Calculate the de-Broglie wavelength of (i) an electron accelerated by a potential difference of 100V and (ii) a particle of mass 0.03 kg moving with a speed of 100m/s Hence show that . wavelength of the particle is not relevant.

To calculate the de Broglie wavelength for both scenarios, we can use the de Broglie wavelength formula, which is given by:

λ = h / p

Here, λ is the wavelength, h is Planck's constant (approximately 6.626 x 10^-34 Js), and p is the momentum of the particle. The momentum (p) can be expressed as the product of mass (m) and velocity (v), so p = mv. Let's break down the calculations for both the electron and the particle step by step.

1. Calculating the de Broglie Wavelength of an Electron

For an electron accelerated through a potential difference (V) of 100 volts, we first need to determine its kinetic energy (KE), which is given by:

KE = eV

Where e is the charge of the electron (approximately 1.602 x 10^-19 coulombs). Thus, the kinetic energy becomes:

KE = (1.602 x 10^-19 C)(100 V) = 1.602 x 10^-17 J

Next, we can relate the kinetic energy to momentum. The kinetic energy can also be expressed in terms of momentum:

KE = p² / (2m)

Rearranging this gives us:

p = √(2mKE)

For an electron, the mass (m) is approximately 9.11 x 10^-31 kg. Plugging in the values:

p = √(2 * (9.11 x 10^-31 kg) * (1.602 x 10^-17 J))

Calculating this yields:

p ≈ 5.34 x 10^-24 kg·m/s

Now, substituting this value into the de Broglie wavelength formula:

λ = h / p = (6.626 x 10^-34 Js) / (5.34 x 10^-24 kg·m/s)

Calculating this gives:

λ ≈ 1.24 x 10^-10 m

2. Calculating the de Broglie Wavelength of a Particle

Now, let's consider a particle with a mass of 0.03 kg moving at a speed of 100 m/s. First, we calculate its momentum:

p = mv = (0.03 kg)(100 m/s) = 3 kg·m/s

Next, we can find the de Broglie wavelength using the same formula:

λ = h / p = (6.626 x 10^-34 Js) / (3 kg·m/s)

Calculating this gives:

λ ≈ 2.21 x 10^-34 m

Comparative Analysis of Wavelengths

Now, let's compare the two wavelengths:

• The wavelength of the electron is approximately 1.24 x 10^-10 m.
• The wavelength of the 0.03 kg particle is approximately 2.21 x 10^-34 m.

From this comparison, it is evident that the wavelength of the particle (0.03 kg) is significantly smaller than that of the electron. In fact, the wavelength of the heavier particle is so small that it becomes negligible in practical terms. This illustrates a key concept in quantum mechanics: as the mass of a particle increases, its de Broglie wavelength decreases, making wave-like behavior less observable.

In summary, while the electron exhibits a measurable wavelength due to its small mass and high velocity, the wavelength of the 0.03 kg particle is so minuscule that it is not relevant for most practical applications. This highlights the importance of mass and speed in determining the wave-particle duality of matter.