Question

What is the wavelength of an electron that is accelerated from rest through a potential difference of \(1.00 \cdot 10^{-5} \mathrm{~V} ?\)

Short answer

Answer: The wavelength of the electron is approximately \(1.08 \cdot 10^{-9}\mathrm{~m}\).

Step-by-step solution

Identify given information

We are given the potential difference that accelerates the electron: \(V = 1.00 \cdot 10^{-5} \mathrm{~V}\). Also, remember the charge of an electron: \(e = 1.60 \cdot 10^{-19} \mathrm{C}\), and its mass: \(m_e = 9.11 \cdot 10^{-31} \mathrm{kg}\).

Calculate the energy gained by the electron

The energy gained by the electron when accelerated from rest through a potential difference (in Joules) is given by the formula: \(E = eV\) Substituting the values, we get \(E = (1.60 \cdot 10^{-19} \mathrm{C})(1.00 \cdot 10^{-5} \mathrm{V}) = 1.60 \cdot 10^{-24} \mathrm{J}\)

Calculate the final velocity of the electron

Since the energy gained by the electron results in its kinetic energy, we can use the formula for kinetic energy: \(E = \frac{1}{2}m_e v^2\) Solving for \(v\), we get: \(v = \sqrt{\frac{2E}{m_e}}\) Now substitute the values for \(E\) and \(m_e\): \(v = \sqrt{\frac{2(1.60 \cdot 10^{-24} \mathrm{J})}{(9.11 \cdot 10^{-31} \mathrm{kg})}} \approx 6.62 \cdot 10^5 \mathrm{m/s}\)

Use de Broglie's formula to find the wavelength

The de Broglie wavelength formula is given by: \(\lambda = \frac{h}{p}\) where \(\lambda\) is the wavelength, \(h\) is Planck's constant (\(h = 6.63 \cdot 10^{-34} \mathrm{Js}\)), and \(p\) is the momentum of the electron. Since the momentum can be denoted as \(p = m_ev\), we can write the formula as: \(\lambda = \frac{h}{m_ev}\) Now, substitute the values for \(h\), \(m_e\), and \(v\): \(\lambda = \frac{6.63 \cdot 10^{-34} \mathrm{Js}}{(9.11 \cdot 10^{-31} \mathrm{kg})(6.62 \cdot 10^5 \mathrm{m/s})} \approx 1.08 \cdot 10^{-9} \mathrm{m}\)

Express the result

The wavelength of the electron that is accelerated from rest through a potential difference of \(1.00 \cdot 10^{-5}\mathrm{~V}\) is approximately \(1.08 \cdot 10^{-9}\mathrm{~m}\).

Key concepts

Electron Wavelength

When we think of waves, we might imagine ripples across a pond or the vibration of a guitar string, but what about the wave-like nature of particles? A key concept in quantum mechanics, originally introduced by physicist Louis de Broglie, is that particles can exhibit wave-like properties. This is a fundamental aspect of nature at small, atomic scales.

The electron wavelength, often referred to as the de Broglie wavelength, pertains to this wave-particle duality. It is the wavelength associated with a moving electron and can be calculated when the electron's momentum is known. The de Broglie wavelength is significant because it provides a way to predict the behavior of particles in quantum mechanics, such as how they spread out or interfere with one another in the same manner that waves do.

For instance, when an electron passes through a narrow aperture, it creates an interference pattern on a screen placed behind the aperture, much like light waves do. This phenomenon is one of the many demonstrations of quantum mechanics in action—an area of physics that may seem counterintuitive to the experiences in our macroscopic world.

Potential Difference

In physics, the concept of potential difference is pivotal for understanding electrical circuits and the behavior of particles within an electric field. Potential difference is the work done per unit charge to move a charge from one point to another in a field, such as an electric field. The unit of measurement for potential difference is the volt (V), with one volt being equivalent to one joule per coulomb.

In the context of our exercise, an electron is released from rest and accelerates due to the potential difference applied to it. This acceleration causes the electron to gain kinetic energy, which consequently affects its momentum and thus, its wavelength. Understanding potential difference is critical in various applications ranging from the basic operation of household electronics to the intricate workings of particle accelerators. It is also a key factor in calculating the energy gained by an electron when accelerated, which directly links to the de Broglie wavelength calculation.

Planck's Constant

The constant of proportion between the energy and frequency of a photon is known as Planck's constant, symbolized by the letter h. Max Planck first introduced this fundamental constant of nature, and it plays an integrating role in quantum mechanics. Its value is approximately 6.63 x 10^-34 joule seconds (Js).

Planck's constant is essential in the realm of quantum mechanics because it sets the scale at which quantum effects become significant. Interestingly, it is also a central component in the equation used to calculate the de Broglie wavelength. For any particle with momentum, including an electron, the equation \( \lambda = \frac{h}{p} \) applies, where \( \lambda \) is the de Broglie wavelength, \( h \) is Planck's constant, and \( p \) is the particle's momentum. This relationship highlights the inseparable linkage between energy (or momentum) and wave-like behavior at the quantum level and is another example of the beautifully complex tapestry woven by the laws of physics.