Chapter 3: Q38E (page 94)

From equations (3-4) to (3-6) obtain equations (3.8). It is easiest to start by eliminating φ between equations (3-4) and (3-5) using cos²φ+sin²φ = 1 The electron speed u may then be eliminated between the remaining equations.

Short Answer

Expert verified

Using the given equations, the following can be obtained is $λ' - λ = \frac{h}{m_{e} c} (1 - \cos θ)$ .

Step by step solution

Given data

$\frac{h}{λ} = \frac{h}{λ'} \cos θ + γ_{u} m_{e} u \cos ϕ \dots . . (1) 0 = \frac{h}{λ'} \sin θ - γ_{u} m_{e} u \sin ϕ \dots . . (2) \frac{h c}{λ} - m_{e} c^{2} = \frac{h c}{λ'} + γ_{u} m_{e} c^{2} \dots . . (3)$

Concept used

Einstein's mass-energy equivalence relation can be expressed as,

E = mc²

Use the equations and solve

Rearranging the above equations (1 and 2) will give:

$γ_{u} m_{e} u \cos ϕ = \frac{h}{λ} - \frac{h}{λ'} \cos θ γ_{u} m_{e} u \sin ϕ = \frac{h}{λ'} \sin θ$

Squaring both will give:

${(γ_{u} m_{e} u)}^{2} \cos^{2} ϕ = {(\frac{h}{λ} - \frac{h}{λ'} \cos θ)}^{2} {(γ_{u} m_{e} u)}^{2} \sin^{2} ϕ = {(\frac{h}{λ'} \sin θ)}^{2}$

Adding both equations will lead to:

${(γ_{u} m_{e} u)}^{2} = {(\frac{h}{λ})}^{2} - \frac{2 h^{2}}{λ λ'} \cos θ + {(\frac{h}{λ'})}^{2}$ …..(4)

The Square of equation 3 will give

${(γ_{u} m_{e} u)}^{2} = {(\frac{h}{λ})}^{2} + {(\frac{h}{λ})}^{2} - \frac{2 h^{2}}{λ λ'} + 2 m_{e} c (\frac{h}{λ} - \frac{h}{λ'}) + m_{e}^{2} c^{2}$ …..(5)

Subtract equation 5 from equation 4

Simplify further,

$\begin{array}{rcl} m_{e}^{2} (γ_{u}^{2} c^{2} - γ_{u}^{2} c^{2}) & = & - \frac{2 h^{2}}{λ λ'} (1 - \cos θ) + 2 m_{e} c (\frac{h}{λ'} - \frac{h}{λ}) + {(m_{e} c)}^{2} \\ \frac{2 h^{2}}{λ λ'} (1 - \cos θ) & = & 2 m_{e} c (\frac{h}{λ'} - \frac{h}{λ}) \\ \frac{h}{λ λ'} (1 - \cos θ) & = & m_{e} c (\frac{1}{λ'} - \frac{1}{λ}) \\ λ - λ' & = & \frac{h}{m_{e} c} (1 - \cos θ) \end{array}$

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From equations (3-4) to (3-6) obtain equations (3.8). It is easiest to start by eliminating φ between equations (3-4) and (3-5) using cos²φ+sin²φ = 1 The electron speed u may then be eliminated between the remaining equations.

Short Answer

Step by step solution

Given data

Concept used

Use the equations and solve

Subtract equation 5 from equation 4

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From equations (3-4) to (3-6) obtain equations (3.8). It is easiest to start by eliminating φ between equations (3-4) and (3-5) using cos2φ+sin2φ = 1 The electron speed u may then be eliminated between the remaining equations.

Short Answer

Step by step solution

Given data

Concept used

Use the equations and solve

Subtract equation 5 from equation 4

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Nuclear Physics

Turning Points in Physics

Electric Charge Field and Potential

Magnetism and Electromagnetic Induction

Astrophysics

Torque and Rotational Motion

Study anywhere. Anytime. Across all devices.

From equations (3-4) to (3-6) obtain equations (3.8). It is easiest to start by eliminating φ between equations (3-4) and (3-5) using cos²φ+sin²φ = 1 The electron speed u may then be eliminated between the remaining equations.