Question

What will be the wavelength of an electron moving with \(\frac{1}{10}\) th of velocity of light? (a) \(2.43 \times 10^{-11} \mathrm{~m}\) (b) \(243 \times 10^{-11} \mathrm{~m}\) (c) \(0.243 \mathrm{~m}\) (d) \(2.43 \times 10^{-4} \mathrm{~m}\)

Short answer

The wavelength of an electron moving with \(\frac{1}{10}\)th of the velocity of light is \(2.43 \times 10^{-11} \mathrm{~m}\).

Step-by-step solution

Understand the concept of de Broglie wavelength

The de Broglie wavelength \(\lambda\) is the wavelength associated with a particle and is given by the formula \(\lambda = \frac{h}{p}\), where \(h\) is the Planck constant \(6.62607015 \times 10^{-34} \mathrm{m^2kg/s}\) and \(p\) is the momentum of the particle.

Calculate the momentum of the electron

The momentum \(p\) of an electron moving with velocity \(v\) is equal to the mass \(m\) of the electron times its velocity \(v\). Given that \(v = \frac{1}{10}c\), where \(c\) is the speed of light \(3 \times 10^8 \mathrm{m/s}\), and the mass of the electron \(m_e\) is approximately \(9.109 \times 10^{-31} \mathrm{kg}\), we can calculate \(p = m_ev\).

Calculate the electron's velocity

The electron's velocity is \(\frac{1}{10}\) times the speed of light, therefore \(v = \frac{1}{10}\times 3 \times 10^8 = 3 \times 10^7 \mathrm{m/s}\).

Plug in the values to find the de Broglie wavelength

Using the values from the previous steps, substitute \(m_e\) and \(v\) into the momentum equation and then use the de Broglie formula: \[\lambda = \frac{h}{p} = \frac{h}{m_ev} = \frac{6.62607015 \times 10^{-34}}{9.109 \times 10^{-31} \times 3 \times 10^7}\]Calculate the value to find the de Broglie wavelength of the electron.

Perform the calculation

Solve the equation from the previous step to find the de Broglie wavelength: \[\lambda = \frac{6.62607015 \times 10^{-34}}{9.109 \times 10^{-31} \times 3 \times 10^7} = 2.43 \times 10^{-11} \mathrm{m}\]The wavelength of the electron is therefore approximately \(2.43 \times 10^{-11} \mathrm{m}\).

Key concepts

Momentum of an Electron

When we talk about the momentum of an electron, we're referring to the movement and mass of the electron as it behaves both as a particle and a wave, an idea central to quantum mechanics. In classical physics, momentum is simply the product of an object's mass and its velocity, described by the equation

\( p = m \times v \).

For electrons, which have a very small mass of approximately \( 9.109 \times 10^{-31} \) kilograms, their momentum becomes significant when moving at high velocities. The higher an electron's velocity, the greater its momentum. Remarkably, this concept allows us to connect an electron's physical movement to its wave-like properties through the de Broglie wavelength equation.

By understanding the momentum, we can dive deeper into the nature of electrons and their behavior on a subatomic level, where they exhibit properties that challenge our classical understanding of physics.

Electron Velocity

The velocity of an electron can greatly vary depending on the forces acting upon it. In our study of elementary particles like electrons, velocity not only denotes speed but also helps in understanding kinetic energy and other wave-like behaviors. For example, the velocity of an electron orbiting an atom can significantly differ from one that is being accelerated in a vacuum.

Within the context of the de Broglie wavelength calculation, electron velocity is especially crucial. As the exercise illustrates, an electron moving at \( \frac{1}{10} \) the speed of light—an exceptional speed in human terms, yet completely plausible on the quantum scale—greatly influences the wavelength associated with that electron. In essence, knowing the electron's velocity allows us to calculate its momentum, which in turn determines the de Broglie wavelength, bridging the gap between particle physics and wave phenomena.

Planck Constant

The Planck constant is a fundamental quantity in quantum mechanics, symbolized by 'h', with a value of \( 6.62607015 \times 10^{-34} \) joule seconds. It is a pivotal factor in quantifying the relationship between energy and frequency of a photon, and it plays a starring role in the renowned equation

\( E = h \times u \),

where \(E\) is energy, and \(u\) is the frequency. This constant also is essential when examining the de Broglie hypothesis, which posits that all matter exhibits wave-like properties. The hypothesis utilizes the Planck constant as a proportionality factor in the de Broglie wavelength formula \( \lambda = \frac{h}{p} \), where \(\lambda\) is the wavelength and \(p\) is the momentum of the particle. The Planck constant's small magnitude underscores the minute scale of quantum phenomena, bridging our macroscopic and quantum understanding of the physical world.