Blackbody Radiation and Planck’s Law
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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?

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

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

    <p>Energy can only have discrete values.</p> Signup and view all the answers

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

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

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

    <p>2.898 × 10^-3 m.K</p> Signup and view all the answers

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

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

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

    <p>Energy is directly proportional to frequency.</p> Signup and view all the answers

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

    <p>The frequency of the incident light.</p> Signup and view all the answers

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

    <p>Electrons exhibit a range of speeds after scattering.</p> Signup and view all the answers

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

    <p>$2d sin \theta = m \lambda$</p> 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?

    <p>It increases as the scattering angle increases.</p> Signup and view all the answers

    What is the momentum of a photon in a vacuum?

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

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

    <p>Different electrons can be accelerated to the same speed.</p> Signup and view all the answers

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

    <p>The amplitude of the light wave.</p> Signup and view all the answers

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

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

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

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

    What did Louis de Broglie contribute to quantum mechanics?

    <p>He introduced the concept of wave-particle duality.</p> Signup and view all the answers

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

    <p>5750 K</p> 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?

    <p>The threshold frequency of the second metal is higher.</p> Signup and view all the answers

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

    <p>504 nm</p> Signup and view all the answers

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

    <p>The quantization of energy</p> Signup and view all the answers

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

    <p>Schrödinger</p> Signup and view all the answers

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

    <p>7.09 x 10–10 m</p> 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?

    <p>70 degrees</p> 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?

    <p>8.0 x 10^6 Hz</p> 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?

    <p>97 c</p> Signup and view all the answers

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

    <p>The proton moves nonrelativistically and the electron must move relativistically</p> 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>$p' = po - p cos ϕ$</p> Signup and view all the answers

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

    <p>The relationship between the wavelengths of the incident and scattered photons</p> Signup and view all the answers

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

    <p>The total energy, including the rest mass energy of the electron</p> Signup and view all the answers

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

    <p>The kinetic energy of the electron</p> 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?

    <p>$E$</p> Signup and view all the answers

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

    <p>It defines relativistic momentum of the electron</p> Signup and view all the answers

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

    <p>The scattering angle of the photon</p> Signup and view all the answers

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

    <p>It increases the wavelength</p> Signup and view all the answers

    What is the momentum of the incident photon expressed as?

    <p>$ rac{hc}{λo}$</p> Signup and view all the answers

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

    <p>Diffraction and interference</p> 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.

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