Engineering Physics PDF

# 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.