Wave Speed Concepts

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Questions and Answers

What significant discovery did Kirchhoff and Bunsen make using spectrum analysis?

They discovered the elements cesium and rubidium. (correct)

They discovered the element hydrogen.

They created the first spectroscope.

They identified the process of radiation.

What phenomenon did Kirchhoff explain regarding the sun's spectrum?

The existence of additional radiation from the sun.

The complete absence of thermal radiation.

The dark lines due to cooler gases absorbing sunlight. (correct)

The presence of more hydrogen than other elements.

What are dark-lined spectra in the context of the sun’s spectrum referred to as?

Reflected spectra.

Continuous spectra.

Absorption spectra. (correct)

Emission spectra.

According to the principles laid out, what happens when a body is in thermodynamic equilibrium?

Its emissive power is related to its absorption coefficient and universal function. Signup and view all the answers

What did the discoveries of cesium and rubidium signify in the field of chemistry?

A new era of seeking undiscovered elements. Signup and view all the answers

Which of the following statements correctly describes the relationship between emission and absorption in thermal radiation?

There is a defined ratio between emissive power and the coefficient of absorption. Signup and view all the answers

What aspect of electromagnetic waves is primarily explored through the particle theory?

Wave properties of particles. Signup and view all the answers

Which concept fundamentally challenges classical physics as discussed in quantum physics?

The uncertainty principle. Signup and view all the answers

What is the relationship between phase speed (vp) and angular frequency (ω)?

vp = ω / k Signup and view all the answers

How is group speed (vg) mathematically related to phase speed (vp) according to the given relations?

vg = vp - λ * (dvP/dλ) Signup and view all the answers

Which equation defines the group speed (vg) in terms of the change in angular frequency (dω) and wavenumber (dk)?

vg = dω/dk Signup and view all the answers

What does Planck's law resolve regarding ultraviolet catastrophe?

The denominator increases faster than the numerator. Signup and view all the answers

In terms of energy (E), what is the relationship of group speed (vg) to particle speed (u)?

vg = dE/dp Signup and view all the answers

What is the relationship between the energy of photons and frequency according to Einstein’s interpretation?

Energy increases linearly with frequency. Signup and view all the answers

Which observation contradicts classical predictions of the photoelectric effect?

Photoemission is instantaneous. Signup and view all the answers

What is the formula for calculating kinetic energy (E) for a classical particle indicated in the provided content?

E = 1/2 mv^2 Signup and view all the answers

How does increasing the intensity of light affect the photoelectric effect?

It leads to more photons but not higher energy photons. Signup and view all the answers

What is the significance of the equation d sin θ = mλ in the context of wave mechanics?

It describes interference in wave patterns. Signup and view all the answers

What is the effect of wavelength (λ) on the group speed (vg) as expressed in the formulas provided?

vg decreases as λ increases Signup and view all the answers

What is the maximum kinetic energy of the most energetic photoelectrons represented as?

KMAX = eΔV Signup and view all the answers

What is the result of the ultraviolet catastrophe as λ approaches 0?

I(λ, T) tends to zero. Signup and view all the answers

Which of the following correctly identifies the phase speed (vp) using frequency (f) and wavelength (λ)?

vp = fλ Signup and view all the answers

What did Planck derive for his constant based on experimental data?

h = 6.626 × 10^-34 J-s Signup and view all the answers

In the context of wave mechanics, what does the term 'wave packet' refer to?

A composite of multiple waves moving at group speed Signup and view all the answers

What happens to the photoemission below the threshold frequency?

No photoemission occurs. Signup and view all the answers

What is the result of substituting expressions for angular frequency in terms of energy (E) into the equation for group speed (vg)?

vg = dE/dp Signup and view all the answers

What phenomenon is described as the electromagnetic radiation emitted by a black body?

Black-body radiation Signup and view all the answers

According to Stefan's Law, what is the relationship between temperature and emitted energy?

Hotter objects emit more energy per unit area than colder objects. Signup and view all the answers

What does Wien's Displacement Law state about wavelength distribution?

It shifts to shorter wavelengths as the black body temperature increases. Signup and view all the answers

What is the formula that corresponds to the Rayleigh-Jeans Law?

I(λ, T) = 2πkB T / λ⁴ Signup and view all the answers

What is the 'ultraviolet catastrophe' in relation to the Rayleigh-Jeans Law?

It describes a divergence of predicted energy output at short wavelengths. Signup and view all the answers

What is the primary assumption of Planck's Law regarding energy emission?

The emission/absorption of energy occurs in integral multiples of hf. Signup and view all the answers

In the context of black-body radiation, what does the symbol 'T' represent?

Temperature of the black body Signup and view all the answers

Which equation describes the intensity of radiation emitted by a black body according to Planck's Law?

I(λ, T) = 2πh c² / λ⁵ (e^(hc/λkT) - 1) Signup and view all the answers

What does the Heisenberg uncertainty principle imply about measuring a particle's properties?

Simultaneous measurements of position and momentum cannot be infinitely accurate. Signup and view all the answers

Using Wien's displacement law, what is the relationship between the temperature of a black body and its peak wavelength?

The peak wavelength decreases as temperature increases. Signup and view all the answers

What is the work function for the second metal if its threshold frequency is higher than the first metal by 40.0%?

2.22 eV Signup and view all the answers

If a blackbody at 7500 K emits photons between 500 nm and 501 nm, how many photons escape per second through a 0.050 mm diameter opening?

1.30 x 10^15/s Signup and view all the answers

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

5750 K Signup and view all the answers

For a photon scattering angle that gives equal kinetic energy to the recoiling electron from a 0.00160 nm photon, what is the scattering angle?

70° 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 does the uncertainty principle predict about the line width produced by the finite lifetime of an excited atom?

It is inversely proportional to the lifetime. Signup and view all the answers

Study Notes

Wave Propagation

Phase Speed (vp): Speed of individual wave crests; calculated as ( v_p = f \lambda ) or ( v_p = \frac{\omega}{k} ).
Group Speed (vg): Speed of the wave packet; expressed by ( v_g = \frac{d\omega}{dk} ) and relates to phase speed via ( v_g = v_p - \lambda \frac{d v_p}{d\lambda} ).

Particle and Group Speed Relation

Particle speed (u) connects with group speed (vg) through kinetic energy equations, leading to ( v_g = v_p - \frac{\lambda}{m} \frac{dp}{d\lambda} ).
Equations define how phase speed ( v_p ) intertwines with wavelengths and kinetic parameters of particles.

Double-Slit Experiment

The classic double-slit experiment demonstrates wave-particle duality through electron beam interference patterns.
The condition for constructive interference is defined by ( d \sin \theta = m \lambda ), where ( m ) is the order number.

Blackbody Radiation Concepts

** Kirchhoff and Bunsen's Discoveries**: Identified elements in the sun’s spectrum using spectroscopic methods, noting absorption lines.
Blackbody definition: An idealized physical body that absorbs all incoming radiation, re-emitting energy based solely on temperature.

Blackbody Radiation Laws

Stefan-Boltzmann Law: Hotter bodies emit more energy, described by ( P = \sigma A e T^4 ).
Wien’s Displacement Law: Describes the shift in peak wavelength as temperature increases: ( \lambda_m T = \text{constant} ).
Rayleigh-Jeans Law: Predicts emission power overestimation at shorter wavelengths, leading to "ultraviolet catastrophe".

Planck’s Law

Introduces the concept that emitted radiation depends on both temperature and frequency, leading to the formula: [ I(\lambda, T) = \frac{2 \pi h c^2}{\lambda^5} \left(\frac{1}{e^{\frac{hc}{\lambda kT}} - 1}\right) ]
Resolved the ultraviolet catastrophe by aligning theoretical results with observed data, demonstrating that ( I(\lambda, T) \rightarrow 0 ) as ( \lambda \rightarrow 0 ).

Photoelectric Effect

Phenomenon where electrons are ejected from metals due to incident light of a certain frequency, with key factors:
- Frequency must surpass threshold for emission; intensity alone does not determine electron energy.
- ( K_{max} = e \Delta V_s ) ties the maximum kinetic energy of ejected electrons to the stopping potential.

Einstein’s Photon Theory

Describes electromagnetic radiation as quantized packets (photons), with energy defined as ( E = hf ).
Higher light intensity equates to more photons rather than increased energy per photon.

Uncertainty Principle

Heisenberg’s principle states that one cannot simultaneously know the exact position and momentum of a particle: ( (\Delta x)(\Delta p_x) \geq \frac{h}{4\pi} ).
An equivalent form relates uncertainty in energy and time: ( (\Delta E)(\Delta t) \geq \frac{h}{4\pi} ).

Solar Characteristics

Sun's Radius: ( 6.96 \times 10^8 ) m.
Total Power Output: ( 3.77 \times 10^{26} ) W.
Using Stefan's Law and Wien's Law, the sun's surface temperature is calculated to be approximately ( 5750 ) K, with maximum emission wavelength around ( 504 ) nm.

Photon Emission Calculations

The number of photons escaping from a blackbody at a specific temperature and wavelength can be quantified using Planck’s Law.
Example calculation yields ( 1.30 \times 10^{15} ) photons/s escaping a small aperture at ( 500 ) nm.

Additional Exercise Results

Stopping potentials and work functions of different metals can be derived based on varying threshold frequencies.
Photon scattering and kinetic energy exchanges can be solved with specific scattering angles and energies.

Sample Problems

Lifetime of an excited atom leading to uncertainty in frequency can be estimated via the uncertainty principle for practical applications in quantum mechanics.

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Description

Explore the concepts of phase speed and group speed in wave mechanics. This quiz will test your understanding of how these two types of speeds are defined and calculated in relation to wave behavior. Perfect for students studying physics topics related to waves.