Stefan Boltzmann Calculator

Stefan-Boltzmann Law Calculator

This calculator determines the total power radiated per unit surface area of a black body, or more generally, the power radiated by an object based on its temperature, surface area, and emissivity, according to the Stefan-Boltzmann law.

(e.g., 273.15 K = 0 °C)

function calculatePower() { var area = parseFloat(document.getElementById("surfaceArea").value); var emissivity = parseFloat(document.getElementById("emissivity").value); var temperature = parseFloat(document.getElementById("temperatureK").value); var stefanBoltzmannConstant = 5.670374419e-8; // W m⁻² K⁻⁴ if (isNaN(area) || area <= 0) { document.getElementById("result").innerHTML = "Please enter a valid surface area greater than 0."; return; } if (isNaN(emissivity) || emissivity 1) { document.getElementById("result").innerHTML = "Please enter a valid emissivity between 0 and 1."; return; } if (isNaN(temperature) || temperature = 0)."; return; } var power = stefanBoltzmannConstant * area * emissivity * Math.pow(temperature, 4); document.getElementById("result").innerHTML = "Total Radiated Power (P): " + power.toExponential(4) + " Watts"; }

Understanding the Stefan-Boltzmann Law

The Stefan-Boltzmann law states that the total energy radiated per unit surface area of a black body across all wavelengths per unit time (also known as the black-body radiant emittance) is directly proportional to the fourth power of the black body's thermodynamic temperature. The law was formulated by Jožef Stefan in 1879 and later derived by Ludwig Boltzmann in 1884 from thermodynamic considerations.

The Formula

The formula for the Stefan-Boltzmann law is:

P = σ * A * ε * T⁴

Where:

• P is the total power radiated (in Watts).
• σ (sigma) is the Stefan-Boltzmann constant, approximately 5.670374419 × 10^-8 W m^-2 K^-4.
• A is the surface area of the emitting body (in square meters, m²).
• ε (epsilon) is the emissivity of the object's surface (a dimensionless number between 0 and 1; 1 for a perfect black body, less than 1 for real objects).
• T is the absolute temperature of the body (in Kelvin, K).

For a perfect black body, emissivity (ε) is 1. For other materials (gray bodies), it is less than 1, representing the ratio of energy radiated by the material to that radiated by a black body at the same temperature.

Temperature in Kelvin

The temperature must be in Kelvin (K). To convert from Celsius (°C) to Kelvin, use the formula: K = °C + 273.15. Absolute zero (0 K) is -273.15 °C.

Example Calculation

Let's say we have an object with a surface area (A) of 0.5 m², an emissivity (ε) of 0.8, and a surface temperature (T) of 500 K.

P = (5.670374419 × 10^-8 W m^-2 K^-4) * 0.5 m² * 0.8 * (500 K)⁴

P = (5.670374419 × 10^-8) * 0.5 * 0.8 * 62,500,000

P ≈ 1417.6 Watts

So, the object would radiate approximately 1417.6 Watts of power.