Problem 24 - Technical Mathematics with Calculus

Question

As light passes through glass or water, its intensity decreases exponentially according to the equation $$I=I_{0} e^{-k x}$$,where \(I\) is the intensity at a depth \(x\) and \(I_{0}\) is the intensity before entering the glass or water. If, for a certain filter glass, \(k=0.500 / \mathrm{cm}\) (which means that each centimeter of filter thickness removes half the light reaching it), find the fraction of the original intensity that will pass through a filter glass \(2.00 \mathrm{cm}\) thick.

Step-by-Step Solution

Answer

About 36.8% of the original intensity passes through the glass.

Step 1: Understanding the Exponential Decay Equation

The given equation for the intensity is an example of exponential decay, where the intensity decreases as it travels through a medium. Here, the equation is of the form \(I = I_0 e^{-kx}\), where \(I\) is the final intensity, \(I_0\) is the initial intensity, \(k\) is the decay constant (in \(cm^{-1}\)), and \(x\) is the thickness of the material in centimeters.

Step 2: Plug in the Given Values

To find the fraction of the original intensity that passes through, plug in the given values for \(k\) and \(x\) into the equation. Here, \(k = 0.500 \ cm^{-1}\) and \(x = 2.00 \ cm\). The equation becomes \(I = I_0 e^{-0.500 \times 2.00}\).

Step 3: Calculate the Exponent

Calculate the value of the exponent \(-kx\) which is \(-0.500 \times 2.00 = -1.00\). The equation now is \(I = I_0 e^{-1.00}\).

Step 4: Evaluate the Exponential Function

Evaluate the exponential function using the calculated exponent. The value of \(e^{-1.00}\) can be found using a calculator. This yields \(e^{-1.00} \approx 0.368\).

Step 5: Calculate the Final Intensity

Now, by substituting the value of \(e^{-1.00}\) back into the equation, we get \(I = I_0 \times 0.368\). This represents the fraction of the initial intensity \(I_0\) that passes through the glass.

Step 6: Express the Fraction of the Original Intensity

Since the problem asks for the fraction of the original intensity, we can report the answer as \(0.368\), meaning about 36.8% of the original light intensity passes through the 2.00 cm thick filter glass.

Key Concepts

Exponential Decay Equation

Understanding the exponential decay equation is crucial when exploring how the intensity of light diminishes as it traverses through materials like glass or water. An exponential decay situation is defined mathematically by an equation of the form \( I = I_0 e^{-kx} \), where \( I \) stands for the final intensity after the light has penetrated a given depth \( x \), and \( I_0 \) denotes the initial intensity before entering the medium.

In this context, the variable \( k \) is known as the decay constant and represents the rate at which the light is attenuated per unit thickness of the material. The base of the exponential function, \( e \), is the mathematical constant approximately equal to 2.71828, which emerges naturally in many areas of mathematics, especially in situations describing growth or decay processes.

The minus sign in the exponent \( -kx \) indicates a decrease in the given quantity, in this case, the intensity of light, as the depth increases. It's worth noting that the strength of this decay is contingent upon the value of \( k \); a larger \( k \) means a more rapid decrease in intensity. By plugging in known values into this equation, we can calculate how much light remains after passing through a specific material – an essential aspect of optics and material science.

Intensity of Light

The intensity of light, represented by \( I \) in the exponential decay equation, pertains to the power carried by light waves or photons per unit area. Intensity is a key concept in both physics and optics as it quantifies the amount of light energy that is received or transmitted through a medium.

When light passes through a transparent or translucent material like glass, its intensity is not constant. Instead, it reduces gradually due to absorption and scattering by the material. This is particularly relevant for applications such as photography, microscopy, and designing lighting systems, where understanding and controlling light intensity is necessary.

For the human eye, the intensity of light is related to brightness, but in scientific terms, it is a measurable quantity that has significant implications for the behavior of light as it interacts with various media. In calculations, the exponential decay of light intensity helps to model scenarios like sunlight passing through the atmosphere, laser beams penetrating biological tissue, and much more.

Decay Constant

The decay constant, symbolized by \( k \) in our equation, is a fundamental parameter expressing the rate at which the exponential decay occurs. It can also be thought of as the inherent 'strength' of the medium's ability to diminish the intensity of light with each unit of thickness.

For the specific example of filter glass, a decay constant of \( 0.500 \/ \text{cm} \) implies that the light intensity is halved (\

reduced by 50%) with each additional centimeter of glass it travels through. The higher the value of \( k \), the faster the intensity of light decreases. Conversely, if the decay constant is small, the light will be able to travel a longer distance before its intensity is significantly affected.

In practice, knowing the decay constant of a material can guide us in choosing the right thickness for lenses or filters to achieve the desired light transmission. For engineers and designers, the decay constant is an indispensable tool in crafting systems where control over light intensity is necessary, such as in solar panels, architectural designs, and various optical instruments.