Engineering Physics PDF

Summary

This document is an engineering physics textbook chapter discussing blackbody radiation. It details experimental observations, theoretical aspects of oscillations, and formulas related to the concept. The text emphasizes blackbody characteristics, radiation formulas, and Planck's quantum hypothesis.

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# Engineering Physics ## 9.2 Blackbody Radiation ### 9.2.1 Blackbody and its characteristics - A blackbody is an idealized object that absorbs all electromagnetic radiation falling on it, regardless of the frequency or angle of incidence. - A blackbody in thermal equilibrium with its surroundi...

# Engineering Physics ## 9.2 Blackbody Radiation ### 9.2.1 Blackbody and its characteristics - A blackbody is an idealized object that absorbs all electromagnetic radiation falling on it, regardless of the frequency or angle of incidence. - A blackbody in thermal equilibrium with its surroundings emits radiation at all frequencies. The emitted radiation is called blackbody radiation. - The spectral distribution of blackbody radiation is a continuous function of the wavelength ­­­­and its shape depends only on the temperature of the blackbody. - The emissive power of a blackbody is independent of the construction material of the cavity. ### 9.2.2 Experimental Observation of Blackbody Radiation - By maintaining a constant temperature of the blackbody, its emissive power (or intensity) can be measured for different wavelengths, which results in spectra as shown in Figure 9.2. - As temperature increases, the wavelength corresponding to the highest emissive power shifts towards lower wavelengths, with a maximum value at a specific wavelength. - The emissive power increases throughout the range of wavelengths except for very low and long wavelengths, for which the emissive power is very low. **Figure 9.2: Experimental results of blackbody radiation for three temperatures T1, T2, and T3.** ### 9.2.3 Theoretical Aspect of Oscillation of Charged Particle - An accelerating charged particle emits electromagnetic radiation. An oscillating charged particle accelerates and thus emits electromagnetic radiation. - Any radiating system consists of charged particles that undergo simple harmonic motion; a blackbody cavity is full of standing waves of electromagnetic radiation of all wavelengths. - The energy is continuously exchanged randomly between atoms through emission and absorption of electromagnetic radiation. - As the temperature increases, new stationary wave modes are generated, and the amplitudes of existing modes also increase. - The energy density of radiation is constant at a certain temperature and increases until equilibrium is reached. - The density of the radiant energy corresponding to radiation with frequencies between v and v-dv (or A and A+ d) can be calculated by the formula: u, dv = n, e, dv - where: - n,dv is the number of oscillating atoms per unit volume that contribute to the radiations of frequencies between v and v-dv (or A and A + d) - e is the average energy of an oscillating atom at an absolute temperature T - The number of modes of electromagnetic stationary waves in a three-dimensional box can be expressed as: n, dv = $\dfrac{8\pi v^2 dv}{c^3}$ - Combining these equations, one can express the energy density of radiations having wavelengths lying between A and A + dλ as: u₁ da = $\dfrac{8\pi ν^2 dv}{c^3}$ ε, dv = $\dfrac{8\pi ν^2 }{c^3}$ ε, dλ - Using the relationship between wavelength and frequency, c = νλ and dv = - c/λ² dλ, we can get: u₁ dλ = $\dfrac{8\pi}{λ^4 } $ ε, dλ - The intensity of emitted radiation (Eλ) is proportional to the energy density (uλ) and can be written as: Eλ = $\dfrac{2πc}{λ^4 } $ ε ### 9.2.4 Rayleigh-Jeans Formula of Radiation - Lord Rayleigh and J.H. Jeans attempted to derive an expression for Eλ using the equipartition theorem of energy. - According to the equipartition theorem, the average energy of an oscillator is given by kT: ε = kT - where: - k is Boltzmann's constant - T is the absolute temperature - Substituting this value of ε into the equations for uλ and Eλ, one can get: uλ dλ = $\dfrac{8\pi kT }{λ^4 }$ dλ Eλ = $\dfrac{2πckT}{λ^4 }$ - This is known as the Rayleigh-Jeans formula and agrees with the results obtained from experiments in the long wavelength region of the electromagnetic spectrum, however, it fails completely at shorter wavelengths. ### 9.2.5 Planck's Radiation Formula and his Quantum Hypothesis - Max Planck proposed that the failure of the Rayleigh-Jeans law is due to the nature of the linear harmonic oscillators, suggesting that their energies are discrete, not continuous. - The energy of an oscillator can only assume values that are multiples of a finite quantum of energy: E = nhv -where: - n is an integer - h is Planck's constant - v is the frequency of the oscillator - This is known as Planck's quantum hypothesis. - The change in the energy of an oscillator can only occur through the absorption or emission of a discrete amount of energy, hv. - Planck calculated the value of Planck's constant, h, to be 6.626 x 10^-34 joules-second, which is a universal constant. - Planck's concept: - a photon is the amount of energy emitted or absorbed in a single step of change, with energy equal to hv. - the change in energy of an oscillator can only take place in discrete amounts. - Using these new postulates, Planck derived a new formula for the intensity distribution of blackbody radiation: uλ dλ = $\dfrac{8\pi hc}{λ^5 }$ $\dfrac{1}{exp(hc/λkT)-1}$ dλ Eλ = $\dfrac{2πhc^2}{λ^5 }$ $\dfrac{1}{exp(hc/λkT)-1}$ - This formula agrees with the experimental results at all wavelengths, resolving the ultraviolet catastrophe. ### 9.2.6 Deductions from Planck's Radiation Formula #### 9.2.6 (a) Wien's displacement law - Wien's displacement law can be derived from Planck's radiation formula. - It establishes that the wavelength corresponding to the maximum intensity of blackbody radiation is inversely proportional to the temperature of the blackbody: λ_m T = constant = 2.898 x 10^-3 mK #### 9.2.6 (b) Stefan-Boltzmann law - Stefan-Boltzmann law states that the total energy density of blackbody radiation is proportional to the fourth power of the absolute temperature: u = aT^4 - where: - a is a constant - This law can be derived from Planck's radiation formula by integrating the energy density over all wavelengths. - The Stefan-Boltzmann constant, σ, can be calculated as: σ = $\dfrac{2π^5 k^4}{15c^2h^3 }$ = 5.67 W m2 K-4 ### 9.2.7 Einstein's Concept Regarding Blackbody Radiation - Einstein proposed that light is quantized and consists of energy packets called photons, which travel at the speed of light. - The energy of a photon is given by: E = hc/λ = hv - where: - h is Planck's constant - c is the speed of light - λ is the wavelength of the photon - v is the frequency of the photon - This concept can be used to derive Planck's radiation formula using statistical mechanics by treating blackbody radiation as a photon gas in thermal equilibrium with the atoms in the cavity walls. - Einstein's concept highlights the duality of light as both a wave and a particle.

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