Chapter 38: Q. 16 (page 1114)

At what speed is an electron's de Broglie wavelength (a) $1.0 pm$ , (b) $1.0 nm$ , (c) $1.0 μ m$ , and (d) $1.0 mm$ ?

Short Answer

Expert verified

(a) $v = 7.27 \times 10^{8} m / s$

(b) $v = 2.77 \times 10^{8} m / s$

(d) $v = 0.727 m / s$

Step by step solution

Part(a) Step 1: Given Information

The electron's de Broglie wave length is $1.0 p m$

Part(a) Step 2: Calculation

We can start the solution with expression for de-Broglie wavelength

$λ = \frac{h}{m v} v = \frac{h}{m λ} v = \frac{6.63 \times 10^{- 34}}{(9.11 \times 10^{- 31}) (1 \times 10^{- 12})} v = 7.27 \times 10^{8} m / s$

This speed is larger than speed of light, so we must calculate relativistic speed:

$λ = \frac{h}{p} p = γ m v \frac{h}{λ} = γ m v γ = \frac{1}{\frac{h}{1 - \frac{v^{2}}{c^{2}}}} \frac{h}{λ} (1 - \frac{v^{2}}{c^{2}}) = \frac{1}{\sqrt{1 - \frac{v^{2}}{c^{2}}}} m v^{2} \frac{h^{2}}{λ^{2}} = v^{2} (m^{2} + \frac{h^{2}}{λ^{2} c^{2}}) v = \frac{h}{λ \cdot \sqrt{(m^{2} + \frac{h^{2}}{λ^{2} c^{2}})}} v = 2.77 \times 10^{8} m / s$

Part(b) Step 1: Given Information

The electron's de Broglie wave length is $1.0 n m$

Part(b) Step 2: Calculation

For $λ = 1 \times 10^{- 9}$ we have:

$λ = \frac{h}{m v} v = \frac{h}{m λ} v = \frac{6.63 \times 10^{- 34}}{(9.11 \times 10^{- 31}) (1 \times 10^{- 9})} v = 7.27 \times 10^{5} m / s$

Part(c) Step 1: Given Information

The electron's de Broglie wave length is $1.0 μ m$

Part(c) Step 2: Calculation

For $λ = 1 \times 10^{- 6}$ we have:

$v = \frac{h}{m λ} v = \frac{6.63 \times 10^{- 34}}{(9.11 \times 10^{- 31}) (1 \times 10^{- 6})} v = 7.27 \times 10^{2} m / s$

Part(d) Step 1: Given Information

The electron's de Broglie wave length is $1.0 m m$

Part(d) Step 2: Calculation

For $λ = 1 \times 10^{- 3}$ we have:

$v = \frac{h}{m λ} v = \frac{6.63 \times 10^{- 34}}{(9.11 \times 10^{- 31}) (1 \times 10^{- 3})} v = 0.727 m / s$

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At what speed is an electron's de Broglie wavelength (a) $1.0 pm$ , (b) $1.0 nm$ , (c) $1.0 μ m$ , and (d) $1.0 mm$ ?

Short Answer

Step by step solution

Part(a) Step 1: Given Information

Part(a) Step 2: Calculation

Part(b) Step 1: Given Information

Part(b) Step 2: Calculation

Part(c) Step 1: Given Information

Part(c) Step 2: Calculation

Part(d) Step 1: Given Information

Part(d) Step 2: Calculation

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At what speed is an electron's de Broglie wavelength (a) 1.0pm, (b) 1.0nm, (c) 1.0μm, and (d)1.0mm ?

Short Answer

Step by step solution

Part(a) Step 1: Given Information

Part(a) Step 2: Calculation

Part(b) Step 1: Given Information

Part(b) Step 2: Calculation

Part(c) Step 1: Given Information

Part(c) Step 2: Calculation

Part(d) Step 1: Given Information

Part(d) Step 2: Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Quantum Physics

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Study anywhere. Anytime. Across all devices.

At what speed is an electron's de Broglie wavelength (a) $1.0 pm$ , (b) $1.0 nm$ , (c) $1.0 μ m$ , and (d) $1.0 mm$ ?