Problem 58 - Algebra and Trigonometry with Analytic Geometry

Question

The energy \(E(x)\) of an electron after passing through material of thickness \(x\) is given by the equation \(E(x)=E_{0} e^{-x / x_{0}}\), where \(E_{0}\) is the initial energy and \(x_{0}\) is the radiation length. (a) Express, in terms of \(E_{0}\), the energy of an electron after it passes through material of thickness \(x_{0}\). (b) Express, in terms of \(x_{0}\), the thickness at which the electron loses \(99 \%\) of its initial energy.

Step-by-Step Solution

Answer

(a) \(E_0 e^{-1}\); (b) \(4.605 x_0\).

Step 1: Analyze Part (a) Expression

For part (a), substitute the given thickness \(x = x_0\) into the equation for energy \(E(x)\). This yields:\[E(x_0) = E_0 e^{-x_0 / x_0} = E_0 e^{-1}.\]Since \(e^{-1}\) simplifies to \(\frac{1}{e}\), the expression for the energy \(E\) after passing through material of thickness \(x_0\) is:\[E(x_0) = \frac{E_0}{e} = E_0 e^{-1}.\]

Step 2: Analyze Part (b) Equation Setup

For part (b), recognize that losing \(99\%\) of the initial energy \(E_0\) means the remaining energy is \(1\%\) or \(0.01 E_0\). Set up the equation:\[0.01 E_0 = E_0 e^{-x/x_0}.\]Divide both sides by \(E_0\) to simplify:\[0.01 = e^{-x/x_0}.\]

Step 3: Solve for Thickness in Part (b)

To isolate \(x/x_0\), take the natural logarithm of both sides:\[\ln(0.01) = -\frac{x}{x_0}.\]Solve for \(x\):\[\frac{x}{x_0} = -\ln(0.01).\]Since \(\ln(0.01) = -4.605\) (approximate value), this implies:\[\frac{x}{x_0} = 4.605.\]Thus, the thickness \(x\) at which the electron loses \(99\%\) of its initial energy is:\[x = 4.605 x_0.\]

Key Concepts

Exponential Decay

Exponential decay is a concept in which a quantity decreases at a rate that is proportional to its current value. It is a common physical phenomenon seen in many areas, such as radioactive decay or attenuation of light through a medium. When we see exponential decay, it follows this general form:

• \( y = y_0 e^{-kt} \)

In this equation, \( y_0 \) is the initial amount or quantity, \( k \) is a positive constant, and \( t \) is the time or, in this case, thickness of material the electron passes through. Exponential decay signifies that, with an increase in time or thickness, the quantity rapidly decreases.
In the context of the exercise, the energy \( E(x) \) of an electron as it travels through a material decreases exponentially, meaning the thicker the material, the more substantial the drop in energy. Here, the constant \( x_0 \), known as the radiation length, determines how quickly this energy is reduced as the electron passes through the material.

Radiation Length

Radiation length is a crucial concept in understanding how beams of particles, such as electrons, interact with matter. It is defined as the distance a high-energy electron travels through a material before its energy is reduced to \( 1/e \) or approximately \( 37\% \) of its initial energy. Radiation length is not only intrinsic to the material but also varies depending on the electron's properties.

• \( x_0 \) symbolizes the radiation length.
• In many applications, it is a parameter used to quantify how easily a material can "stop" a beam of particles.

In the problem, the radiation length \( x_0 \) is used to compute the thickness needed for energy attenuation. Essentially, every radiation length absorbed reduces the particle's energy to around \( 37\% \) of its previous value. For example, when an electron passes through a thickness \( x_0 \) of the material, its energy reduces to \( E_0 e^{-1} \), showing a direct application of exponential decay.

Natural Logarithm

The natural logarithm, typically represented as \( \ln \), is a critical concept when dealing with exponential equations. It is the inverse function of the exponential function \( e^x \). Natural logarithms are particularly useful for solving equations where the unknown variable is an exponent.

• \( \ln(e^x) = x \)
• It is often used to linearize exponential equations, making them easier to solve.

In the given exercise, the use of the natural logarithm provides a way to solve for thickness \( x \) when the electron's energy needs to decrease to a certain level. By taking the natural logarithm of both sides of an equation like \( 0.01 = e^{-x/x_0} \), we can transform the problem into a linear form, allowing straightforward algebraic manipulation to find \( x \). This conversion is crucial for interpreting problems involving exponential decay, as it provides the tools needed to isolate variables effectively.