Dual Nature — Advanced Multi-Step Practice (Hard)

Quick Recap — Advanced Dual Nature (Multi-step)

• Photoelectric numbers: $KE_{max}=\dfrac{1240}{\lambda\text{(nm)}}-\phi\text{(eV)}$; stopping potential $V_0=\dfrac{KE_{max}}{e}$; threshold $\lambda_0=\dfrac{1240}{\phi}$.
• Photoelectron speed: $v_{max}=\sqrt{\dfrac{2\,KE_{max}}{m}}$.
• de Broglie ratios: same $KE$ or same $V$: $\dfrac{\lambda_1}{\lambda_2}=\sqrt{\dfrac{m_2}{m_1}}$ (for equal charge); e.g. electron vs proton $\approx\sqrt{1836}\approx43$.
• Two-wavelength work function: from $eV_1=\dfrac{hc}{\lambda_1}-\phi$ and $eV_2=\dfrac{hc}{\lambda_2}-\phi$, subtract to eliminate $\phi$.

Worked mini-example. 300 nm light ($E=\dfrac{1240}{300}\approx4.13$ eV) on a metal of $\phi=2.13$ eV gives $KE_{max}=2.0$ eV and a stopping potential of 2.0 V.