Photoelectric Effect

$KE_{\max} = h\nu - \phi \quad [ML^2T^{-2}] \quad \text{J or eV}$

$eV_0 = h\nu - \phi \implies V_0 = \frac{h\nu - \phi}{e} \quad [ML^2T^{-3}A^{-1}] \quad \text{V}$

$\nu_0 = \frac{\phi}{h} \quad [T^{-1}] \quad \text{Hz} \qquad \lambda_0 = \frac{hc}{\phi} = \frac{1240}{\phi(\text{eV})} \text{ nm} \quad [L] \quad \text{m}$

$KE_{\max} \text{ from wavelengths:} \quad KE_{\max} = hc\left(\frac{1}{\lambda} - \frac{1}{\lambda_0}\right) \quad [ML^2T^{-2}]$

Photon Properties

$E = h\nu = \frac{hc}{\lambda} = \frac{1240}{\lambda(\text{nm})} \text{ eV} \quad [ML^2T^{-2}]$

$p = \frac{h}{\lambda} = \frac{E}{c} = \frac{h\nu}{c} \quad [MLT^{-1}] \quad \text{kg m/s}$

de Broglie Wavelength

$\lambda = \frac{h}{mv} = \frac{h}{p} \quad [L] \quad \text{m}$

$\lambda = \frac{h}{\sqrt{2m \cdot KE}} \quad [L] \qquad \lambda_e = \frac{1.227}{\sqrt{V}} \text{ nm (electrons only)}$

$\lambda = \frac{h}{\sqrt{2mqV}} \quad \text{(any particle, charge q)} \qquad \lambda_{\text{thermal}} = \frac{h}{\sqrt{3mkT}} \quad [L]$

Ratios and Comparisons

$\frac{\lambda_1}{\lambda_2} = \sqrt{\frac{m_2 q_2}{m_1 q_1}} \text{ (same V)} \qquad \frac{\lambda_e}{\lambda_p} \approx 42.8 \qquad \frac{\lambda_e}{\lambda_\alpha} \approx 121$

Key Constants

$h = 6.63 \times 10^{-34} \text{ J·s} \quad [ML^2T^{-1}] \qquad hc = 1240 \text{ eV·nm}$

$e = 1.6 \times 10^{-19} \text{ C} \quad m_e = 9.1 \times 10^{-31} \text{ kg} \quad m_p = 1.67 \times 10^{-27} \text{ kg}$