Question

An electron is accelerated through an electric potential to a kinetic energy of \(18.6 \mathrm{keV}\). What is its characteristic wavelength? [Hint: Recall that the kinetic energy of a moving object is \(E=\frac{1}{2} m v^{2}\), where \(m\) is the mass of the object and \(\nu\) is the speed of the object.]

Short answer

The characteristic wavelength of the electron is approximately \(2.816 × 10^{-11}\) meters.

Step-by-step solution

Convert the kinetic energy into SI units

The given kinetic energy is 18.6 keV, which should be converted to SI units (joules). To do this, multiply the given energy by \(1.6 × 10^{-19}\) J/eV. \[18.6\,\text{keV} × 1.6 × 10^{-16}\,\text{J/keV} = 2.976 × 10^{-15}\,\text{J}\]

Calculate the speed of the electron

We can find the speed of the electron by rearranging the kinetic energy formula: \[v = \sqrt{\frac{2E}{m}}\] For an electron, the mass \(m\) is approximately \(9.11 × 10^{-31}\) kg. Plug in the mass and the energy we calculated in Step 1: \[v = \sqrt{\frac{2 × 2.976 × 10^{-15}\,\text{J}}{9.11 × 10^{-31}\,\text{kg}}} = 2.580 × 10^7\,\text{m/s}\]

Calculate the momentum of the electron

The momentum of the electron can be calculated using the formula \(p = mv\). Substitute the mass and velocity values into the formula: \[p = (9.11 × 10^{-31}\,\text{kg}) (2.580 × 10^7\,\text{m/s}) = 2.353 × 10^{-23}\,\text{kg m/s}\]

Calculate the characteristic wavelength using de Broglie's formula

Now we can determine the characteristic wavelength using the de Broglie formula \(\lambda = \frac{h}{p}\), where \(h\) is Planck's constant \((6.626 × 10^{-34}\,\text{Js})\). Substitute the values into the formula: \[\lambda = \frac{6.626 × 10^{-34}\,\text{Js}}{2.353 × 10^{-23}\,\text{kg m/s}} = 2.816 × 10^{-11}\,\text{m}\] The characteristic wavelength of the electron is approximately \(2.816 × 10^{-11}\) meters.

Key concepts

Kinetic Energy Conversion

When dealing with physics problems involving electrons, you often need to convert kinetic energy from kiloelectron volts (keV) to joules (J). This conversion is necessary because many formulas used in physics require SI units. To convert keV to J, use the conversion factor: 1 eV = \(1.6 \times 10^{-19}\) J. For example, if an electron is accelerated to a kinetic energy of 18.6 keV, multiply by \(1.6 \times 10^{-16}\) J/keV to obtain a result of \(2.976 \times 10^{-15}\) J. This energy, now in joules, is ready to be used in further calculations.

De Broglie Wavelength

The concept of de Broglie wavelength links the wave-like behavior of particles to their momentum. Louis de Broglie proposed that every particle has an associated wavelength, which is a component of the wave-particle duality of nature. The de Broglie wavelength \(\lambda\) is given by the formula: \[\lambda = \frac{h}{p}\]where \(h\) is Planck's constant, and \(p\) denotes momentum. This formula allows you to calculate the wavelength of any particle, assuming its momentum is known. When applied to electrons, this principle reveals characteristic aspects of their motion at atomic scales. For example, for an electron with a calculated momentum \(2.353 \times 10^{-23}\, \text{kg m/s}\), its de Broglie wavelength would be \(2.816 \times 10^{-11}\, \text{m}\).

Momentum Calculation

Momentum is a crucial aspect of calculating an electron's de Broglie wavelength. In physics, momentum \(p\) is the product of an object's mass \(m\) and velocity \(v\), given by the equation:\[p = mv\]To determine an electron's momentum, first calculate its speed using the kinetic energy formula:\[v = \sqrt{\frac{2E}{m}}\]Where \(E\) is kinetic energy, and \(m\) is the electron's mass. For an electron with \(E = 2.976 \times 10^{-15}\, \text{J}\), its speed \(v\) calculates to \(2.580 \times 10^{7}\, \text{m/s}\). Multiplying by the electron's mass \(9.11 \times 10^{-31}\, \text{kg}\) results in momentum \(p = 2.353 \times 10^{-23}\, \text{kg m/s}\).

Planck's Constant

Planck's constant \(h\) plays a key role in quantum mechanics and the calculation of the de Broglie wavelength. It is a fundamental physical constant that describes the size of quanta. The value of Planck's constant is approximately \(6.626 \times 10^{-34}\, \text{Js}\), a number you’ll use frequently in calculations related to wave-particle duality.In determining the de Broglie wavelength, Planck's constant is essential. It enables us to relate a particle's wavelength to its momentum through the formula:\[\lambda = \frac{h}{p}\]In this context, Planck's constant helps bridge the gap between classical and quantum physics, showing how momentum influences wave-like properties. Understanding this constant's role is crucial for mastering quantum mechanics and comprehending the behavior of particles like electrons.