Chapter 38: Q82P (page 1185)

Derive Eq. 38-11, the equation for the Compton shift, from Eqs. 38-8, 38-9, and 38-10 by eliminating v and $θ$ .

Short Answer

Expert verified

The equation 38-11 is derived as follows:

$λ^{'} - λ = Δ λ = \frac{h}{m c} (1 - \cos ϕ)$

Step by step solution

Introduction

Consider the equation 38-8

$\frac{h}{m} (\frac{1}{λ} - \frac{1}{λ'}) + 1 = \frac{v}{\sqrt{1 - {(\frac{v}{c})}^{2}}}$

Consider the equation 38-9

$\frac{h}{m λ} - \frac{h}{m λ'} \cos ϕ = \frac{v}{\sqrt{1 - {(\frac{v}{})}^{2}}} \cos θ$

Consider the equation 38-10

$\frac{h}{m λ'} \sin ϕ = \frac{v}{\sqrt{1 - {(\frac{v}{c})}^{2}}} \sin θ$

Determine the derivation

On squaring and adding the Equation 38-9 and Equation 38-10. From the equation $\sin^{2} θ + \cos^{2} θ = 1$ solve as:

${(\frac{h}{m})}^{2} [{(\frac{1}{λ} - \frac{1}{λ'} \cos ϕ)}^{2} + {(\frac{1}{λ'} \sin ϕ)}^{2}] = \frac{v^{2}}{1 - {(\frac{v}{c})}^{2}} {(\frac{h}{m})}^{2} [{(\frac{1}{λ} - \frac{1}{λ'} \cos ϕ)}^{2} + {(\frac{1}{λ'} \sin ϕ)}^{2}] = - c^{2} (1 - \frac{1}{1 - {(\frac{v}{c})}^{2}}) {(\frac{h}{m c})}^{2} [{(\frac{1}{λ} - \frac{1}{λ'} \cos ϕ)}^{2} + {(\frac{1}{λ'} \sin ϕ)}^{2}] + 1 = (\frac{1}{1 - {(\frac{v}{c})}^{2}})$

Now squaring the Equation 38-8, and on comparing the above equation.

${(\frac{h}{m c} (\frac{1}{λ} - \frac{1}{λ'}) + 1)}^{2} = (\frac{v^{2}}{1 - {(\frac{v}{c})}^{2}}) {(\frac{h}{m c} (\frac{1}{λ} - \frac{1}{λ'}) + 1)}^{2} = {(\frac{h}{m c})}^{2} [{(\frac{1}{λ} - \frac{1}{λ'} \cos ϕ)}^{2} + {(\frac{1}{λ'} \sin ϕ)}^{2}] + 1$

On solving the above equation we will get this equation that is known as Equation 38-11

$λ' - λ = ∆ λ = \frac{h}{m c} (1 - \cos ϕ)$

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Derive Eq. 38-11, the equation for the Compton shift, from Eqs. 38-8, 38-9, and 38-10 by eliminating v and $θ$ .

Short Answer

Step by step solution

Introduction

Determine the derivation

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Derive Eq. 38-11, the equation for the Compton shift, from Eqs. 38-8, 38-9, and 38-10 by eliminating v and θ.

Short Answer

Step by step solution

Introduction

Determine the derivation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Torque and Rotational Motion

Conservation of Energy and Momentum

Particle Model of Matter

Mechanics and Materials

Rotational Dynamics

Circular Motion and Gravitation

Study anywhere. Anytime. Across all devices.

Derive Eq. 38-11, the equation for the Compton shift, from Eqs. 38-8, 38-9, and 38-10 by eliminating v and $θ$ .