To determine the de Broglie wavelength associated with electrons accelerated by a voltage of 50 kV, we can use the de Broglie wavelength formula, which relates the wavelength to the momentum of a particle. The formula is given by:
Calculating the de Broglie Wavelength
The de Broglie wavelength (λ) can be calculated using the equation:
λ = h / p
Where:
- h is Planck's constant (approximately 6.626 x 10-34 m2 kg/s)
- p is the momentum of the electron
First, we need to find the momentum of the electron. The kinetic energy (KE) gained by the electron when accelerated through a potential difference (V) is given by:
KE = eV
Where:
- e is the charge of the electron (approximately 1.602 x 10-19 C)
- V is the accelerating voltage (50,000 V in this case)
Substituting the values, we find:
KE = (1.602 x 10-19 C) * (50,000 V) = 8.01 x 10-15 J
Next, we can relate the kinetic energy to momentum using the equation:
KE = p² / (2m)
Where m is the mass of the electron (approximately 9.11 x 10-31 kg). Rearranging gives:
p = √(2m * KE)
Now substituting the values:
p = √(2 * (9.11 x 10-31 kg) * (8.01 x 10-15 J))
Calculating this yields:
p ≈ 1.34 x 10-24 kg m/s
Now we can find the de Broglie wavelength:
λ = h / p = (6.626 x 10-34 m2 kg/s) / (1.34 x 10-24 kg m/s)
Calculating this gives:
λ ≈ 4.94 x 10-10 m or 0.494 nm
Resolving Power Comparison
Now, let’s compare the resolving power of an electron microscope with that of an optical microscope using yellow light. The resolving power (d) can be estimated using the Rayleigh criterion:
d = 1.22 * λ / NA
Where NA is the numerical aperture of the microscope. For optical microscopes using yellow light (around 580 nm), the wavelength is much larger than that of the electrons. For example:
- Wavelength of yellow light: λ ≈ 580 nm = 5.8 x 10-7 m
- Wavelength of electrons: λ ≈ 0.494 nm = 4.94 x 10-10 m
Given that the numerical apertures of both types of microscopes can be similar, the significant difference in wavelength leads to a much higher resolving power for the electron microscope. The resolving power is inversely proportional to the wavelength, meaning that as the wavelength decreases, the resolving power increases.
In summary, the electron microscope, with its de Broglie wavelength of approximately 0.494 nm, has a resolving power that is vastly superior to that of an optical microscope using yellow light, which has a wavelength of about 580 nm. This allows electron microscopes to resolve much finer details at the atomic level, making them invaluable in fields such as materials science and biology.
