2.2

Question 1

Specific Intensity Definition

Answer 1

Describes radiation field at any
- point r
- time t
- direction n̂((θ, φ))
I_ν(r, t, n̂)

Question 2

Specific Intensity Formula

Answer 2

I_ν(r, t, n̂) cosθ dA dt dΩ dν = dE_ν dν
- Specific intensity through area dA
- dE_ν [W Hz^-1] amount of radiation energy
- θ angle between surface normal and n̂

Question 3

Blackbody Radiation Formula

Answer 3

B_ν dν = (2πh/c²) * (ν³ dν) / (exp(hν/k_BT) - 1)

Divide both sides by dν and B_ν(T) is the specific intensity of blackbody radiation

Question 4

Radiation Flux Definition

Answer 4

F_ν

= total energy of radiation coming from all directions per unit area per unit time
= dE_νdν/(dA dt) and integrate over all solid angles
= Intensity flowing through area dA per unit time dt

Question 5

Radiation Flux Formula

Answer 5

• Integration of I over all solid angles:

F_ν = ∫ I_ν cosθ dΩ

• F_ν = net flux = flux up and flux down = F_ν = F_ν⁺ + F_ν^-
• F_ν = 0 for isotropic radiation field inside star or cavity
• F_ν = F_ν⁺ and F_ν^- = 0 outside star or cavity

Question 6

Integration of a function over all solid angles

Answer 6

• Integration of I over all solid angles:

∫ I_ν dΩ
= ∫^2π _{φ = 0} ∫^π _{θ = 0} (I_ν(θ, φ) sinθ) dθ dφ

Question 7

Total Radiation Flux Formula

Answer 7

F = ∫ F_ν dν

Question 8

Energy Density Formula

Answer 8

• radiation I_ν transverses distance c · dt
• fills cylinder c · dt · cosθ · dA
• dE_ν / (c · dt · cosθ · dA) = (I_ν / c) dΩ

→ energy density = integration for all directions:

u_ν = ∫ (I_ν /c) dΩ

• u_ν = non-isotropic energy density.

Question 9

Planck Radiation Energy Density

Answer 9

• energy density for isotropic Planck-radiation

U_ν = (4π/c) · B_ν (T) = (8πh/c³) · (ν³ dν) / (exp(hν/k_B T) - 1)

• valid for thermodynamic equilibrium inside star or cavity
• radiation is isotropic

Question 10

Total Energy Density for Planck Radiation

Answer 10

U = ∫ U_ν dν = a T⁴

• σ Stefan-Boltzmann constant
• a = 4σ / c

Question 11

Stefan-Boltzmann Constant

Answer 11

σ = 5.67 * 10^-8W/(m²K⁴)

Question 12

Radiation Pressure Formula

Answer 12

P_ν = (1/c) ∫ I_ν cos²θ dΩ

• hν / c = momentum transfer per photon on absorbing material
• dP_ν = (I / c) dΩ = radiation pressure
• I_ν cos θ dΩ = radiation through surface dA
• I_νcos²θ dΩ = momentum component perpendicular to surface dA

Question 13

Radiation pressure for isotropic Planck radiation

Answer 13

• I_ν = B_ν
• integration over all directions

P_ν = (B_ν / c) ∫cos²θ dΩ = 4π / 3 · B_ν / c = 1/3 · U_ν

Question 14

Total Radiation Pressure

Answer 14

P_R = U / 3 = (a / 3) T⁴
- radiation pressure has same units as energy density