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Chapter 10: Problem 20

What is the de Broglie wavelength? What happens to the de Broglie wavelength of an electron when its speed is increased?

Short Answer

Expert verified
Answer: When the speed of an electron is increased, its de Broglie wavelength becomes shorter.

Step by step solution

01

Deriving the de Broglie wavelength formula

According to the de Broglie hypothesis, every particle has a wave-like nature, and its wavelength is given by the de Broglie wavelength, which is related to its momentum. The momentum (p) of any particle can be calculated as the product of its mass (m) and velocity (v): p = m * v According to the Planck's relation, the momentum of a particle can be related to its wavelength (λ) using Planck's constant (h): p = h/λ Therefore, we can relate the de Broglie wavelength to the mass and velocity of a particle as: λ = h/(m * v)
02

Analyzing the effect of speed on the de Broglie wavelength

From the formula derived above, λ = h/(m * v), we can see that the de Broglie wavelength of a particle depends inversely on its velocity. If the speed of the electron is increased, then its velocity (v) will also increase. As a result, the denominator of the equation will become larger. Given that Planck's constant (h) and the mass of the electron (m) remain constant, an increase in the velocity (v) will cause a decrease in the de Broglie wavelength (λ) of the electron. This means that when the speed of the electron is increased, its de Broglie wavelength becomes shorter.

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