Planck's Law

Planck's Law Problems and Solutions

Problem 1:

Calculate the spectral radiance of a blackbody at a temperature of 3000 K at a wavelength of 700 nm using Planck's law.

Given:

• Planck's constant, h = 6.626×10⁻³⁴ J·s
• Speed of light, c = 3×10⁸ m/s
• Wavelength, λ = 700×10⁻⁹ m
• Boltzmann's constant, kB = 1.381×10⁻²³ J/K
• Temperature, T = 3000 K

Planck’s Law (wavelength form):

B(λ, T) = [2hc² / λ⁵] × 1 / (exp(hc / λkBT) - 1)

Answer: B(λ, T) ≈ 7.52 × 10¹⁰ W·sr⁻¹·m⁻³

Problem 2:

Determine the spectral radiance of a blackbody at a temperature of 6000 K at a frequency of 6×10¹⁴ Hz using Planck's law.

Given:

• Planck's constant, h = 6.626×10⁻³⁴ J·s
• Speed of light, c = 3×10⁸ m/s
• Frequency, ν = 6×10¹⁴ Hz
• Boltzmann's constant, k = 1.381×10⁻²³ J/K
• Temperature, T = 6000 K

Planck’s Law (frequency form):

B(ν, T) = [2hν³ / c²] × 1 / (exp(hν / kT) - 1)

Answer: B(ν, T) ≈ 2.64 × 10⁻⁸ W·sr⁻¹·m⁻²·Hz⁻¹

Problem 3:

Find the wavelength at which the spectral radiance of a blackbody at a temperature of 4500 K is maximized using Wien's displacement law.

Given:

• Wien's constant, b = 2.897×10⁻³ m·K
• Temperature, T = 4500 K

Wien’s Displacement Law:

λmax = b / T

Answer: λmax ≈ 644 nm