1) An electron moving with a speed of 2.84E+6 m/s has the same momentum as a photon. Calculate the de Broglie wavelength of the electron.
2) Calculate the wavelength of the photon.
To calculate the de Broglie wavelength of the electron, we can use the de Broglie equation, which states that the wavelength (λ) of a particle is inversely proportional to its momentum (p):
λ = h / p
where:
- λ is the wavelength
- h is Planck's constant (approximately 6.626 x 10^-34 J·s)
- p is the momentum
1) To find the momentum of the electron, we can use the equation:
p = m * v
where:
- p is the momentum
- m is the mass of the electron (approximately 9.109 x 10^-31 kg)
- v is the velocity of the electron
Given:
- v = 2.84 x 10^6 m/s
Calculating the momentum of the electron:
p = (9.109 x 10^-31 kg) * (2.84 x 10^6 m/s)
p = 2.59 x 10^-24 kg·m/s
Now, we can substitute the value of the momentum into the de Broglie equation:
λ = (6.626 x 10^-34 J·s) / (2.59 x 10^-24 kg·m/s)
λ ≈ 2.561 x 10^-10 m
Therefore, the de Broglie wavelength of the electron is approximately 2.561 x 10^-10 meters.
2) To calculate the wavelength of the photon, we can use the equation:
λ = c / f
where:
- λ is the wavelength
- c is the speed of light (approximately 3.00 x 10^8 m/s)
- f is the frequency of the photon
However, the frequency of the photon was not given. In order to calculate the wavelength, we need to know either the frequency or the energy of the photon. If you have the energy or frequency of the photon, please provide that information so I can help you calculate the wavelength.
To solve these questions, we can use the equation:
wavelength = h / momentum
where "wavelength" is the de Broglie wavelength, "h" is the Planck's constant (6.626E-34 J s), and "momentum" is the momentum of the particle.
1) To find the de Broglie wavelength of the electron, we can use the given information that it has the same momentum as a photon. Since momentum is conserved, we can equate the momentum of the electron to the momentum of the photon:
momentum of electron = momentum of photon
mass of electron * velocity of electron = energy of photon / speed of light
mass of electron * velocity of electron = (Plank's constant / wavelength of photon) * speed of light
(9.1E-31 kg) * (2.84E+6 m/s) = (6.626E-34 J s) / wavelength of photon
wavelength of photon = (6.626E-34 J s) / [(9.1E-31 kg) * (2.84E+6 m/s)]
Simplifying the above equation, we find:
wavelength of photon = 3.65E-9 meters or 3.65 nanometers
So, the wavelength of the photon is approximately 3.65 nanometers.
2) To calculate the wavelength of the photon, we can use the given equation:
wavelength = h / momentum
We know that the momentum of the photon is equal to the momentum of the electron, which we calculated in the previous step:
wavelength of photon = (6.626E-34 J s) / [(9.1E-31 kg) * (2.84E+6 m/s)]
Simplifying the above equation, we find:
wavelength of photon = 2.41E-9 meters or 2.41 nanometers
So, the de Broglie wavelength of the electron is approximately 2.41 nanometers.