Calculate the wavelength of a beam of electrons that have been accelerated through a potential difference of 10 000 V. Use the formula λ = h/p (Planck's constant, h = 6.626x10^-34 J.s^-1, charge on electron, e = 1.60x10^-19 C, mass of electron, m_e = 9.11x10^-31 kg.)

Derivation of the equation:

The momentum of the electrons is p = m_e v, where v is the velocity attained by the electrons after being accelerated, and m_e is the mass of the electron.

The kinetic energy of the electrons is ½m_e v², which we can rewrite as p²/2m_e. This equals eV, the energy gained by accelerating the electrons through a potential difference, V, where e is the charge on the electron. Thus,

Since p²/2m_e = eV,

p² = 2e m_eV

and p = (2e m_eV)^½

Substituting for p in λ = h/p gives λ = h/(2e m_eV)^½ = 1.12x10^-9V^-½

Solution:

Now apply the equation, where V = 10 000V, and we get λ = 1.12x10^-9V^-½ = 1.12x10^-9 ÷ 100 = 1.12x10^-11 m = 1.12x10^-2 nm.