Radiation Formulas

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

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 ╬╗ to (╬╗ + d╬╗) per second per unit area to the wavelength spread i.e.,
e_╬╗ = \(\frac{d Q_{\lambda}}{d \lambda}\)
and
e = \(\int_{0}^{\infty}\)e_╬╗d╬╗

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.
a_╬╗ = 1
and r = t = 0
FerriΓÇÖs ideal Black body:

9. KirchoffΓÇÖs law

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

10. StefanΓÇÖs law

╧â is StefanΓÇÖs constant = 5.67 ├ù 10^-8 W/m² – K⁴
Net loss of energy per second per unit area = ╧â (T⁴ – T₀⁴)
For a black surface of area A the net rate of loss of heat
–\(\frac{d Q}{d t}\) = ╧âA (T⁴ – T₀⁴) J/s
For a surface of emissivity e
–\(\frac{d Q}{d t}\) = ╧âAe (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. NewtonΓÇÖs law of cooling

Rate of cooling ∝ Excess of temperature
–\(\frac{d Q}{d t}\) = K (╬╕ – ╬╕₀), K is cooling constant.
or \(\frac{d \theta}{d t}=\frac{K}{m s}\)(╬╕ – ╬╕₀) = k'(╬╕ – ╬╕₀)

Time to cool from ╬╕₁ to ╬╕₂
t = \(\frac{2.303}{\mathrm{K}^{\prime}}\)log₁₀\(\left(\frac{\theta_{1}-\theta_{0}}{\theta_{2}-\theta_{0}}\right)\)

If variation of 0 from ╬╕₁ to ╬╕₂ can be treated as linear then
\(\frac{\Delta \theta}{\Delta t}=\frac{\theta_{1}-\theta_{2}}{t}=K^{\prime}\left[\frac{\theta_{1}+\theta_{2}}{2}-\theta_{0}\right]\)

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

E_╬╗ – ╬╗ curve has a maximum at ╬╗ = ╬╗_m.
Area between E_╬╗ – ╬╗, curve and ╬╗ axis is proportional to T⁴.

13. WienΓÇÖs displacement law

or λ_m ∝ \(\frac{1}{T}\)
or ╬╗_m T = b (WienΓÇÖs constant)
b = 2.93 × 10^-3m-K
or v_m ∝ T
v_m = b’T, b’= \(\frac{c}{b}\)

14. Laws of distribution

WienΓÇÖs law:
E_╬╗ d╬╗ = \(\frac{A}{\lambda^{5}}\) e^-a/╬╗T d╬╗.

Rayleigh – JeanΓÇÖs law:
E_╬╗ d╬╗ = \(\frac{8 \pi \mathrm{kT}}{\lambda^{4}}\) d╬╗

PlanckΓÇÖs law:
E_╬╗ d╬╗ = \(\frac{8 \pi \mathrm{hc}}{\lambda^{5}} \frac{1}{\left[\mathrm{e}^{\mathrm{hc} / \lambda \mathrm{kT}}-1\right]}\) d╬╗
PlanckΓÇÖs law reduces to WeinΓÇÖs law at short wavelengths and to Rayeigh- Jeans law at long wavelengths.

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 ├ù 10³ W/m³

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 T_p is inversely proportional to the square root of its distance upon the sun i.e., T_p ∝ \(\frac{1}{\sqrt{d}}\).

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