Find the kinetic energy of an electron whose de Broglie wavelength is the same as that of a 100 keV x-ray.

Sure, let's solve this step by step.

Step 1: Understand the problem We are given that the de Broglie wavelength of an electron is the same as that of a 100 keV x-ray. We need to find the kinetic energy of the electron.

Step 2: Recall the de Broglie wavelength formula The de Broglie wavelength (λ) is given by the formula λ = h/p, where h is Planck's constant and p is the momentum of the particle.

Step 3: Recall the energy-wavelength relationship for a photon The energy E of a photon is given by the formula E = hc/λ, where c is the speed of light.

Step 4: Calculate the wavelength of the x-ray We are given that the energy of the x-ray is 100 keV, which is 100,000 eV. Converting this to joules (since the other constants are in SI units), we get E = 100,000 eV * 1.6 x 10^-19 J/eV = 1.6 x 10^-14 J. Using the energy-wavelength relationship, we can solve for λ: λ = hc/E = (6.626 x 10^-34 J*s * 3 x 10^8 m/s) / 1.6 x 10^-14 J = 1.24 x 10^-10 m.

Step 5: Calculate the momentum of the electron Since the de Broglie wavelength of the electron is the same as that of the x-ray, we can use the de Broglie wavelength formula to solve for the momentum of the electron: p = h/λ = 6.626 x 10^-34 Js / 1.24 x 10^-10 m = 5.34 x 10^-24 kgm/s.

Step 6: Calculate the kinetic energy of the electron The kinetic energy K of a particle is given by the formula K = p^2/2m, where m is the mass of the particle. For an electron, m = 9.11 x 10^-31 kg. Plugging in the values, we get K = (5.34 x 10^-24 kg*m/s)^2 / (2 * 9.11 x 10^-31 kg) = 1.57 x 10^-17 J.

Step 7: Convert the kinetic energy to eV Since the original energy was given in keV, we should convert the kinetic energy to the same units for consistency. 1 J = 6.24 x 10^18 eV, so K = 1.57 x 10^-17 J * 6.24 x 10^18 eV/J = 98 keV.

So, the kinetic energy of the electron is approximately 98 keV.