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Thermal Physics
Planck's Radiation Formula
Planck's radiation law describes the spectral energy density of electromagnetic radiation emitted by a black body in thermal equilibrium at a definite temperature. It successfully resolves the ultraviolet catastrophe by introducing the quantum hypothesis of energy emission and absorption.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Basic understanding of black body radiation, statement of Planck's formula, limiting cases yielding Rayleigh-Jeans law and Wien's displacement law, and Stefan-Boltzmann law.
PGT extension
Detailed derivation of Planck's formula using photon gas statistics or density of states in a cavity, frequency and wavelength distribution functions, and derivation of Wien's displacement constant.
Prerequisites
Concept of Temperature
Kinetic theory of gases
Law of equipartition of energy
Black Body Cavity and Standing Waves
A black body cavity (Hohlraum) maintains thermal equilibrium through continuous absorption and emission of radiation by the walls.
Radiation inside the cavity forms standing electromagnetic waves satisfying boundary conditions at the walls.
The number of independent modes of vibration per unit volume in the frequency range ν to ν + dν is given by g(ν)d$\nu = (8\pi\nu^{2}/c^{3})d\nu$.
Planck's Quantum Hypothesis
Electromagnetic energy is not emitted or absorbed continuously, but in discrete packets called quanta or photons.
The energy of each oscillator or quantum is restricted to discrete values given by $E = h\nu$, where h is Planck's constant.
Average energy of an oscillator at absolute temperature T is determined using Maxwell-Boltzmann statistics as = hν / (exp(hν/kT) - 1).
Wien's Distribution Law as a Limit
At very high frequencies or low temperatures where hν >> kT, the exponential term exp(hν/kT) >> 1.
Planck's formula reduces to Wien's radiation formula: U(ν)d$\nu = (8\pih\nu^{3}/c^{3}) exp(-h\nu/kT) d\nu$.
Wien's law accurately predicts spectral radiance in the short-wavelength (ultraviolet) region.
Rayleigh-Jeans Law as a Limit
At very low frequencies or high temperatures where hν << kT, the exponential term can be expanded as exp(hν/kT) ≈ 1 + hν/kT.
Planck's formula reduces to the classical Rayleigh-Jeans law: U(ν)d$\nu = (8\pi\nu^{2}kT/c^{3})d\nu$.
The Rayleigh-Jeans law fails at high frequencies, predicting infinite energy density, known as the ultraviolet catastrophe.
Derivations
Derivation of Planck's Radiation Formula
Conditions: Isotropic black body radiation inside a cubic cavity of volume V in thermal equilibrium at temperature T
Determine the number of electromagnetic cavity modes per unit volume in frequency interval ν to ν + dν as N(ν)d$\nu = \frac{8pi\nu^2}{c^3} d\nu$.
Apply the Maxwell-Boltzmann distribution to find the probability of an oscillator having energy $E_n = nh\nu$, giving the mean energy = \frac{su$m_{n=0}^{\infty} (nh\nu) e^{-nh\nu/kT}{\sum_{n=0}^{\infty} e^{-nh\nu/kT}}$.
Evaluate the summation to obtain the mean oscillator energy = \frac{h\nu}{e^{h\nu/kT} - 1}.
Multiply the density of modes by the average energy to obtain the spectral energy density U(\nu)d\nu = N(\nu)langle E \rangle d\nu.
Result: U(\nu)d$\nu = \frac{8pi h \nu^3}{c^3} \frac{1}{e^{h\nu/kT} - 1} d\nu$
Wavelength Form of Planck's Law
Conditions: Transformation of spectral energy density from frequency domain to wavelength domain using $\nu = c/\lambda$
Express frequency in terms of wavelength: \nu = \frac{c}{lambda}, hence d$\nu = -\frac{c}{\lambda^2} dlambda (ignoring negative sign$ for magnitude).
Substitute ν and dν into the frequency distribution formula U(ν)dν.
Simplify the resulting expression by collecting powers of λ and c.
$$U(\nu)d\nu = \frac{8pi h \nu^3}{c^3} \frac{1}{e^{h\nu/kT} - 1} d\nu$$
U(ν) = energy density, h = Planck's constant, $\nu = frequency$, c = speed of light, k = Boltzmann constant, T = absolute temperature J m³ s (for U(ν)dν)
Valid for all frequencies and temperatures
Planck's Law (Wavelength Form)
$$U(\lambda)dlambda = \frac{8pi hc}{\lambda^5} \frac{1}{e^{hc/\lambda k T} - 1} dlambda$$
$\lambda = wavelength$, h = Planck's constant, c = speed of light, k = Boltzmann constant, T = absolute temperature J m&sup-3; (for U(λ)dλ)
Valid for all wavelengths and temperatures
Density of States (Frequency)
$$g(\nu)d\nu = \frac{8pi\nu^2}{c^3} d\nu$$
g(ν) = number of modes per unit volume, $\nu = frequency$, c = speed of light m&sup-3; s
Standing waves in a 3D rectangular cavity
Average Energy of Quantum Oscillator
$$\langle E \rangle = \frac{h\nu}{e^{h\nu/kT} - 1}$$
⟨E$⟩ = average energy$, h = Planck's constant, $\nu = frequency$, k = Boltzmann constant, T = absolute temperature J
Thermal equilibrium at temperature T
Rayleigh-Jeans Law
$$U_RJ(\nu)d\nu = \frac{8pi\nu^2 kT}{c^3} d\nu$$
$U_RJ = classical energy density$, k = Boltzmann constant, T = temperature J m&sup-3;
$U_W = energy density$, h = Planck's constant, $\nu = frequency$ J m&sup-3;
hν >> kT (High frequency limit)
Wien's Displacement Law
$$\lambda_{max} T = b$$
λmax = wavelength of maximum emission, T = temperature, $b = Wien's constant (\approx 2.898 \times 10^{-3} m K)$ m K
Derived by setting dU/d$\lambda = 0 from Planck's formula$
Stefan-Boltzmann Law from Planck's Law
$$E = \sigma T^4$$
E = total emissive power, $\sigma = Stefan-Boltzmann constant$, T = temperature W m&sup-2;
Integration of Planck's formula over all frequencies from 0 to ∞
Graphs and experiments
Black Body Spectral Radiance Curves
Draw: Curves of spectral energy density U(λ) versus wavelength λ for different absolute temperatures T, showing a distinct peak shifting toward shorter wavelengths as temperature increases.
Interpret: Each curve peaks at a specific wavelength λmax governed by Wien's displacement law. The area under each curve increases steeply with temperature (T&sup4;). At long wavelengths, all curves converge, matching the Rayleigh-Jeans prediction.
Exam use: Frequently asked in conceptual questions regarding the shift of peak wavelength with temperature and the failure of classical physics at short wavelengths.
Comparison of Planck's Law with Classical Laws
Draw: A plot of U(λ) vs λ showing experimental data points fitting Planck's curve perfectly, while the Rayleigh-Jeans curve diverges to infinity as λ approaches zero.
Interpret: Demonstrates the ultraviolet catastrophe resolved by Planck's quantum assumption, proving classical equipartition fails for high-frequency modes.
Exam use: Tested via MCQ matching where students identify why Rayleigh-Jeans law fails at short wavelengths.
Fery's Black Body Furnace
Draw: Schematic diagram of a double-walled hollow spherical enclosure with a conical projection opposite to the aperture and a small pinhole.
Interpret: Radiation entering the pinhole undergoes multiple internal reflections with negligible chance of escaping, making it an almost perfect black body absorber.
Exam use: Standard apparatus question in viva voce and descriptive exam sections for experimental realization of black bodies.
Solved numericals
TGT core
Calculate the energy of a photon of frequency 5 × 10¹&sup4; Hz. (Given $h = 6.63 \times 10^{-34} J s)$
Use Planck's energy relation: E = h\nu.
Substitute the given values: $E = (6.63 \times 10^{-34} \text{ J s}) \times (5 \times 10^{14} \text{ Hz})$.
Using Wien's displacement law, find the temperature of a stellar surface if its maximum spectral emission occurs at wavelength λmax = 500 nm. (Take $b = 2.898 \times 10^{-3} m K)$
State Wien's displacement law: lambda_{max} T = b.
Rearrange to solve for temperature: $T = \frac{b}{\lambda_{max}}$.
Assumption: Radiation inside the cavity consists of standing electromagnetic waves interacting with atomic oscillators whose energies are quantized as $E = nh\nu$. Limitation: The derivation assumes thermal equilibrium and ideal cavity walls, which is an idealized model approximated by practical black bodies.
Common misconceptions
Common error: Planck's formula predicts infinite energy at high frequencies.
Correct: Planck's formula drops exponentially to zero at high frequencies.
The denominator factor exp(hν/kT) - 1 grows exponentially as frequency increases, suppressing high-frequency emission and avoiding the ultraviolet catastrophe.
Common error: Wien's law is valid for all wavelengths.
Correct: Wien's law is only a valid approximation at short wavelengths (high frequencies).
At long wavelengths, the exponential term can no longer dominate, causing Wien's law to deviate significantly from experimental observations.
Quick revision
Planck's radiation law unifies Wien's law and Rayleigh-Jeans law through the quantum hypothesis $E = h\nu$.
Planck's Law (Frequency Form): U(\nu)d$\nu = \frac{8pi h \nu^3}{c^3} \frac{1}{e^{h\nu/kT} - 1} d\nu$
Planck's Law (Wavelength Form): U(lambda)dlambd$a = \frac{8pi hc}{\lambda^5} \frac{1}{e^{hc/\lambda k T} - 1} dlambda$
Must remember
Planck's radiation formula gives the energy density inside a black body cavity as $U(\nu)d\nu = \frac{8\pi h \nu^3}{c^3} \frac{1}{e^{h\nu/kT} - 1} d\nu$.
The number of independent modes of vibration per unit volume in the frequency range $\nu$ to $\nu + d\nu$ is $g(\nu)d\nu = \frac{8\pi\nu^2}{c^3}d\nu$.
The average energy of a quantum oscillator at absolute temperature $T$ is given by $\langle E \rangle = \frac{h\nu}{e^{h\nu/kT} - 1}$.
Rayleigh-Jeans law is the low-frequency limit ($h\nu \ll kT$) of Planck's formula and leads to the ultraviolet catastrophe.
Wien's distribution law is the high-frequency limit ($h\nu \gg kT$) of Planck's formula, valid in the ultraviolet region.
Planck's quantum hypothesis states that electromagnetic energy is emitted or absorbed in discrete packets called quanta, $E = h\nu$.
Glossary
HohlraumAn isothermal black body cavity used in laboratory settings to study thermal radiation.
Ultraviolet CatastropheThe classical prediction that total energy radiated by a black body at short wavelengths approaches infinity.
Planck's Constant ($h$)Fundamental physical constant relating photon energy to its frequency, approximately $6.626 \times 10^{-34} \text{ J s}$.
Energy Density ($U(\nu)$)Energy of radiation per unit volume per unit frequency interval.
Density of States ($g(\nu)$)The number of standing electromagnetic wave modes per unit volume per unit frequency interval.
Predicts infinite energy (Ultraviolet Catastrophe)
Predicts exponential decay to zero
Practice Quiz
Choose an option to reveal the explanation.
1. What is the correct expression for the number of modes of vibration per unit volume in a frequency range $\nu$ to $\nu + d\nu$ inside a 3D black body cavity?
Explanation: The density of states for standing electromagnetic waves inside a 3D rectangular cavity taking polarization into account is given by $g(\nu)d\nu = \frac{8\pi\nu^2}{c^3}d\nu$.
2. According to Planck's quantum hypothesis, the energy of each oscillator is restricted to discrete values given by:
Explanation: Planck's quantum hypothesis states that energy is quantized in discrete packets where the energy of each oscillator is $E = h\nu$.
3. What is the average energy $\langle E \rangle$ of a quantum oscillator in thermal equilibrium at absolute temperature $T$?
Explanation: Using Maxwell-Boltzmann statistics, the average energy of a quantum oscillator is $\langle E \rangle = \frac{h\nu}{e^{h\nu/kT} - 1}$.
4. Under which condition does Planck's radiation formula reduce to the classical Rayleigh-Jeans law?
Explanation: At very low frequencies or high temperatures where $h\nu \ll kT$, the exponential term can be expanded as $e^{h\nu/kT} \approx 1 + \frac{h\nu}{kT}$, reducing Planck's formula to the Rayleigh-Jeans law.
5. Under which condition does Planck's radiation formula reduce to Wien's radiation law?
Explanation: At very high frequencies or low temperatures where $h\nu \gg kT$, the exponential term $e^{h\nu/kT} \gg 1$, leading to Wien's law: $U(\nu)d\nu = \frac{8\pi h\nu^3}{c^3}e^{-h\nu/kT}d\nu$.
6. What is the primary failure of the classical Rayleigh-Jeans law known as?
Explanation: The Rayleigh-Jeans law predicts infinite energy density at high frequencies (short wavelengths), a failure known as the ultraviolet catastrophe.
7. The dimension/unit of energy density $U(\nu)d\nu$ is:
Explanation: Energy density represents energy per unit volume, which has SI units of Joules per cubic meter ($\text{J m}^{-3}$).
8. Planck's radiation formula in terms of wavelength $\lambda$ contains a denominator factor proportional to:
Explanation: The wavelength form of Planck's law is $U(\lambda)d\lambda = \frac{8\pi hc}{\lambda^5} \frac{1}{e^{hc/\lambda kT} - 1} d\lambda$, containing $\lambda^5$ in the denominator.
9. What happens to Planck's formula at extremely high frequencies where $\nu \to \infty$?
Explanation: Due to the exponential term in the denominator of Planck's formula, the energy density drops rapidly to zero at high frequencies, avoiding the ultraviolet catastrophe.
10. Which statistical distribution is used to determine the average energy of oscillators in Planck's derivation?
Explanation: Planck utilized Maxwell-Boltzmann statistics to calculate the average energy of the discrete energy states of oscillators.
11. Wien's displacement law can be derived from Planck's radiation formula by:
Explanation: Wien's displacement law ($\lambda_{\max} T = b$) is derived by differentiating the wavelength form of Planck's formula with respect to $\lambda$ and setting $\frac{dU}{d\lambda} = 0$.
12. Integration of Planck's radiation formula over all frequencies from $0$ to $\infty$ yields which famous law?
Explanation: Integrating Planck's energy density over all frequencies gives the total emissive power $E = \sigma T^4$, which is the Stefan-Boltzmann law.
13. In Planck's law, what does the constant $h$ represent?
Explanation: The symbol $h$ represents Planck's fundamental action constant.
14. A black body cavity is also known as:
Explanation: A black body cavity is termed a Hohlraum (German for 'hollow space').
15. Why did classical physics fail to explain black body radiation across all wavelengths?
Explanation: Equipartition theorem in classical physics assigned $kT$ to every standing wave mode independently of frequency, leading to infinite energy at high frequencies.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. The spectral energy density of black body radiation at very low frequencies is proportional to:
Explanation: At low frequencies, Planck's law reduces to the Rayleigh-Jeans law: $U(\nu)d\nu \propto \nu^2 d\nu$.
2. In the high-frequency limit, Planck's radiation formula transforms into:
Explanation: When $h\nu \gg kT$, Planck's formula simplifies to Wien's radiation law.
3. The average energy of a Planck oscillator at temperature $T$ for high temperatures ($kT \gg h\nu$) approaches:
Explanation: For $kT \gg h\nu$, expanding the denominator $e^{h\nu/kT} - 1 \approx \frac{h\nu}{kT}$, giving $\langle E \rangle \approx kT$.
4. What is the dimensionality of the density of states $g(\nu)d\nu$ in a 3D cavity in terms of units?
Explanation: Density of states per unit volume per unit frequency has units of $\text{m}^{-3} / (\text{s}^{-1}) = \text{m}^{-3} \text{ s}$.
5. Which of the following laws correctly explains the entire black body spectrum from zero to infinite frequency?
Explanation: Planck's Radiation Law is valid across all frequencies and bridges the Rayleigh-Jeans and Wien regimes.
6. The product $\lambda_{\max} T$ in Wien's displacement law is:
Explanation: $\lambda_{\max} T = b$ where $b$ is Wien's displacement constant, which is a universal constant.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. If the absolute temperature of a black body is doubled, the peak wavelength $\lambda_{\max}$ corresponding to maximum emission will:
Explanation: According to Wien's displacement law $\lambda_{\max} T = b$, $\lambda_{\max}$ is inversely proportional to $T$. If $T$ is doubled, $\lambda_{\max}$ is halved.
2. The spectral energy density $U(\nu)$ according to Planck's formula has a maximum at a frequency proportional to:
Explanation: By differentiating Planck's frequency distribution with respect to $\nu$ and setting it to zero, the peak frequency $\nu_{\max}$ is directly proportional to temperature $T$.
3. In the derivation of Planck's law, the number of standing wave modes $g(\nu)d\nu$ inside a cavity of volume $V$ depends on frequency as:
Explanation: The density of states in frequency for a 3D cavity is proportional to $\nu^2$ ($g(\nu)d\nu = \frac{8\pi\nu^2}{c^3}d\nu$).
4. Calculate the average energy of a quantum oscillator when $h\nu \gg kT$.
Explanation: When $h\nu \gg kT$, $\langle E \rangle = \frac{h\nu}{e^{h\nu/kT} - 1} \approx h\nu e^{-h\nu/kT}$.
5. Why does the Rayleigh-Jeans formula fail at short wavelengths?
Explanation: Classical equipartition assigns $kT$ energy to each vibrational mode. Since high-frequency modes are infinite in number, the integrated energy diverges.
6. Which of the following expressions represents Planck's law in frequency form?
Explanation: Option A is the correct frequency form of Planck's radiation formula.
7. The Stefan-Boltzmann constant $\sigma$ can be expressed in fundamental physical constants by integrating Planck's formula. Its temperature dependence in total emissive power is:
Explanation: Total emissive power $E = \sigma T^4$, proving a fourth-power temperature dependence.
8. In the limit $h \to 0$ (classical limit), Planck's distribution formula transforms into:
Explanation: Taking the limit as $h \to 0$ in Planck's formula gives $e^{h\nu/kT} - 1 \approx \frac{h\nu}{kT}$, which directly yields the classical Rayleigh-Jeans law.
9. What is the physical significance of the denominator term $(e^{h\nu/kT} - 1)$ in Planck's formula?
Explanation: The term $(e^{h\nu/kT} - 1)^{-1}$ arises naturally from the statistical mechanics of indistinguishable bosons (photons) or quantized oscillators.
10. If a black body's temperature is increased from $T$ to $2T$, the total radiant energy emitted per second per unit area increases by a factor of:
Explanation: By Stefan-Boltzmann law ($E = \sigma T^4$), if $T$ becomes $2T$, emissive power becomes $(2T)^4 = 16T^4$, an increase by a factor of 16.