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Radiation Formulas

Major Formulas needed to understand the Topic Radiation are listed in an organized manner here. Take the help of the Radiation Formulas List and know the logic behind the concepts easily. Try to apply the Formulae in Radiation Cheat Sheet during your homework and arrive at the solutions quickly. Radiation Formulae covers everything like Spectral emissive power, Emissive power, Thermal radiation, Reflection, absorption and transmission coefficients, etc. For any other help on related concepts, you can have a quick look at our Physics Formulas existing.

Important Radiation Formulae

1. Radiation

Process of transfer of energy by electromagnetic waves.

2. Thermal radiation

Electromagnetic waves corresponding to infra-red region.

3. Reflection, absorption and transmission coefficients

r = \(\frac{Q_{r}}{Q}\), a = \(\frac{Q_{a}}{Q}\) and t = \(\frac{Q_{t}}{Q}\)
r + a + t = 1
Radiation formulas img 1

4. Emissive power

Total amount of radiations emitted per second per unit area.
E = Q/A.t

5. Spectral emissive power

Ratio of amount of radiations emitted in a given range of wavelength λ to (λ + dλ) per second per unit area to the wavelength spread i.e.,
eλ = \(\frac{d Q_{\lambda}}{d \lambda}\)
e = \(\int_{0}^{\infty}\)eλ
Radiation formulas img 2

6. Emissivity

Ratio of emissive power of a given surface to emisive power of a black body.

7. Spectral absorptive power

In a given range of wavelength, d-f. is ratio of amount of radiations absorbed to amount of radiations incident.

8. Perfectly black body

Which absorbs all incident radiation of all wavelengths.
aλ = 1
and r = t = 0
Ferri’s ideal Black body:
Radiation formulas img 3

9. Kirchoff’s law

\(\frac{\mathrm{e}_{\lambda}}{\mathrm{a}_{\lambda}}\) = constant = Emissive power of a black body.

10. Stefan’s law

σ is Stefan’s constant = 5.67 × 10-8 W/m2 – K4
Net loss of energy per second per unit area = σ (T4 – T04)
For a black surface of area A the net rate of loss of heat
–\(\frac{d Q}{d t}\) = σA (T4 – T04) J/s
For a surface of emissivity e
–\(\frac{d Q}{d t}\) = σAe (T4 – T04) J/s
Rate of fall of temperature
\(-\frac{\mathrm{d} \theta}{\mathrm{dt}}=\frac{\sigma \mathrm{Ae}\left(\mathrm{T}^{4}-\mathrm{T}_{0}^{4}\right)}{\mathrm{msJ}}{ }^{\circ} \mathrm{C} / \mathrm{s}\)

11. Newton’s law of cooling

Rate of cooling ∝ Excess of temperature
–\(\frac{d Q}{d t}\) = K (θ – θ0), K is cooling constant.
or \(\frac{d \theta}{d t}=\frac{K}{m s}\)(θ – θ0) = k'(θ – θ0)
Radiation formulas img 4
Time to cool from θ1 to θ2
t = \(\frac{2.303}{\mathrm{K}^{\prime}}\)log10\(\left(\frac{\theta_{1}-\theta_{0}}{\theta_{2}-\theta_{0}}\right)\)
Radiation formulas img 5
If variation of 0 from θ1 to θ2 can be treated as linear then
\(\frac{\Delta \theta}{\Delta t}=\frac{\theta_{1}-\theta_{2}}{t}=K^{\prime}\left[\frac{\theta_{1}+\theta_{2}}{2}-\theta_{0}\right]\)
Radiation formulas img 6
Specific heat of liquids
\(\frac{\left(\mathrm{W}+\mathrm{m}_{\mathrm{L}} \mathrm{s}_{\mathrm{L}}\right)\left(\theta_{1}-\theta_{2}\right)}{\mathrm{t}_{1}}=\frac{\left(\mathrm{W}+\mathrm{m}_{\mathrm{w}} \mathrm{s}_{\mathrm{w}}\right)\left(\theta_{1}-\theta_{2}\right)}{\mathrm{t}_{2}}\)

12. Spectral distribution of radiant energy

Eλ – λ curve has a maximum at λ = λm.
Area between Eλ – λ, curve and λ axis is proportional to T4.

13. Wien’s displacement law

or λm ∝ \(\frac{1}{T}\)
or λm T = b (Wien’s constant)
b = 2.93 × 10-3m-K
or vm ∝ T
vm = b’T, b’= \(\frac{c}{b}\)
Radiation formulas img 7
Radiation formulas img 8

14. Laws of distribution

Wien’s law:
Eλ dλ = \(\frac{A}{\lambda^{5}}\) e-a/λT dλ.

Rayleigh – Jean’s law:
Eλ dλ = \(\frac{8 \pi \mathrm{kT}}{\lambda^{4}}\) dλ

Planck’s law:
Eλ dλ = \(\frac{8 \pi \mathrm{hc}}{\lambda^{5}} \frac{1}{\left[\mathrm{e}^{\mathrm{hc} / \lambda \mathrm{kT}}-1\right]}\) dλ
Planck’s law reduces to Wein’s law at short wavelengths and to Rayeigh- Jeans law at long wavelengths.
Radiation formulas img 9

15. Solar constant

S = Radiant energy per minute per cm2 by earth at the mean distance from sun
= 1.94 cal/cm2-mt
Solar constant in MKS units
S = 1.38 × 103 W/m3
Radiation formulas img 10
Temperature of sun
T = \(\left(\frac{\mathrm{S}}{\sigma}\right)^{1 / 4} \cdot\left(\frac{\mathrm{d}}{\mathrm{R}_{\mathrm{s}}}\right)^{1 / 2}\) = 5800 K
Equilibrium temperature of a planet Tp is inversely proportional to the square root of its distance upon the sun i.e., Tp ∝ \(\frac{1}{\sqrt{d}}\).
Radiation formulas img 11

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