Derive an expression for the de Broglie Wavelength of an electron moving under potential difference of V volt.

To derive the expression for the de Broglie wavelength of an electron moving under a potential difference of V volts, we first need to understand the relationship between the kinetic energy gained by the electron and the potential difference. When an electron is accelerated through a potential difference, it gains kinetic energy equal to the work done on it by the electric field.

Kinetic Energy of the Electron

The kinetic energy (KE) acquired by the electron can be expressed as:

KE = eV

Here, e is the charge of the electron (approximately \(1.6 \times 10^{-19}\) coulombs), and V is the potential difference in volts. This equation tells us that the kinetic energy of the electron is directly proportional to the potential difference through which it is accelerated.

Relating Kinetic Energy to Momentum

Next, we need to relate the kinetic energy to the momentum of the electron. The momentum (p) of an object is given by the equation:

p = mv

where m is the mass of the electron and v is its velocity. However, for an electron moving at significant fractions of the speed of light, we should consider relativistic effects. For simplicity, we can start with the classical approximation, which is valid at lower speeds.

Finding the Velocity

From the kinetic energy equation, we can express the velocity of the electron. The kinetic energy can also be written in terms of momentum:

KE = \frac{p^2}{2m}

Setting the two expressions for kinetic energy equal gives:

eV = \frac{p^2}{2m}

Rearranging this equation to solve for momentum, we find:

p = \sqrt{2meV}

De Broglie Wavelength

Now that we have an expression for momentum, we can use de Broglie's hypothesis, which states that the wavelength (λ) associated with a particle is inversely proportional to its momentum:

λ = \frac{h}{p}

where h is Planck's constant (approximately \(6.626 \times 10^{-34}\) Js). Substituting our expression for momentum into this equation gives:

λ = \frac{h}{\sqrt{2meV}}

Final Expression

Thus, the de Broglie wavelength of an electron moving under a potential difference of V volts is:

λ = \frac{h}{\sqrt{2meV}}

Summary

This expression shows how the wavelength of an electron is influenced by its kinetic energy, which is determined by the potential difference through which it has been accelerated. As the potential difference increases, the kinetic energy increases, leading to a shorter wavelength, which is a fundamental concept in quantum mechanics, illustrating the wave-particle duality of matter.