Planck’s Law

Introduction

Planck’s Law is a revolutionary principle that marked the birth of quantum mechanics. It explains how objects emit electromagnetic radiation based on their temperature, providing a detailed description of blackbody radiation. This law has transformed our understanding of energy, light, and the behavior of matter at the atomic scale.

Variables & Units

The fundamental form of Planck’s Law is:

B(λ, T) = (2hc²)/(λ⁵) × 1/(exp(hc/(λkT)) – 1)

• B(λ, T): Spectral radiance, measured in watts per steradian per cubic meter (W·sr⁻¹·m⁻³).
• λ (Wavelength): Measured in meters (m).
• T (Temperature): Measured in Kelvin (K).
• h (Planck’s Constant): Approximately 6.626×10⁻³⁴ J·s.
• c (Speed of Light): Approximately 3×10⁸ m/s.
• k (Boltzmann’s Constant): Approximately 1.381×10⁻²³ J/K.

Consistent SI units are essential to accurately calculate the spectral radiance of a black body.

The Fundamental Equation

Planck’s Law is mathematically expressed as:

B(λ, T) = (2hc²)/(λ⁵) × 1/(exp(hc/(λkT)) – 1)

This equation describes the intensity of radiation emitted at different wavelengths by a black body in thermal equilibrium.

How to Use Planck’s Law

Applying Planck’s Law involves the following steps:

1. Determine the Temperature: Identify the absolute temperature (T) of the black body.
2. Select the Wavelength: Choose the wavelength (λ) at which you wish to calculate the spectral radiance.
3. Substitute into the Equation: Plug the values into the equation B(λ, T) = (2hc²)/(λ⁵) × 1/(exp(hc/(λkT)) – 1).
4. Calculate B(λ, T): Solve the equation to find the spectral radiance at that wavelength.

Example Problems

Example 1: Calculating Spectral Radiance

Problem: Calculate the spectral radiance of a black body at 5000 K at a wavelength of 500 nm. (Note: 500 nm = 5×10⁻⁷ m)

B(λ, T) = (2 × 6.626×10⁻³⁴ × (3×10⁸)²) / ( (5×10⁻⁷)⁵ ) × 1/(exp((6.626×10⁻³⁴×3×10⁸)/(5×10⁻⁷×1.381×10⁻²³×5000))-1)

Explanation: Substituting the values and solving the equation yields the spectral radiance at 500 nm for a 5000 K black body.

Example 2: Determining the Peak Wavelength

Problem: While Wien’s Displacement Law is typically used to find the wavelength of maximum emission, Planck’s Law provides the full spectral distribution from which the peak can be determined numerically for a black body at a given temperature.

(This problem involves analyzing the spectral curve defined by Planck’s Law to find its maximum value.)

Explanation: By evaluating B(λ, T) over a range of wavelengths for a black body at a specific temperature, you can determine the peak wavelength.

Practical Applications

Planck’s Law has numerous applications, including:

• Astrophysics: Determining the temperature and energy output of stars and other celestial bodies.
• Thermal Imaging: Designing sensors that detect infrared radiation for temperature measurement.
• Material Science: Analyzing the emission properties of materials to determine their thermal characteristics.
• Climate Science: Studying Earth’s radiation budget and the effects of greenhouse gases on energy balance.

Advanced Concepts

Beyond its basic form, Planck’s Law leads to many advanced insights:

• Quantum Mechanics: Planck’s Law was pivotal in the development of quantum theory by introducing the concept of energy quanta.
• Blackbody Radiation: It provides a complete description of the spectral distribution of radiation from an ideal black body.
• Energy Quantization: The law implies that electromagnetic energy is emitted in discrete packets, a key concept in modern physics.

Frequently Asked Questions

Conclusion

Planck’s Law revolutionized our understanding of radiation and energy. By describing the spectral distribution of blackbody radiation, it laid the foundation for quantum mechanics and has numerous applications in astrophysics, thermal imaging, and material science.