Answer:We can use the de Broglie wavelength equation and the energy equation for photons to solve for the potential difference required for electrons to have the same wavelength and energy as the given X-ray.
(a) To have the same wavelength as an X-ray of wavelength 0.220 nm, we can use the de Broglie wavelength equation:
λ = h/p = h/(mv)
where λ is the wavelength, h is Planck's constant, p is the momentum, m is the mass of the particle, and v is the velocity of the particle.
For an electron with the same wavelength as an X-ray of wavelength 0.220 nm, we can assume it has a velocity close to the speed of light since it has such a small mass. Thus, we can use the relativistic energy equation for photons:
E = pc = hv
where E is the energy of the photon and c is the speed of light.
Setting the two equations equal to each other, we get:
hv = h/(m√(1-(v^2/c^2))) v
Simplifying, we get:
v = c √(1 - (m c^2 / E)^2)
Substituting the values given, we get:
v = c √(1 - (9.109 x 10^-31 kg x (3.00 x 10^8 m/s)^2 / (0.220 x 10^-9 m x 2 x 1.60 x 10^-19 J/eV))^2) = 2.76 x 10^8 m/s
Using the velocity and the de Broglie wavelength equation, we can solve for the momentum of the electron:
λ = h/p
p = h/λ = 6.63 x 10^-34 J s / (0.220 x 10^-9 m) = 3.02 x 10^-25 kg m/s
Now, we can use the momentum and the energy equation for a charged particle accelerated through a potential difference to solve for the potential difference required to give the electron this momentum:
E = (p^2/2m) + qV
where E is the kinetic energy, p is the momentum, m is the mass of the electron, q is the charge of the electron, and V is the potential difference.
Solving for V, we get:
V = (E - (p^2/2m))/q = ((9.109 x 10^-31 kg x (2.76 x 10^8 m/s)^2)/2 - (3.02 x 10^-25 kg m/s)^2/(2 x 9.109 x 10^-31 kg)) / (1.60 x 10^-19 C) = 507 V
Therefore, electrons must be accelerated through a potential difference of 507 V to have the same wavelength as an X-ray of wavelength 0.220 nm.
(b) To have the same energy as the X-ray in part (a), we can use the energy equation for photons:
E = hc/λ
where E is the energy of the photon, h is Planck's constant, c is the speed of light, and λ is the wavelength of the photon.
Substituting the values given, we get:
E = (6.63 x 10^-34 J s x 3.00 x 10^8 m/s) / (0.220 x 10^-9 m x 2) = 1.51 x 10^-15 J
Now, we can use the energy equation for a charged particle accelerated through a potential difference to solve for the potential difference required to give the electron this energy:
E = q
Step-by-step explanation: