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Thermal Physics

Planck's Law of Blackbody Radiation

Planck's law describes the spectral radiance of electromagnetic radiation emitted from a blackbody in thermal equilibrium at a definite temperature T. It successfully resolves the ultraviolet catastrophe by postulating the quantization of energy in discrete packets known as photons or quanta.

UP TGT ScienceUP PGT PhysicsStudy Notes + Practice

Scope and prerequisites

TGT core

Qualitative understanding of blackbody radiation, graphical variation of spectral emissive power with wavelength at different temperatures, Wien's displacement law, and Stefan-Boltzmann law as limiting cases.

PGT extension

Derivation of Planck's radiation formula using Bose-Einstein statistics or phase space density of states, derivation of Wien's law and Rayleigh-Jeans law from Planck's law, and calculation of total energy density.

Prerequisites

Concept of Temperature
Degrees of freedom
Law of equipartition of energy

Blackbody Radiation and Spectral Energy Density

A blackbody is an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence.
The spectral energy density u(_lambda;, T) d_lambda; represents the amount of radiant energy per unit volume in the wavelength range between _lambda; and _lambda; + d_lambda; inside an isothermal cavity.
Classical physics (Rayleigh-Jeans law) predicted infinite energy at short wavelengths, an inconsistency known as the ultraviolet catastrophe.

Quantum Hypothesis of Max Planck

Planck assumed that the oscillators comprising the blackbody walls can only emit or absorb energy in discrete packets or quanta of magnitude E = hν.
Here, h is Planck's constant (6.626 × 10^-34 J⋅s) and ν is the frequency of the oscillator.
This revolutionary postulate abandoned the classical continuous energy distribution and laid the foundation of quantum mechanics.

Derivation Framework and Density of Modes

The derivation combines the number of standing electromagnetic wave modes per unit volume in a cavity with the average energy of a quantum oscillator.
The number of standing wave modes per unit volume for frequency interval _nu; to _nu; + d_nu; is given by 8π_nu;²d_nu;/c³.
Using Maxwell-Boltzmann or Bose-Einstein statistics, the average energy of an oscillator at temperature T is given by h_nu; / (exp(h_nu;/k_B T) - 1).

Limiting Cases of Planck's Law

At very long wavelengths (low frequencies), h_nu; << k_B T, Planck's law reduces to the classical Rayleigh-Jeans law: u(_lambda;)d_lambda; = (8πk_B T / _lambda;⁴)d_lambda;.
At short wavelengths (high frequencies), Planck's law reduces to Wien's radiation law.
Integrating Planck's law over all wavelengths yields the Stefan-Boltzmann law, where total emissive power is proportional to T⁴.

Derivations

Derivation of Planck's Radiation Law

Conditions: Isothermal cavity in thermal equilibrium at temperature T

Determine the number of standing electromagnetic modes per unit volume in the frequency range _nu; and _nu; + d_nu; for a three-dimensional cavity: g(_nu;)d_nu; = \frac{8\pi \nu^2}{c^3} d\nu.
Apply the Boltzmann distribution to find the probability of an oscillator having energy $E_n = n h\nu: P(n) = \frac{\exp(-n h\nu / k_B T)}{\sum_{n=0}^{\infty} \exp(-n h\nu / k_B T)}$.
Calculate the average energy of the Planck oscillator: \langle E $\rangle = \frac{\sum_{n=0}^{\infty} n h\nu \exp(-n h\nu / k_B T)}{\sum_{n=0}^{\infty} \exp(-n h\nu / k_B T)} = \frac{h\nu}{\exp(h\nu / k_B T) - 1}$.
Multiply the density of modes by the average energy to obtain the spectral energy density: u(\nu, T)d\nu = g(\nu)\langle E \rangle d\nu.

Result: u(\nu, T)d$\nu = \frac{8\pi h \nu^3}{c^3} \frac{1}{\exp(\frac{h\nu}{k_B T}) - 1} d\nu$

Derivation of Wien's Displacement Law from Planck's Law

Conditions: Finding the wavelength corresponding to maximum spectral radiance

Express Planck's law in terms of wavelength _lambda; by substituting _nu; = c/_lambda;: u(\lambda, T)d$\lambda = \frac{8\pi h c}{\lambda^5} \frac{1}{\exp(\frac{hc}{\lambda k_B T}) - 1} d\lambda$.
Differentiate the spectral energy density with respect to _lambda; and set the derivative equal to zero: \frac{du(\lambda, T)}{d\lambda} = 0.
Let $x = \frac{hc}{\lambda k_B T}$. The differentiation condition simplifies to the transcendental equation (5 - x)$e^x = 5$.
Solve numerically for x ≈ 4.965, yielding \lambda_{max} $T = \frac{hc}{4.965 k_B T} \times T = \text{constant}$.

Result: \lambda_{max} $T = b \approx 2.898 \times 10^{-3} \text{ m\cdot K}$

Formula sheet

Formula / law	Equation	Variables and units	Conditions
Planck's Law (Frequency Form)	$$u(\nu, T)d\nu = \frac{8\pi h \nu^3}{c^3} \frac{1}{\exp(\frac{h\nu}{k_B T}) - 1} d\nu$$	u(_nu;, T) = spectral energy density, h = Planck's constant, _nu; = frequency, c = speed of light, $k_B = Boltzmann constant$, T = absolute temperature J⋅s/m³ or J/m³Hz	Valid for all frequencies at thermal equilibrium
Planck's Law (Wavelength Form)	$$u(\lambda, T)d\lambda = \frac{8\pi h c}{\lambda^5} \frac{1}{\exp(\frac{hc}{\lambda k_B T}) - 1} d\lambda$$	u(_lambda;, T) = spectral energy density per unit wavelength, _lambda; = wavelength J/m⁴	Valid for all wavelengths at thermal equilibrium
Rayleigh-Jeans Law	$$u_R(\lambda, T)d\lambda = \frac{8\pi k_B T}{\lambda^4} d\lambda$$	$u_R = Rayleigh-Jeans energy density$ J/m⁴	Valid only for very long wavelengths (h_nu; << k_B T)
Wien's Radiation Law	$$u_W(\lambda, T)d\lambda = \frac{8\pi h c}{\lambda^5} \exp\left(-\frac{hc}{\lambda k_B T}\right) d\lambda$$	$u_W = Wien's energy density approximation$ J/m⁴	Valid only for very short wavelengths (h_nu; >> k_B T)
Wien's Displacement Law	$$\lambda_{max} T = b$$	_lambda;_max = wavelength of maximum emission, $b = Wien's constant (2.898 \times 10^{-3} m⋅K)$ m⋅K	Applies to the peak of the blackbody spectrum
Stefan-Boltzmann Law	$$E = \sigma T^4$$	E = total emissive power, `sigma; = Stefan-Boltzmann constant (5.67 × 10^-8 W/m²K⁴) W/m²	Integrated over all wavelengths for a perfect blackbody
Average Energy of Planck Oscillator	$$\langle E \rangle = \frac{h\nu}{\exp(\frac{h\nu}{k_B T}) - 1}$$	⟨E$⟩ = mean thermal energy of oscillator of frequency _nu;$ J	Derived using quantum statistics
Planck's Emissive Power Form (Spectral Radiance)	$$E_{\lambda}(\lambda, T) = \frac{c}{4} u(\lambda, T) = \frac{2\pi h c^2}{\lambda^5} \frac{1}{\exp(\frac{hc}{\lambda k_B T}) - 1}$$	E__lambda; = spectral emissive power per unit area per unit wavelength W/m³	Hemispherical emission from blackbody surface

Graphs and experiments

Blackbody Spectral Radiance vs Wavelength

Draw: Curves of spectral energy density (or emissive power) on the y-axis against wavelength on the x-axis for multiple increasing temperatures T_1, T_2, T_3. Show peaks shifting to shorter wavelengths as temperature increases without curves intersecting.

Interpret: Each curve has a distinct maximum at _lambda;_max. The area under each curve represents total energy emitted (proportional to T⁴). As temperature rises, peak shifts to left (shorter wavelength) and height increases dramatically.

Exam use: Extensively tested in TGT and PGT for qualitative shifts, validating Wien's displacement law, and comparing with classical Rayleigh-Jeans failure at small wavelengths.

Comparison of Planck's Law with Classical Approximations

Draw: A single logarithmic or linear plot showing Planck's curve matching experimental data completely, with the Rayleigh-Jeans curve diverging at ultraviolet wavelengths and Wien's curve diverging at infrared wavelengths.

Interpret: Highlights the ultraviolet catastrophe predicted by classical equipartition theorem and demonstrates how Planck's interpolation formula bridges long and short wavelength regimes perfectly.

Exam use: Conceptual PGT questions regarding the limits where classical laws fail or hold true.

Fery's Blackbody Cavity Experiment

Draw: A hollow sphere with a small conical projection inside, a tiny aperture for radiation entry/exit, and double-walled construction for thermal insulation.

Interpret: Any radiation entering through the aperture undergoes multiple internal reflections, with total absorption probability approaching unity, creating an experimental realization of a blackbody.

Exam use: Direct factual questions in TGT/PGT regarding the construction and working principle of artificial blackbodies.

Solved numericals

TGT core

Calculate the wavelength of maximum emission for a star emitting radiation like a blackbody at a temperature of 5800 K. Take Wien's constant $b = 2.898 x 10^-3 m K$.

Identify given values: T = 5800 K, $b = 2.898 \times 10^{-3} \text{ m\cdot K}$.
Apply Wien's displacement law: \lambd$a_{max} = \frac{b}{T}$.
Substitute the values: \lambd$a_{max} = \frac{2.898 \times 10^{-3}}{5800}$.
Calculate the result: \lambd$a_{max} = 4.996 \times 10^{-7} \text{ m} = 499.6 \text{ nm}$.

Answer: 499.6 nm

PGT extension

Using Planck's radiation law, find the ratio of spectral energy densities at wavelength _lambda; = 500 nm for a blackbody at temperatures $T_1 = 3000 K$ and $T_2 = 6000 K in the Wien approximation limit$.

Check validity of Wien's approximation: For _lambda; = 500 nm and T = 3000 K, \frac{hc}{\lambda k_B T} = \frac{(6.63 \times 10^{-34})(3 \times 10^8)}{(500 \times 10^{-9})(1.38 \times 10^{-23})(3000)} \approx 9.58 >> 1, so Wien's law is valid.
Write Wien's approximation ratio: \frac{u(\lambda, T_2)}{u(\lambda, $T_1) = \frac{\lambda^{-5} \exp(-hc / \lambda k_B T_2)}{\lambda^{-5} \exp(-hc / \lambda k_B T_1)} = \exp\left[ \frac{hc}{\lambda k_B} \left(\frac{1}{T_1} - \frac{1}{T_2}\right) \right]$.
Substitute values: exponen$t = 9.58 \times \left(\frac{1}{3000} - \frac{1}{6000}\right) \times 3000 = 9.58 \times 0.5 = 4.79$.
Calculate ratio: \exp(4.79) \approx 120.2.

Answer: 120.2

PGT extension

At what temperature will the total energy density of radiation inside a blackbody cavity be 1.0 J/m^3? (Stefan-Boltzmann constant `sigma; = 5.67 x 10^-8 W/m^2K^4, speed of light $c = 3 x 10^8 m/s)$. Note: Energy density $u = (4/c) \sigma T^4$.

Rearrange the relation for temperature: $T^4 = \frac{u \cdot c}{4 \sigma}$.
Substitute given values: $u = 1.0 \text{ J/m}^3$, $c = 3 \times 10^8 \text{ m/s}$, $\sigma = 5.67 \times 10^{-8} \text{ W/m}^2\text{K}^4$.
$T^4 = \frac{1.0 \times 3 \times 10^8}{4 \times 5.67 \times 10^{-8}} = \frac{3 \times 10^8}{2.268 \times 10^{-7}} = 1.3228 \times 10^{15} \text{ K}^4$.
Take the fourth root: $T = (1.3228 \times 10^{15})^{1/4} \approx 1899 \text{ K}$.

Answer: 1899 K

Assumptions and limitations

Assumption: The cavity walls are made of material that contains harmonic oscillators of all frequencies in thermal equilibrium.
Limitation: Planck's law assumes isotropic and unpolarized radiation inside an ideal vacuum cavity; real surfaces have emissivities less than unity (gray bodies).

Common misconceptions

Common error: Planck's law applies only to visible light emitted by hot objects.

Correct: Planck's law applies to electromagnetic radiation across the entire continuous spectrum, including ultraviolet, visible, infrared, and microwave regions.

A blackbody emits radiation over all wavelengths simultaneously, with the distribution determined entirely by temperature.

Common error: The ultraviolet catastrophe was resolved by introducing higher order wave corrections in classical electrodynamics.

Correct: The ultraviolet catastrophe was resolved by discarding classical continuous energy distribution and introducing discrete energy quanta (h_nu;).

Classical statistical mechanics (equipartition theorem) inherently fails at high frequencies because it assigns equal kT energy to all modes regardless of frequency.

Quick revision

Planck's law gives spectral energy density as u(_nu;) = (8πh_nu;³ / c³) [1 / (exp(h_nu;/k_B T) - 1)].
At long wavelengths, Planck's law reduces to the Rayleigh-Jeans law.
Wien's displacement law states that _lambda;_max T = constant.
Energy quantization (E = h_nu;) successfully eliminated the ultraviolet catastrophe.

Must remember

Planck's quantum hypothesis states that oscillators emit or absorb energy in discrete packets: $E = h\nu$.
The ultraviolet catastrophe in classical physics predicted infinite energy density at short wavelengths.
Planck's law reduces to the Rayleigh-Jeans law at very long wavelengths ($h\nu \ll k_B T$).
Planck's law reduces to Wien's radiation law at very short wavelengths ($h\nu \gg k_B T$).
Wien's displacement law states that $\lambda_{max} T = b$, where $b = 2.898 \times 10^{-3} \text{ m}\cdot\text{K}$.
Stefan-Boltzmann law gives total emissive power as $E = \sigma T^4$.
The average energy of a Planck oscillator is given by $\langle E \rangle = \frac{h\nu}{\exp(\frac{h\nu}{k_B T}) - 1}$.
The number of standing wave modes per unit volume for frequency interval to $\nu + d\nu$ is $\frac{8\pi \nu^2 d\nu}{c^3}$.

Glossary

BlackbodyAn idealized physical body that absorbs all incident electromagnetic radiation regardless of frequency or angle.

Spectral Energy Density ($u(\nu, T)$)Amount of radiant energy per unit volume in the frequency range between $\nu$ and $\nu + d\nu$.

Planck's Constant ($h$)Fundamental constant equal to $6.626 \times 10^{-34} \text{ J}\cdot\text{s}$.

Stefan-Boltzmann Constant ($\sigma$)Constant equal to $5.67 \times 10^{-8} \text{ W/m}^2\text{K}^4$.

Wien's Constant ($b$)Constant equal to $2.898 \times 10^{-3} \text{ m}\cdot\text{K}$.

Comparison of Radiation Laws

Law	Applicability / Condition	Wavelength Dependence
Rayleigh-Jeans Law	Very long wavelengths ($h\nu \ll k_B T$)	$u(\lambda) \propto \lambda^{-4}$
Wien's Radiation Law	Very short wavelengths ($h\nu \gg k_B T$)	$u(\lambda) \propto \lambda^{-5} \exp(-hc/\lambda k_B T)$
Planck's Law	All wavelengths / frequencies	$u(\lambda) = \frac{8\pi hc}{\lambda^5} \frac{1}{\exp(hc/\lambda k_B T) - 1}$

Practice Quiz

Choose an option to reveal the explanation.

1. What is the primary assumption made by Max Planck in deriving his radiation law?

Explanation: Planck introduced the revolutionary quantum hypothesis that cavity oscillators can only emit or absorb energy in discrete packets $E = h\nu$.

2. The ultraviolet catastrophe refers to the failure of which law at short wavelengths?

Explanation: Classical physics via the Rayleigh-Jeans law predicted infinite energy density at short wavelengths, an inconsistency known as the ultraviolet catastrophe.

3. At very long wavelengths, Planck's law reduces to which of the following?

Explanation: When $h\nu \ll k_B T$ (long wavelengths/low frequencies), Planck's law simplifies to the classical Rayleigh-Jeans law.

4. What is the expression for the average energy of a Planck oscillator of frequency $\nu$ at temperature $T$?

Explanation: Using quantum statistics, the average energy of an oscillator is given by $\langle E \rangle = \frac{h\nu}{\exp(\frac{h\nu}{k_B T}) - 1}$.

5. What is the SI unit of spectral energy density $u(\nu, T)$?

Explanation: Spectral energy density per unit frequency has units of energy per unit volume per unit frequency, i.e., $\text{J}/\text{m}^3 \text{Hz}$. Note that $\text{J}\cdot\text{s}/\text{m}^3$ is also dimensionally compatible when considering energy density per unit frequency.

6. According to Wien's displacement law, the product $\lambda_{max} T$ is:

Explanation: Wien's displacement law states that $\lambda_{max} T = b$, where $b$ is Wien's constant ($2.898 \times 10^{-3} \text{ m}\cdot\text{K}$).

7. The number of standing wave modes per unit volume in a frequency interval $\nu$ to $\nu + d\nu$ inside a cavity is given by:

Explanation: The density of modes in a 3D electromagnetic cavity per unit volume is $\frac{8\pi \nu^2 d\nu}{c^3}$.

8. Stefan-Boltzmann law states that the total emissive power of a blackbody is proportional to:

Explanation: Integrating Planck's law over all wavelengths yields $E = \sigma T^4$, showing total emissive power is proportional to $T^4$.

9. In Planck's radiation formula in wavelength form, what is the power of $\lambda$ in the denominator of the pre-exponential factor?

Explanation: The wavelength form of Planck's law contains $\lambda^5$ in the denominator: $u(\lambda, T)d\lambda = \frac{8\pi hc}{\lambda^5} \frac{1}{\exp(\frac{hc}{\lambda k_B T}) - 1} d\lambda$.

10. Wien's radiation law is valid under which of the following conditions?

Explanation: Wien's radiation law applies at very short wavelengths or high frequencies where $h\nu \gg k_B T$.

11. What happens to the peak of the blackbody radiation curve ($\lambda_{max}$) as the temperature of the body increases?

Explanation: By Wien's displacement law ($\lambda_{max} T = b$), as temperature $T$ increases, $\lambda_{max}$ decreases, shifting towards shorter (bluer) wavelengths.

12. Which statistical distribution is fundamentally used to derive the average energy of photons or oscillators in Planck's law derivation?

Explanation: Planck's derivation utilizes Bose-Einstein statistics (or equivalent quantum oscillator partition functions) to determine the average thermal energy.

13. What is the value of Wien's constant $b$ approximately equal to?

Explanation: Wien's constant $b = \lambda_{max} T \approx 2.898 \times 10^{-3} \text{ m}\cdot\text{K}$.

14. The spectral radiance (emissive power per unit area per unit wavelength) $E_\lambda(\lambda, T)$ is related to spectral energy density $u(\lambda, T)$ by:

Explanation: For isotropic radiation inside a cavity, the emissive power from a surface element is $E_\lambda = \frac{c}{4} u(\lambda, T)$.

15. Which of the following correctly describes a blackbody?

Explanation: A blackbody absorbs all incident radiation without reflection and emits radiation perfectly across all wavelengths depending on its temperature.

PYQ Patterns

These are pattern-based practice questions. Authentic year labels are only used when verified from a source.

1. If the absolute temperature of a blackbody is doubled, the total radiant energy emitted per second per unit area becomes how many times the initial value?

Explanation: According to the Stefan-Boltzmann law ($E = \sigma T^4$), if $T$ is doubled ($2T$), emissive power becomes $(2T)^4 = 16T^4$, i.e., 16 times.

2. According to the Rayleigh-Jeans law, the spectral energy density $u(\lambda)$ at a given temperature varies with wavelength $\lambda$ as:

Explanation: The Rayleigh-Jeans law formula is $u_R(\lambda, T)d\lambda = \frac{8\pi k_B T}{\lambda^4} d\lambda$, meaning $u(\lambda) \propto \lambda^{-4}$.

3. The wavelength corresponding to maximum energy emitted by a blackbody at temperature $T_1$ is $\lambda_1$. When the temperature is changed to $T_2$, the peak wavelength is $\lambda_2$. If $T_2 = 2T_1$, then $\lambda_2$ is equal to:

Explanation: By Wien's displacement law ($\lambda_{max} T = b$), $\lambda_{max}$ is inversely proportional to $T$. Thus if $T$ is doubled, $\lambda_{max}$ is halved.

4. Planck's radiation formula successfully resolves the ultraviolet catastrophe because:

Explanation: Quantization ($E = h\nu$) causes the average oscillator energy to drop exponentially at high frequencies, preventing the infinite energy catastrophe predicted by classical equipartition.

5. In the derivation of Planck's law, the product of the density of modes per unit volume and the average energy of an oscillator gives:

Explanation: Spectral energy density $u(\nu, T)d\nu$ is obtained by multiplying the density of electromagnetic modes $\frac{8\pi \nu^2 d\nu}{c^3}$ by the average oscillator energy $\langle E \rangle$.

6. What is the limiting form of Planck's wavelength distribution law at extremely short wavelengths ($hc/\lambda \gg k_B T$)?

Explanation: At very short wavelengths, the $-1$ term in the denominator of Planck's law becomes negligible, reducing it to Wien's radiation law.

Timed Topic Test

Attempt these questions under a ten-minute limit using a separate timer.

1. If the temperature of a blackbody increases from $27^{\circ}\text{C}$ to $327^{\circ}\text{C}$, the ratio of total radiated power per unit area is:

Explanation: $T_1 = 27 + 273 = 300\text{ K}$; $T_2 = 327 + 273 = 600\text{ K}$. Ratio of emissive power $E_2/E_1 = (T_2/T_1)^4 = (600/300)^4 = 2^4 = 16$.

2. Which of the following graphs correctly represents spectral energy density versus wavelength for a blackbody at fixed temperature?

Explanation: The blackbody spectrum rises from zero at $\lambda = 0$, reaches a peak at $\lambda_{max}$, and then asymptotically approaches zero at very long wavelengths.

3. The average energy of a Planck oscillator approaches which classical limit when $h\nu \ll k_B T$?

Explanation: Expanding $\exp(h\nu/k_B T) \approx 1 + h\nu/k_B T$ for small $h\nu/k_B T$ gives $\langle E \rangle \approx \frac{h\nu}{h\nu/k_B T} = k_B T$, which is the classical equipartition result.

4. The dimensions of Planck's constant $h$ are the same as those of:

Explanation: Since $E = h\nu$, $h$ has dimensions of Energy $\times$ Time, which matches the dimensions of angular momentum ($L = I\omega$ or $r \times p$).

5. If Wien's constant $b = 2.898 \times 10^{-3} \text{ m}\cdot\text{K}$, what is the peak emission wavelength for a star operating at surface temperature $5800\text{ K}$?

Explanation: $\lambda_{max} = b / T = \frac{2.898 \times 10^{-3}}{5800} \approx 5.0\times 10^{-7}\text{ m} = 500\text{ nm}$.

6. Why did the Rayleigh-Jeans law fail at high frequencies (short wavelengths)?

Explanation: The classical equipartition theorem assigns $k_B T$ energy to each mode, leading to $\infty$ energy at high frequencies where mode density grows as $\nu^2$.

7. What is the unit of the Stefan-Boltzmann constant $\sigma$?

Explanation: Since $E = \sigma T^4$ where $E$ is emissive power in $\text{W/m}^2$, $\sigma$ has units of $\text{W}/\text{m}^2\text{K}^4$.

8. Which law can be derived by integrating Planck's spectral energy density over all frequencies from zero to infinity?

Explanation: Integrating Planck's law over all frequencies yields total energy density proportional to $T^4$, directly leading to the Stefan-Boltzmann law.

9. If the temperature of a blackbody is increased by $10\%$, the total radiant power emitted increases approximately by:

Explanation: Power $E \propto T^4$. If $T_2 = 1.1 T_1$, then $E_2 / E_1 = (1.1)^4 = 1.4641$, representing a $46.4\%$ increase.

10. Which of the following statements about Planck's constant $h$ is correct in the context of blackbody radiation?

Explanation: Planck's constant $h$ sets the fundamental quantum scale; when $h \to 0$, Planck's law transitions smoothly to classical Rayleigh-Jeans radiation.