Blackbody Radiation and Planck’s Law

Questions and Answers

What does the area under each curve in black body radiation represent?

The wavelength distribution

The temperature of the black body

The power at a specific frequency

The amount of energy emitted (correct)

According to Stefan’s Law, how does the total power of emitted radiation change with temperature?

It increases with the square root of temperature.

It decreases linearly with temperature.

It remains constant regardless of temperature.

It increases with the fourth power of temperature. (correct)

What does Wien’s Displacement Law state about the peak wavelength distribution?

It depends solely on the type of black body.

It shifts to longer wavelengths with increased temperature.

It remains unchanged with temperature.

It shifts to shorter wavelengths as temperature increases. (correct)

What is the main issue associated with the Rayleigh-Jeans Law?

It demonstrates energy output diverging toward infinity at short wavelengths. Signup and view all the answers

What does Planck's Law modify about the energy of oscillators?

Energy can only have discrete values. Signup and view all the answers

In the context of the black body radiation, which statement is accurate regarding Rayleigh-Jeans model?

The average energy associated with each wavelength is the same. Signup and view all the answers

What is the constant involved in Wien's Displacement Law?

2.898 × 10^-3 m.K Signup and view all the answers

What does the term 'ultraviolet catastrophe' refer to in the context of black body radiation?

The failure of classical physics to explain black body radiation. Signup and view all the answers

What is the relationship between the energy of a photon and its frequency?

Energy is directly proportional to frequency. Signup and view all the answers

In the context of the photoelectric effect, what determines the maximum kinetic energy of photoelectrons?

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

How does the Compton effect differ from classical predictions regarding electron behavior?

Electrons exhibit a range of speeds after scattering. Signup and view all the answers

What is the condition for constructive interference of X-rays when diffracted by crystals?

$2d sin \theta = m \lambda$ Signup and view all the answers

What happens to the wavelength of X-rays when they are scattered by electrons according to the Compton effect?

It increases as the scattering angle increases. Signup and view all the answers

What is the momentum of a photon in a vacuum?

It is equal to the energy divided by the speed of light. Signup and view all the answers

Which of the following statements regarding electromagnetic waves and their interaction with electrons is false?

Different electrons can be accelerated to the same speed. Signup and view all the answers

Which factor contributes to the intensity of light rather than the energy of individual photons?

The amplitude of the light wave. Signup and view all the answers

What does the Heisenberg uncertainty principle indicate about measuring a particle's position and momentum?

It is fundamentally impossible to measure both simultaneously with infinite accuracy. Signup and view all the answers

Which equation relates energy and time in the context of uncertainty?

(ΔE)(Δt) ≥ h / 4π Signup and view all the answers

What did Louis de Broglie contribute to quantum mechanics?

He introduced the concept of wave-particle duality. Signup and view all the answers

What is the surface temperature of the Sun, assuming it emits as a black body?

5750 K Signup and view all the answers

If the stopping potential for photoelectrons from one metal is larger compared to another, what can be inferred about the threshold frequency?

The threshold frequency of the second metal is higher. Signup and view all the answers

What is the value of λmax for the Sun given its surface temperature?

504 nm Signup and view all the answers

What aspect of electromagnetic radiation does Einstein's theory fundamentally address?

The quantization of energy Signup and view all the answers

Which physicist is known for developing the partial differential equation for the wave function of particles?

Schrödinger Signup and view all the answers

What is the wavelength of an electron with a kinetic energy of 3.0 eV?

7.09 x 10–10 m Signup and view all the answers

What scattering angle results in the recoiling electron having kinetic energy equal to the energy of the scattered photon?

70 degrees Signup and view all the answers

Using the uncertainty principle, what is the line width Δf produced by the finite lifetime of an excited atom of 1.0 x 10–8 s?

8.0 x 10^6 Hz Signup and view all the answers

What is the velocity of an electron confined inside an atomic nucleus of diameter 2 x 10–15 m according to the uncertainty principle?

97 c Signup and view all the answers

If an electron and a proton are both confined within the same nucleus, how do their velocities differ?

The proton moves nonrelativistically and the electron must move relativistically Signup and view all the answers

What is the relationship between the momentum of the incident photon and the scattered photon in terms of their wavelengths?

$p' = po - p cos ϕ$ Signup and view all the answers

In the context of Compton scattering, what does the Compton shift equation represent?

The relationship between the wavelengths of the incident and scattered photons Signup and view all the answers

According to the conservation of energy in Compton scattering, what is conserved?

The total energy, including the rest mass energy of the electron Signup and view all the answers

What does the term $K = E - mc^2$ specifically represent?

The kinetic energy of the electron Signup and view all the answers

Which variable in the equation $E = p^2 c^2 + m^2 c^4$ represents the total relativistic energy of an electron?

$E$ Signup and view all the answers

What is the significance of the momentum $p = eta m v$, where $eta = rac{1}{ ext{sqrt}(1 - rac{v^2}{c^2})}$?

It defines relativistic momentum of the electron Signup and view all the answers

In the equations provided, what does the variable $ heta$ represent?

The scattering angle of the photon Signup and view all the answers

What effect does increasing the scattering angle $ heta$ have on the wavelength of the scattered photon?

It increases the wavelength Signup and view all the answers

What is the momentum of the incident photon expressed as?

$rac{hc}{λo}$ Signup and view all the answers

What type of phenomena does light exhibit that confirms its wave nature?

Diffraction and interference Signup and view all the answers

Study Notes

Blackbody Radiation and Planck’s Hypothesis

Black-body radiation refers to electromagnetic radiation emitted by an idealized physical object.
Every object emits electromagnetic waves at any temperature, governed by thermal radiation.
Stefan’s Law indicates that total power (P) emitted increases with temperature:
- Formula: ( P = \sigma A e T^4 )

Key Laws of Radiation

Wien’s Displacement Law states that peak wavelength (( \lambda_m )) is inversely proportional to temperature (T):
- Relationship: ( \lambda_m T = 2.898 \times 10^{-3} , \text{m} \cdot \text{K} )
Rayleigh-Jeans Law describes emitted intensity ( I(\lambda, T) ) for long wavelengths:
- Formula: ( I(\lambda, T) = \frac{2 \pi c k_B T}{\lambda^4} )
- It leads to the ultraviolet catastrophe for short wavelengths.

Planck’s Law

Planck’s Law describes intensity for blackbody radiation:
- Formula: ( I(\lambda,T) = \frac{2 \pi h c^2}{\lambda^5} \cdot \frac{1}{e^{\frac{hc}{\lambda k_BT}} - 1} )
Assumes discrete energy values for oscillators within cavity walls, providing foundational support for quantum mechanics.

Einstein’s Interpretation of Electromagnetic Radiation

Energy packets known as photons carry discrete energy:
- Energy per photon: ( E = hf )
Photon intensity correlates with the number of photons, not energy.
Einstein's photoelectric equation relates max kinetic energy (Kmax) of ejected electrons and incident light frequency:
- Equation: ( K_{max} = hf - \phi )

Compton Effect

Occurs when X-rays scatter off free electrons, resulting in a change in wavelength related to scattering angle.
Classical predictions include oscillation of electrons and radiation pressure.

Diffraction of X-Rays by Crystals

Crystals behave as 3D gratings for X-rays.
Condition for constructive interference: ( 2d \sin \theta = m \lambda )

Conservation Principles

Conservation of energy and momentum apply in scattering events with formulations guiding particle interactions.

Compton Shift

The Compton shift quantifies wavelength change due to scattering:
- Formula: ( \lambda' - \lambda_0 = \frac{h}{m_ec}(1 - \cos \theta) )

Dual Nature of Light

Light demonstrates both wave and particle properties, evidenced by diffraction and interference phenomena.

Heisenberg Uncertainty Principle

It is fundamentally impossible to measure both position and momentum of a particle with absolute precision:
- Relation: ( \Delta x \cdot \Delta p_x \ge \frac{h}{4\pi} )
A corresponding relation exists for energy and time:
- Relation: ( \Delta E \cdot \Delta t \ge \frac{h}{4\pi} )

Historical Timeline

Early 1900s contributions from Planck (quantized energy), Einstein (PE effect), de Broglie (wave-particle duality), Schrödinger (wave function), and Heisenberg (uncertainty principle).

Example Problems

Problems include calculating surface temperature of the Sun, determining work functions, and understanding photon scattering, illustrating applications of principles discussed.

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Description

Test your knowledge on blackbody radiation and Planck's hypothesis with this quiz. Explore key concepts including Stefan's Law, Wien's Displacement Law, and the Rayleigh-Jeans Law. Understand the implications of these laws in modern physics.