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Chapter 3: Problem 20

From theoretical considerations, it is known that light from a certain star should reach Earth with intensity I0. However, the path taken by the light from the star to Earth passes through a dust cloud, with absorption coefficient 0.1/light-year. The light reaching Earth has intensity 1>2 I0. How thick is the dust cloud? (The rate of change of light intensity with respect to thickness is proportional to the intensity. One light-year is the distance traveled by light during 1 yr.)

Short Answer

Expert verified
The thickness of the dust cloud is \(- \frac{\ln(1/2)}{0.1}\) light-years.

Step by step solution

01

Understanding the problem

We know the original intensity \(I_0\) of the light, the absorption coefficient \(k=0.1/light-year\), and the observed intensity \(I=1/2I_0\) on Earth. We need to find the thickness \(x\) of the dust cloud.
02

Apply the formula for exponential decay

The intensity of the light after passing through the dust cloud can be expressed with a formula for exponential decay: \(I = I_0 \cdot e^{-kx}\), where \(I\) is the final intensity, \(I_0\) is the initial intensity, \(e\) is the base of natural logarithms, \(k\) is the absorption coefficient, and \(x\) is the thickness of the cloud. Given \(I=1/2I_0\) and \(k=0.1\), we want to solve for \(x\).
03

Solve the equation

Substitute the given values into the equation: \(1/2I_0= I_0 \cdot e^{-0.1x}\). Dividing both sides by \(I_0\) and taking the natural logarithm of both sides gives \(\ln(1/2)=-0.1x\). Solve this for \(x\) to find the thickness of the dust cloud.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Light Intensity
Light intensity refers to the amount of light energy that passes through a certain area. At the source, such as a star, this intensity is strong and is often denoted as \(I_0\), which represents the initial or original intensity. As light travels across distances, especially through mediums like dust clouds, its intensity can decrease.
  • The decrease in intensity happens due to absorption and scattering of light particles which is described by the exponential decay model.
  • This means that the intensity of light reduces as it travels through materials that absorb or block part of the light.
  • In our problem, the light from a star passes through a dust cloud before reaching Earth, leading to a decrease in its intensity to \(\frac{1}{2} I_0\).
The phenomenon where light becomes dimmer because of obstruction or absorption is a critical concept for understanding how light behaves in different conditions. Understanding this decay helps us determine how thick the obstacle, like a dust cloud, might be.
Absorption Coefficient
The absorption coefficient is a measure of how much light is absorbed by a material as it passes through. With units typically given in terms of distance like per light-year, it denotes how quickly the light intensity decreases. In our exercise, the absorption coefficient \(k\) is given as 0.1 per light-year.
  • This coefficient is essential in determining how transparent or opaque a medium is.
  • A high absorption coefficient means more light is absorbed, resulting in a significant decrease in light intensity.
  • By using the absorption coefficient in equations, we can predict how much of the light will be absorbed over a given distance through the material.
The relationship between absorption coefficient and light intensity loss is mathematically described using exponential functions, which provides a way to calculate the impact on intensity when passing through a material like the dust cloud.
Natural Logarithms
Natural logarithms are logarithms with a base of \(e\), which is approximately 2.71828. They are fundamental in various areas of mathematics and physics due to their unique properties and natural occurrence in growth and decay processes.
  • In the context of light absorption, the natural logarithm helps solve exponential decay equations.
  • When light intensity after passing through a medium is modeled by \(I = I_0 \cdot e^{-kx}\), the natural logarithm is used to linearize the equation, making it easier to solve.
  • In our problem, by taking the natural logarithm on both sides of the equation \(\frac{1}{2} = e^{-0.1x}\), it simplifies the calculation by allowing us to isolate \(x\), the thickness of the dust cloud.
Understanding natural logarithms is crucial for dealing with exponential changes since they make complex exponential decay problems more manageable to solve.

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