When an electron is accelerated by a voltage difference, the...

When an electron is accelerated by a voltage difference, the kinetic energy acquired by the electron equals the voltage times the charge on the electron. Thus, one volt imparts a kinetic energy of $1.602 \times 10^{-19}$ volt-coulombs, or $1.602 \times$ $10^{-19} \mathrm{~J}$. What is the wavelength for electrons accelerated by $1.00 \times 10^{4}$ volts?

Key Concepts

Voltage-Induced Kinetic Energy

This concept explains that when a charged particle such as an electron is accelerated by a voltage difference, the work done on the electron (i.e., the energy imparted to it) is given by the product of the charge and the voltage. This energy is converted into kinetic energy, meaning that as electrons are accelerated, their speed increases as a result of the energy given to them by the electric field.

De Broglie Wavelength

The de Broglie wavelength is a central idea in quantum mechanics which posits that particles such as electrons exhibit wave-like properties. According to de Broglie, the wavelength of a particle is inversely proportional to its momentum, and this principle is fundamental in explaining phenomena like electron diffraction and interference.

Kinetic Energy-Momentum Relationship

This key concept connects classical mechanics to quantum ideas by expressing the relationship between a particle's kinetic energy and its momentum. Specifically, for non-relativistic speeds, the kinetic energy can be written as (1/2)mv², and the momentum is given by mv. This allows one to relate kinetic energy to momentum through the formula p = ?(2mK), which is essential for calculating the de Broglie wavelength.

Planck's Constant

Planck's constant is a fundamental physical constant that appears in many quantum mechanical equations, including the expression for the de Broglie wavelength. It sets the scale at which quantum effects become significant and is a key factor in quantifying the wave-like behavior of particles. Its presence in the wavelength formula underscores the quantization of energy and momentum in the microscopic world.

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