Through what potential difference must electrons be accelerated if they are to have:
(a) the same wavelength as an x ray of wavelength $0.220$ nm; and
(b) the same energy as the x ray in part (a)?

Verified step by step guidance

1. Step 1: To find the potential difference for part (a), start by using the de Broglie wavelength formula for electrons: $\lambda =\frac{h}{p}$, where $\lambda$ is the wavelength, $h$ is Planck's constant, and $p$ is the momentum of the electron.
2. Step 2: Relate the momentum $p$ to the kinetic energy of the electron using $\sqrt{2meK}$, where $me$ is the mass of the electron and $K$ is its kinetic energy. Substitute this into the de Broglie equation to express $K$ in terms of the wavelength.
3. Step 3: Use the relationship between kinetic energy and potential difference: $K=eV$, where $e$ is the charge of the electron and $V$ is the potential difference. Solve for $V$ in terms of the given wavelength $\lambda$.
4. Step 4: For part (b), calculate the energy of the x-ray photon using the formula $E=\frac{h}{c}$, where $c$ is the speed of light. This energy will be equal to the kinetic energy of the electron.
5. Step 5: Use the relationship $K=eV$ again to find the potential difference $V$ for part (b), substituting the energy of the x-ray photon for $K$.

Key Concepts

De Broglie Wavelength

The De Broglie wavelength is a fundamental concept in quantum mechanics that relates the wavelength of a particle to its momentum. It is given by the formula λ = h/p, where λ is the wavelength, h is Planck's constant, and p is the momentum of the particle. For electrons, this means that their wavelength can be calculated based on their velocity and mass, allowing us to compare it to the wavelength of x-rays.

Energy of Electrons

The energy of an electron can be determined using the equation E = qV, where E is the energy, q is the charge of the electron, and V is the potential difference through which the electron is accelerated. This relationship shows that the energy gained by an electron is directly proportional to the potential difference, which is crucial for understanding how to match the energy of electrons to that of x-rays.

Photon Energy

The energy of a photon, such as an x-ray, is given by the equation E = hc/λ, where E is the energy, h is Planck's constant, c is the speed of light, and λ is the wavelength. This relationship indicates that shorter wavelengths correspond to higher energy photons. Understanding this concept is essential for determining the potential difference required for electrons to match the energy of x-ray photons.