To what speed must an electron be accelerated for it to have a wavelenght of 3.0cm?
To what speed must an electron be accelerated for it to have a de Broglic
wavelength of 5.0 cm
The "de Broglie wavelength" of an object with mass m and velocity v is
lambda = h/(m v)
where h is Planck's constant and m is the mass. This assumes that v is much less than c; otherwise use the relativistc momentum in the denominator.
Rearrange that equation and solve for velocity. I assume you know the values for h and the elctron mass.
To calculate the speed at which an electron must be accelerated to have a specific wavelength, we can use the de Broglie wavelength equation. The equation relates the wavelength (λ) of a particle to its momentum (p) and mass (m). Here is the formula:
λ = h / p
Where:
λ is the wavelength
h is the Planck's constant (6.62607015 × 10^-34 J·s)
p is the momentum of the particle
Since we are given the wavelength (λ) of 3.0 cm, we need to convert it to meters.
1 cm = 0.01 meters
So, λ = 3.0 cm = 0.03 meters
Now, we can rearrange the equation to solve for the momentum (p) using the following steps:
1. Substitute the term for λ in the equation:
0.03 meters = h / p
2. Rearrange the equation to solve for momentum (p):
p = h / 0.03 meters
3. Plug in the values for Planck's constant (h):
p = (6.62607015 × 10^-34 J·s) / 0.03 meters
4. Calculate the momentum (p):
p ≈ 2.209 x 10^-32 kg·m/s
Now, we can find the speed of the electron by using the momentum (p) and the mass of an electron (m). The equation for speed (v) is:
v = p / m
The mass of an electron is approximately 9.10938356 × 10^-31 kg. We can now calculate the speed:
v = (2.209 x 10^-32 kg·m/s) / (9.10938356 × 10^-31 kg)
v ≈ 0.024 m/s
Therefore, the electron must be accelerated to a speed of approximately 0.024 m/s to have a wavelength of 3.0 cm.