Question

Equation \((3-1)\) expresses Planck's spectral energy density as an energy per range \(d f\) of frequencies. Quite of ten, it is more convenient to express it as an energy per range \(d \lambda\) of wavelengths, By differentiating \(f=c / \lambda,\) we find that df \(=-c / \lambda^{2} d \lambda\). Ignoring the minus sign (we are interested only in relating the magnitudes of the ranges \(d f\) and \(d \lambda\) ). show that, in terms of wavelength. Planck's formula is \(\frac{d U}{d \lambda}=\frac{8 \pi V h c}{e^{h c / \lambda k_{B} T}-1} \frac{1}{\lambda^{5}}\)

Short answer

\(\frac{dU}{d \lambda} = \frac{8 \pi V h c}{e^{hc / \lambda k_{B}T}-1} \frac{1}{\lambda^5}\). The rule of negative signs when switching \(df\) to \(d\lambda\) has been settled by the absolute value, which reflects the physical significance of the magnitudes of the ranges. It took advantage of the differentiation of the reciprocal function to obtain \(\lambda^5\) in the denominator.

Step-by-step solution

Start with the given formula

Planck's spectral energy density is given as \(dU = \frac{8\pi V h f^3}{e^{hf/k_{B}T}-1} df\)

Equate and replace

Substitute the given values and write the conversion in terms of frequency \(f\), \(df = -c / \lambda^{2} d \lambda\) to get \(dU = \frac{8 \pi V h (c / \lambda)^3}{e^{hc / \lambda k_{B}T}-1} (-c / \lambda^{2} d \lambda)\)

Simplify the Equation

After simplifying the above equation, it is observed that \(dU = \frac{8 \pi V h c^4}{\lambda^5 (e^{hc / \lambda k_{B}T}-1)} d\lambda\)

Finalize the Equation

Rearrange the equation into the standard form of Planck's formula, which gives us \(\frac{dU}{d \lambda} = \frac{8 \pi V h c}{e^{hc / \lambda k_{B}T}-1} \frac{1}{\lambda^5}\)

Key concepts

Spectral Energy Distribution

The concept of spectral energy distribution (SED) is fundamental for understanding how energy is emitted across different wavelengths or frequencies by various objects in physics and astronomy. An object's SED is a representation of the amount of energy it emits at each wavelength or frequency and is crucial for examining the properties of light-emitting sources, such as stars or black bodies.

An SED can be plotted with energy density on the y-axis and frequency or wavelength on the x-axis. This graph provides a visual insight into which part of the electromagnetic spectrum the object is most radiantly efficient. By studying an SED, scientists can infer the temperature, composition, and motion of the object in question. When it comes to blackbody radiation, Planck's spectral energy density formula is often used to calculate the energy emitted per unit frequency or wavelength interval by a perfect blackbody.

Frequency to Wavelength Conversion

Frequency and wavelength are two essential characteristics of waves, including electromagnetic radiation. The frequency (\f) of a wave refers to the number of wave cycles that pass a point per unit of time, typically measured in hertz (Hz), whereas the wavelength (\f\begin{small}\(\backslash\)lambda\f\begin{small}) is the distance between successive wave peaks, measured in meters (m) for electromagnetic waves.

The relationship between frequency and wavelength is inversely proportional and is dictated by the equation \f\begin{small}f = c / \begin{small}\(\backslash\)lambda\begin{small}\f\begin{small}, where \begin{small}c\f\begin{small} is the speed of light. This relationship is crucial in converting Planck's spectral energy density from a frequency basis to a wavelength basis. Understanding this conversion allows for the comparison of measurements taken in different units and aids in computational simplification across various physics fields, including optics and quantum mechanics.

Blackbody Radiation

Blackbody radiation is a cornerstone concept in thermodynamics and quantum mechanics. A blackbody is an idealized physical body that perfectly absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. Consequently, it also emits radiation with maximum efficiency at every frequency.

The spectral energy density of a blackbody is described by Planck's law, which provides the amount of electromagnetic radiation emitted by a blackbody in thermal equilibrium at a given temperature, per unit area, per unit solid angle, and within a given frequency. This emission is characterized by the blackbody's temperature, and the SED peaks at a wavelength inversely related to the temperature (Wien's Displacement Law). The quantization of energy and the concept of photons are integral to the explanation of blackbody radiation, leading to the development of quantum mechanics.

Quantum Mechanics

Quantum mechanics is the branch of physics that deals with the behaviors and interactions of particles at microscopic scales, where the classical laws of physics are no longer sufficient. Max Planck's introduction of the concept of quantization of energy is seen as the birth of quantum mechanics. He proposed that energy is emitted or absorbed in discrete units called quanta.

Planck's spectral energy density formula is a groundbreaking application of quantum mechanics. It incorporates the idea that electromagnetic waves can exhibit particle-like properties, leading to the conclusion that light can be absorbed or emitted in quantized packets (photons). This quantization resolved the classical ultraviolet catastrophe problem by correctly describing the emitted radiation from black bodies. Quantum mechanics radically changed our understanding of fundamental processes at the smallest scales, profoundly affecting fields such as chemistry, material science, and electronics.