Physics Chapter: Wien's and Stefan-Boltzmann Laws

Questions and Answers

What does the Stefan-Boltzmann law describe?

The proportionality constant for the law is variable.

The energy radiated per unit surface area increases with the square of temperature.

The total energy radiated per unit surface area is proportional to the fourth power of absolute temperature. (correct)

The total energy radiated by a blackbody is constant.

What is the value of the Stefan constant (σ)?

5.67 × 10−8 J/(s m² K⁴) (correct)

2.05 × 10−8 J/(s m² K⁴)

7.35 × 10−8 J/(s m² K⁴)

3.14 × 10−8 J/(s m² K⁴)

In the context of blackbody radiation, how is energy density related to intensity?

Energy density equals intensity multiplied by 4.

Energy density is directly proportional to the intensity.

The intensity is the square of the energy density.

Intensity is calculated by dividing energy density by 4. (correct)

Which equation correctly represents the total energy radiated by a blackbody?

E = ∫ Eλ dλ from 0 to ∞ Signup and view all the answers

What does the variable x represent in the context provided?

The ratio of the temperature to the wavelength. Signup and view all the answers

What does the equation $e^x (5 - x) = 5$ represent in this context?

A transcendental equation that cannot be solved analytically Signup and view all the answers

Which method is used to solve the transcendental equation in this context?

Graphical solution by plotting functions Signup and view all the answers

What is the value of $x$ at the point of intersection of the two curves?

4.96 Signup and view all the answers

What does Wien’s displacement law describe in this context?

The maximum wavelength of radiation emitted by a black body Signup and view all the answers

Which of the following is true about the equation $h c / (K_B x) = constant$?

It relates energy to the maximum wavelength and temperature Signup and view all the answers

In the expression $d ext{z}/d ext{λ} = 0$ for $ ext{λ} = ext{λ}_m$, what does this imply?

The maximum wavelength occurs at that value of $ ext{λ}$ Signup and view all the answers

What graphical functions are plotted to find the solution for $x$?

$y = 1 - e^{-x}$ and $y = x/5$ Signup and view all the answers

What occurs as $λ$ approaches $λ_m$ according to the equation provided?

The derivative of the function equals zero Signup and view all the answers

What relationship does the photoelectric current have with the intensity of the incident radiation?

It is directly proportional to the intensity. Signup and view all the answers

What phenomenon describes the emission of electrons from a metal surface when exposed to light?

Photoelectric effect Signup and view all the answers

What determines the maximum speed of the emitted photoelectron in the photoelectric effect?

The frequency of the incident light. Signup and view all the answers

What is the work function of a material in the context of the photoelectric effect?

The minimum energy needed to emit photoelectrons. Signup and view all the answers

What does quantum theory suggest about the emission of radiation from a black body?

Radiation is emitted only in discrete packets called quanta. Signup and view all the answers

Which law describes the distribution of radiation emitted by a black body?

Planck's law of blackbody radiation Signup and view all the answers

Which equation represents the derived form of Stefan-Boltzmann's law from Planck’s law?

E = 15h^3c^2 / (2π^4K_B^4T^4) Signup and view all the answers

What is the limitation of classical theory in explaining black body radiation?

It predicts extreme energy emission at shorter wavelengths. Signup and view all the answers

In the derivation of Stefan-Boltzmann's law, what constant is denoted by σ?

Stefan's constant. Signup and view all the answers

At what condition will a metal surface emit electrons in the photoelectric effect?

When the light's frequency exceeds the threshold frequency. Signup and view all the answers

Which of the following statements about photoelectrons is true?

Electromagnetic radiation of appropriate frequency can cause their emission. Signup and view all the answers

How does the number of photoelectrons emitted relate to the frequency of the incident light?

It is independent of frequency. Signup and view all the answers

What range of wavelengths does ultraviolet light occupy in relation to the electromagnetic spectrum?

Shorter than visible light. Signup and view all the answers

Who proposed the theory that radiation is emitted in discrete packets of energy?

Max Planck Signup and view all the answers

What value represents the Stefan constant (σ) derived from components of Planck’s law?

5.67 × 10^−8 J/(sm^2K^4) Signup and view all the answers

What is the term for the frequency below which no electrons are emitted in the photoelectric effect?

Threshold frequency Signup and view all the answers

What does the limit of $u_{\lambda} d\lambda$ approach as $\lambda$ tends to infinity?

$8\pi K_{B} T$ Signup and view all the answers

According to Wien's displacement law, what happens to the peak of the radiation curve when the temperature increases?

It moves to shorter wavelengths Signup and view all the answers

What is considered constant in the expression for the energy density $u_{\lambda}$ when deducing Wien's displacement law?

The numerator in Planck’s law Signup and view all the answers

What is the relationship given by Planck's law for the energy density of radiation maximizing?

Maximum when denominator is minimum Signup and view all the answers

What can be deduced from the expression when higher-order terms are neglected while using the exponential function?

Only the first term is significant Signup and view all the answers

How is the product of the maximum wavelength and temperature characterized in Wien's displacement law?

It remains constant Signup and view all the answers

What is the approximation made for $exp(\lambda K_{B} T)$ when $\lambda$ is small?

Approximate it to a linear term Signup and view all the answers

Which of these equations is derived from the limit discussed in Planck's law?

$u_{\lambda} = \frac{8\pi h c}{\lambda^5( exp( \frac{h c}{\lambda K_{B} T})-1)}$ Signup and view all the answers

Study Notes

Wien's Displacement Law

The denominator in the equation for blackbody radiation is expressed as ( \frac{hc}{\lambda^5 (e^{\frac{hc}{\lambda k_B T}} - 1)} ).
At the peak wavelength ( \lambda_m ), the derivative ( \frac{dz}{d\lambda} = 0 ) leads to a transcendental equation.
The solution is derived graphically where intersections yield ( x \approx 4.96 ) when solving ( e^x (5 - x) = 5 ).
Consequently, Wien’s displacement law establishes ( \lambda_m T = \text{constant} ), specifically ( \lambda_m T = 0.0029 , \text{m K} ).

Stefan-Boltzmann Law

The total energy radiated by a blackbody per unit surface area is proportional to the fourth power of its absolute temperature ( T ).
Proportionality constant ( \sigma ) (Stefan-Boltzmann constant) is ( 5.67 \times 10^{-8} , \text{J/(s m}^2 \text{K}^4) ).
Total emitted energy ( E ) across all wavelengths is given by ( E = \int_0^\infty E_\lambda d\lambda ).
The relationship between energy density and intensity is defined by ( E_\lambda = \frac{u_\lambda}{4} ).
Deriving ( E ) involves integrals related to Planck's law which show ( E = \sigma T^4 ).

Photoelectric Effect

Electrons are emitted from a metal surface when illuminated with electromagnetic radiation exceeding a threshold frequency, termed photoelectrons.
Photoelectric current is proportional to radiation intensity but independent of frequency.
Maximum speed of emitted photoelectrons depends solely on the frequency of the incident light.
Each material requires a minimum energy (work function ( \phi_0 )) to emit photoelectrons, known as the ultraviolet catastrophe, as classical models failed to account for experimental blackbody radiation results.
Planck's theory of energy quanta led to the derivation of a correct distribution curve for blackbody radiation.

Key Characteristics of Photoemission

The number of emitted photoelectrons correlates with the incident radiation's intensity.
The maximum kinetic energy of emitted electrons is frequency-dependent.
Each material has a work function necessary for photoemission, representing a significant deviation from classical radiation theories.

Planck's Law of Blackbody Radiation

Planck's law describes the spectral distribution of electromagnetic radiation emitted by a blackbody, resolving errors of classical physics.
The law relies on energy being emitted in quantized packets, aligning theoretical predictions with experimental findings.
The equation effectively outlines the distribution of radiation correlating to various wavelengths and temperatures.

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Description

Test your understanding of Wien's Displacement Law and the Stefan-Boltzmann Law in this quiz. Dive into the mathematics of blackbody radiation and explore the relationships between temperature and wavelength, as well as the total energy radiated by blackbodies. Perfect for physics students looking to reinforce their knowledge.