Question

Light Through an Absorbing Medium: Sunlight passes through a glass filter. Each millimeter of glass absorbs \(20 \%\) of the light passing through it. What percentage of the original sunlight will remain after passing through \(5.0 \mathrm{mm}\) of the glass?

Short answer

After passing through 5 mm of glass, 32.768% of the original sunlight remains.

Step-by-step solution

Understand the Absorption Per Layer

Identify that each millimeter of glass absorbs 20% of the light that reaches it, which means 80% of the light passes through each millimeter. This can be represented by a transmission factor of 0.8 (since 80% = 0.8).

Calculate the Transmission Through Multiple Layers

Since the light passes through 5 millimeters of glass, the transmission factor needs to be raised to the power of 5 to find the total percentage of light transmitted. Use the formula: transmission after 5 mm = (0.8)^5.

Perform the Calculation

Calculate the total transmission after 5 mm: (0.8)^5 = (0.8)*(0.8)*(0.8)*(0.8)*(0.8) = 0.32768.

Convert to Percentage

To find the percentage, multiply the result by 100: 0.32768 * 100 = 32.768%.

Key concepts

Exponential Decay

When discussing the light passing through a transparent material, such as glass, exponential decay is the mathematical principle that explains how the intensity of light decreases as it travels through an absorptive medium. Think of this concept like a sponge soaking up water—the sponge represents the glass, and the water represents light. Just as a sponge can only hold so much water, glass can only let through a certain amount of light before some of it is absorbed.

The term 'exponential decay' describes a process where the quantity decreases at a rate proportional to its current value. For the light passing through glass, each millimeter represents a point at which the light's intensity is reduced by a certain percentage. This reduction happens in a multiplicative manner, meaning we multiply the current intensity by a fixed fraction to determine the new intensity after passing through each additional millimeter.

Percentage Calculation

Percentage calculation is a fundamental mathematical skill that comes in handy when quantifying the light intensity remaining after passing through an absorbing medium. To relate this to our daily life, think of a shopping discount. If an item is marked with a 20% discount, we know that we'll pay 80% of the original price. Similarly, if each millimeter of glass absorbs 20% of the light, 80% remains.

To find out how much light is left after passing through multiple layers of glass, we calculate the cumulative effect of the 20% reduction across those layers. This is where we use exponential decay as we repeatedly apply the 80% transmission factor, multiplying it by itself for each millimeter of glass the light travels through.

Properties of Materials

Understanding the properties of materials is crucial when studying light absorption. Transparent materials, such as the glass in our exercise, have specific characteristics that determine how they interact with light. Transparency denotes the ability of light to pass through a material without being scattered or absorbed significantly.

However, no material is perfectly transparent, and some light is invariably absorbed. The amount of light a material absorbs depends on factors such as its thickness, the wavelength of the light, and the material's intrinsic absorption properties. In this case, the glass's property of absorbing 20% of the light per millimeter showcases how material properties directly affect light transmission and can be quantified to predict behavior.

Mathematical Modelling

Mathematical modelling is a powerful tool used to represent real-world situations using mathematical concepts and language. It involves creating a mathematical representation of a system to analyze and make predictions. In our example, the system is the passage of light through glass and its subsequent absorption.

By creating a model that uses an exponential function to describe the decay of light intensity with each millimeter of glass it passes through, we can predict how much light remains after travelling through any thickness. Our model uses the percentage of light absorbed (20%) to determine the transmission factor (80% or 0.8), and then applies this factor repeatedly to model the exponential decay. Through mathematical modelling, we turn a physical phenomenon into a predictable and calculable system.