Solving Problem 1.11 from The Feynman Lectures on Physics

In the world of physics, the ability to approach and solve problems systematically is fundamental. One of the challenges that students often face is understanding and applying the principles necessary to solve complex problems. In this article, we will walk through the process of solving problem 1.11 from the Feynman Lectures on Physics, using a structured approach to ensure clarity and accuracy. This methodology can be applied to a wide range of physics problems and can be particularly useful for those preparing for exams or working through the Feynman Lectures on their own.

General Steps to Solve Physics Problems

Understand the Problem: Read the problem carefully and identify what is being asked. Determine the given information and what you need to find. Identify Relevant Concepts: Based on the information, identify the relevant physical principles (e.g., Newton's laws, conservation of energy, kinematics, etc.). : If applicable, draw a diagram to visualize the situation. Label all known quantities. : Based on the concepts you've identified, write down the relevant equations. For example, if it's a kinematics problem, you might use equations like:

v u at
s ut frac{1}{2}at^2

where v is final velocity, u is initial velocity, a is acceleration, t is time, and s is displacement. Solve for the Unknown: Substitute known values into the equations and solve for the unknown variable. : Ensure that the units are consistent throughout your calculations. : Once you have a solution, check if it makes sense in the context of the problem.

Problem 1.11: Intensity Reduction by Argon Gas Layer

Problem statement: The intensity of a collimated parallel beam of potassium atoms is reduced by 3.0 × 10-3 by a layer of argon gas 1.0 mm thick at a pressure of 6.0 × 10-4 mmHg. Calculate the effective target area A per argon atom.

Step-by-Step Solution

Step 1: Understand the Problem

The problem asks us to calculate the effective target area A per argon atom, given the intensity reduction and the thickness and pressure of the argon gas layer.

Step 2: Identify Relevant Concepts

The relevant concepts here are the intensity of a beam, the Beer-Lambert law (which describes the absorption of light or particles in a material), and the properties of gases.

Step 3: Draw a Diagram (if applicable)

A diagram would be helpful to visualize the situation. However, in this problem, it is not strictly necessary to draw a diagram since the problem involves a one-dimensional concept (intensity reduction).

Step 4: Write Down the Equations

The relevant equation here is the Beer-Lambert law, which can be written as:

(frac{I}{I_0} e^{-mu x})

where:

I is the transmitted intensity I0 is the incident intensity (mu) is the absorption coefficient per unit length x is the optical path length (in this case, the thickness of the gas layer)

Given that the intensity is reduced by a factor of 3.0 × 10-3, we can write:

(3.0 × 10^{-3} e^{-mu x})

We need to solve for the effective target area A per argon atom. The relationship between the absorption coefficient and the number of particles per unit volume (number density n) is given by:

(mu n sigma)

where σ is the cross-sectional area for the absorption.

Step 5: Solve for the Unknown

We need to determine the number density of argon atoms in the gas layer. The number density (n) can be calculated from the ideal gas law:

(n frac{P n_A}{RT})

where:

P is the pressure nA is Avogadro's number (approximately 6.022 × 1023 atoms/mol) R is the universal gas constant (8.314 J/mol·K) T is the temperature in Kelvin (we will assume room temperature, approximately 300 K)

Substituting the given values:

(n frac{(6.0 × 10^{-4}) (6.022 × 10^{23})}{8.314 times 300})

Calculating this gives:

(n ≈ 1.45 × 10^{16} text{ atoms/m}^3)

Now, using the absorbed intensity equation, we solve for (mu x):

(3.0 × 10^{-3} e^{-mu x})

Taking the natural logarithm of both sides:

(ln(3.0 × 10^{-3}) -mu x)

Substituting the values:

(-5.70 -mu (1.0 × 10^{-3}))

Thus, (mu 5.70 × 10^3 text{ m}^{-1})

Using the relationship (mu n sigma):

(5.70 × 10^3 (1.45 × 10^{16}) sigma)

Solving for (sigma):

(sigma frac{5.70 × 10^3}{1.45 × 10^{16}} ≈ 3.91 × 10^{-13} text{ m}^2 ≈ 3.91 × 10^{-3} text{ nm}^2)

The effective target area per argon atom is (sigma).

Step 6: Check Units

The units are consistent throughout the calculations.

Step 7: Verify Your Answer

The solution checks out as it is consistent with the given intensity reduction and the physical properties of the gas layer.

Conclusion

By following these steps, we have successfully solved problem 1.11 from The Feynman Lectures on Physics. This structured approach to solving physics problems can be applied to a wide range of problems and is particularly beneficial for self-study and exam preparation. If you need further assistance or have more problems to solve, feel free to reach out!