used, simplifying the ratio \( \frac{E_1}{E_2} \) reduces to \( \frac{6000}{3000} = 2 \), reflecting two times the energy for \( 3000 \) Å compared to \( 6000 \) Å.

Photon Wavelength

Photon wavelength, denoted by \( \lambda \), is the distance between successive crests of a wave. It is a key determinant of a photon's energy. In visible light, shorter wavelengths correlate with blue and violet colors, whereas longer wavelengths are associated with reds and oranges.

The formula \( E = \frac{hc}{\lambda} \) shows that photon energy increases as the wavelength decreases. This relationship allows scientists and students to understand how light's color affects its energy, paving the way for practical applications in technologies like lasers and spectroscopy. Photon wavelength is central to translating electromagnetic waves, such as light, into either scientific inquiry or functional energy solutions.

Speed of Light

The speed of light, symbolized as \( c \), represents one of the most fundamental constants in physics, with a precise value of approximately \( 3 \times 10^8 \) meters per second. Light speed is not just about velocity; it defines the framework for understanding the transmission of electromagnetic energy across space.

In the context of photon energy, the speed of light is used in conjunction with Planck's constant to determine how quickly energy is delivered by photons over their wavelength. This is crucial for calculating the energy in processes that span various fields such as telecommunications, energy, and even the nature of the universe. Understanding this speed also provides insight into how information could be transmitted over large distances instantaneously within the cosmic scale, respecting the ultimate speed limit set by light.