an electron beam emerges from an accelerator with K.E of 100eV. what is its de- broglie wavelength?

To find the de Broglie wavelength of an electron beam with a kinetic energy of 100 eV, we can use the de Broglie wavelength formula, which relates the wavelength to the momentum of a particle. The formula is given by:

De Broglie Wavelength Formula

The de Broglie wavelength (λ) is calculated using the equation:

λ = h / p

where:

• h is Planck's constant (approximately 6.626 x 10^-34 Js)
• p is the momentum of the particle

Calculating Momentum

First, we need to determine the momentum of the electron. The kinetic energy (K.E) of the electron is given by:

K.E = (1/2)mv²

However, for relativistic particles or when dealing with electrons at higher energies, it is often more convenient to use the relationship between kinetic energy and momentum:

K.E = p² / (2m)

Rearranging this gives us:

p = √(2m * K.E)

Mass of the Electron

The mass of an electron (m) is approximately 9.11 x 10^-31 kg. We also need to convert the kinetic energy from electron volts to joules. Since 1 eV = 1.602 x 10^-19 J, we have:

K.E = 100 eV = 100 x 1.602 x 10^-19 J = 1.602 x 10^-17 J

Calculating the Momentum

Now we can substitute the values into the momentum formula:

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

Calculating this gives:

p ≈ √(2.917 x 10^-47 kg·J) ≈ 5.4 x 10^-24 kg·m/s

Finding the De Broglie Wavelength

Now that we have the momentum, we can find the de Broglie wavelength:

λ = h / p = (6.626 x 10^-34 Js) / (5.4 x 10^-24 kg·m/s)

Calculating this yields:

λ ≈ 1.23 x 10^-10 m

Final Result

Thus, the de Broglie wavelength of the electron beam with a kinetic energy of 100 eV is approximately 1.23 x 10^-10 meters, or about 0.123 nanometers. This wavelength is on the order of atomic dimensions, which is why electrons can exhibit wave-like behavior in quantum mechanics.