To accurately determine emissivity from transmission percentage and wavelength, we rely on a few key theoretical principles. These principles explain how energy interacts with a material’s surface and how we can mathematically express those interactions. The primary concept is Kirchhoff’s Law of Thermal Radiation, which ties together emissivity, transmission, reflection, and absorption.

#### 1. **Kirchhoff’s Law of Thermal Radiation**

Kirchhoff’s Law states that, for any material in thermal equilibrium, the emissivity ($ε(λ)$) at a given wavelength is equal to its absorptivity ($α(λ)$) at the same wavelength. In simpler terms, the amount of energy a material emits as thermal radiation is directly proportional to the amount it absorbs.

For a material that is in thermal equilibrium, the sum of the energy reflected, transmitted, and absorbed must equal the total incident energy. This can be expressed as:

**α(λ)+T(λ)+R(λ)=1**Where:

- α(λ) is the absorptivity, which represents the fraction of energy absorbed by the material.
- T(λ) is the transmission percentage, representing the fraction of energy that passes through the material.
- R(λ) is the reflection percentage, representing the fraction of energy that is reflected off the material.

Since Kirchhoff’s Law tells us that emissivity (ε(λ) is equal to absorptivity ($)$), we can write the equation as:

**ε(λ)=1−T(λ)−R(λ)**This equation provides the basis for calculating emissivity once we have values for transmission and reflection. Essentially, by knowing how much energy is transmitted and reflected, we can infer how much is absorbed, and thus how much is emitted as radiation.

#### 2. **Transmission and Reflection**

Transmission and reflection are two key measurable properties that allow us to determine emissivity. Let’s break them down:

**Transmission (T(λ))**: This is the percentage of incident radiation that passes through the material without being absorbed or reflected. In some cases, especially with opaque materials, transmission might be zero, meaning the material either reflects or absorbs all the energy.
**Reflection ($)$**: This is the percentage of incident radiation that is reflected off the material’s surface. Reflection is also wavelength-dependent, with some materials reflecting more in certain parts of the electromagnetic spectrum (such as the visible or infrared range).

By measuring these two quantities, we can calculate the absorptivity α(λ), which is the key to finding emissivity.

#### 3. **Wavelength Dependency**

Emissivity is not a constant value for most materials but varies across different wavelengths of radiation. For instance, a material might exhibit high emissivity in the infrared spectrum but low emissivity in the visible spectrum. This is why emissivity is often expressed as a function of wavelength ε(λ).

To determine emissivity at a specific wavelength, it is crucial to measure transmission and reflection at that same wavelength. As different applications may focus on different parts of the spectrum (e.g., infrared thermography in the 8–14 µm range), choosing the appropriate wavelength for analysis is essential.

#### 4. **Absorptivity and Emissivity Relationship**

The relationship between absorptivity and emissivity is central to the determination of emissivity. Kirchhoff’s Law simplifies this relationship:

**ε(λ)=α(λ)**Where absorptivity can be calculated using:

**α(λ)=1−T(λ)−R(λ)**This formula shows that once transmission and reflection percentages are known, the remaining portion of the energy must be absorbed by the material, and thus, emissivity can be determined.

#### 5. **Practical Formula for Emissivity Determination**

Bringing everything together, the final formula to determine emissivity from transmission and reflection at a given wavelength is:

**ε(λ)=1−T(λ)−R(λ)**This equation provides a practical approach to calculating emissivity when transmission and reflection data are available. The key is ensuring that these properties are measured accurately at the same wavelength.

#### 6. **Simplified Case: Opaque Materials**

For opaque materials, the transmission percentage T(λ) is often negligible (i.e., T(λ)≈0). In these cases, the emissivity can be simplified to:

**ε(λ)=1−R(λ)**This is common in materials such as metals or dense ceramics, where most of the incident radiation is either absorbed or reflected, and little to no transmission occurs.

## III. How to Determine Emissivity from Transmission Percentage and Wavelength