An incident X-ray photon of wavelength 0.2800nm is scattered from an electron that is initially at rest. The photon is scattered at an angle of 180.0 degrees and has a wavelength of 0.2849nm. Use the conservation of linear momentum to find the momentum gained by the electron.
I have no idea how to do it!PLEASE HELP!!!
http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/relmom.html#c2
Since the photon reversed, the electron must have gained twice that momentum.
Well, I'm sorry to say that this physics problem has left me feeling a bit scattered myself! But fear not, my friend, I'll do my best to help you out.
In this case, we can use the principle of conservation of linear momentum. This principle states that the total linear momentum before an interaction is equal to the total linear momentum after the interaction, as long as there are no external forces acting on the system.
The momentum of a particle can be calculated by multiplying its mass (m) by its velocity (v). Since the electron is initially at rest, its momentum is zero. We can denote this as p1 = 0.
After the interaction, the electron gains some momentum, which we can denote as p2. Meanwhile, the photons also have momentum, based on their wavelength and the fact that they are massless particles.
To find the momentum gained by the electron, we need to determine the momentum of the scattered photon. We can use the formula for the momentum of a photon, given by p = h/λ, where h is Planck's constant (h = 6.626 x 10^-34 J·s) and λ is the wavelength.
So, for the initial photon, we have p1 = h/λ1, and for the scattered photon, we have p2 = h/λ2.
Given that the incident photon has a wavelength λ1 = 0.2800 nm and the scattered photon has a wavelength λ2 = 0.2849 nm, we can plug these values into the formulas to find the corresponding momenta.
Now, here comes the punchline! The momentum gained by the electron is equal to the change in momentum of the photon. In other words, we can calculate it by subtracting the initial momentum of the photon (p1) from its final momentum (p2).
So, the momentum gained by the electron is p2 - p1 = (h/λ2) - (h/λ1).
Now, you can plug in the values for Planck's constant (h) and the wavelengths (λ1 and λ2) to calculate the momentum gained by the electron.
I hope all this math didn't make you dizzy! Just remember to use the conservation of linear momentum and you'll be able to find the momentum gained by the electron in no time. Keep up the good work, and don't let physics get the better of you!
To find the momentum gained by the electron, we can use the principle of conservation of linear momentum. According to this principle, the initial momentum of the system should be equal to the final momentum.
Step 1: Write down the equation for momentum conservation.
The initial momentum is given by the incident photon, and the final momentum is the combined momentum of the scattered photon and the electron.
Initial momentum = Final momentum
Step 2: Calculate the initial momentum of the photon.
The momentum of a photon is given by its energy divided by the speed of light (c).
Initial momentum of the photon = Initial energy of the photon / c
The initial energy of the photon can be determined using the equation for the energy of a photon:
E = hc / λ
Where:
E = energy of the photon
h = Planck's constant (6.626 x 10^-34 Js)
c = speed of light (3 x 10^8 m/s)
λ = wavelength of the incident photon
Step 3: Calculate the final momentum of the photon.
The final momentum of the photon can be calculated using the same equation as the initial momentum.
Final momentum of the photon = Final energy of the photon / c
The final energy of the photon can be determined using the wavelength of the scattered photon.
E = hc / λ
Where:
E = energy of the photon
h = Planck's constant (6.626 x 10^-34 Js)
λ = wavelength of the scattered photon
Step 4: Calculate the momentum gained by the electron.
The momentum gained by the electron can be calculated by subtracting the final momentum of the photon from its initial momentum.
Momentum gained by the electron = Initial momentum of the photon - Final momentum of the photon
Now let's plug in the given values and solve the equations.
Given:
Initial wavelength of the photon (λ1) = 0.2800 nm = 0.2800 x 10^-9 m
Final wavelength of the photon (λ2) = 0.2849 nm = 0.2849 x 10^-9 m
Planck's constant (h) = 6.626 x 10^-34 Js
Speed of light (c) = 3 x 10^8 m/s
Step 5: Calculate the initial momentum of the photon.
Initial energy of the photon = hc / λ1
Initial momentum of the photon = Initial energy of the photon / c
Step 6: Calculate the final momentum of the photon.
Final energy of the photon = hc / λ2
Final momentum of the photon = Final energy of the photon / c
Step 7: Calculate the momentum gained by the electron.
Momentum gained by the electron = Initial momentum of the photon - Final momentum of the photon
By plugging in the given values and solving the equations, we can find the momentum gained by the electron.
To find the momentum gained by the electron, we can use the conservation of linear momentum. According to this principle, the total momentum before and after the interaction remains the same.
In this case, the initial momentum of the system is zero since the electron is initially at rest. The final momentum will be the momentum of the scattered photon and the momentum gained by the electron.
The momentum of a particle is given by the formula: p = mv, where p is the momentum, m is the mass, and v is the velocity.
Since the electron is at rest initially, its momentum is zero: p_electron_initial = 0.
The momentum of a photon can be calculated using the equation: p_photon = h/λ, where p_photon is the momentum of the photon, h is Planck's constant (h = 6.626 x 10^-34 J·s), and λ is the wavelength of the photon.
Using this formula, the momentum of the incident photon is: p_photon_initial = h/λ_initial.
The momentum of the scattered photon is: p_photon_final = h/λ_final.
According to the conservation of linear momentum, the total initial momentum equals the total final momentum.
So, we have: p_electron_initial + p_photon_initial = p_photon_final + p_electron_final.
Since the initial momentum of the electron is zero, the equation can be simplified to: p_photon_initial = p_photon_final + p_electron_final.
Now, let's substitute the values given in the problem.
λ_initial = 0.2800 nm = 0.2800 x 10^-9 m.
λ_final = 0.2849 nm = 0.2849 x 10^-9 m.
Plugging the values into the momentum equations, we can solve for p_photon_initial, p_photon_final, and p_electron_final.
p_photon_initial = (6.626 x 10^-34 J·s) / (0.2800 x 10^-9 m)
p_photon_final = (6.626 x 10^-34 J·s) / (0.2849 x 10^-9 m)
Finally, we can solve for p_electron_final by rearranging the equation:
p_electron_final = p_photon_initial - p_photon_final.
Plug in the values and calculate the momentum gained by the electron.