Chapter 6: Problem 113

What is the wavelength of an electron that is moving at a \(3 \%\) of the speed of light?

Short Answer

Expert verified

The wavelength of the electron moving at 3% of the speed of light is \( 8 \times 10^{-11} \) m.

Step by step solution

Calculate the Speed of The Electron

First, calculate the speed of the electron, given that it is 3% of the speed of light. The speed of light, labeled \( c \), is approximately \( 3 \times 10^8 \) meters per second (m/s). Therefore, the speed (v) of the electron is \( 0.03 \times c = 0.03 \times 3 \times 10^8 \; m/s = 9 \times 10^6 \; m/s \).

Use de Broglie's formula

Next, apply de Broglie's formula which is \( \lambda = \frac{h}{mv} \), where \( \lambda \) is the wavelength, \( h \) is Planck’s constant, \( m \) is the mass of an electron, and \( v \) is the speed of the electron. Plug the known values into this formula. The mass of an electron is \( 9.1 \times 10^{-31} \) kg and Planck's constant is \( 6.62607015 \times 10^{-34} \; m^2 \cdot kg/s \).

Calculate the Wavelength

Now, substitute the values into the formula: \( \lambda = \frac{6.62607015 \times 10^{-34}}{ 9.1 \times 10^{-31} \times 9 \times 10^6} = 8 \times 10^{-11} \; m \). So the wavelength of the electron moving with 3% the speed of light is \( 8 \times 10^{-11} \; m \).

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What is the wavelength of an electron that is moving at a \(3 \%\) of the speed of light?

Short Answer

Step by step solution

Calculate the Speed of The Electron

Use de Broglie's formula

Calculate the Wavelength

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