Modern Physics: Quick Revision

A brief note on Modern Physics. Read it before a test. Try each question at the end before you open its answer.

The big ideas

• Light comes in photons: $E = h\nu = \dfrac{hc}{\lambda}$, with $hc = 1240\ \text{eV nm}$.
• Photoelectric effect: $KE_{\max} = h\nu - \phi = eV_0$. Below the threshold no electrons leave; brightness changes how many, not how fast.
• Bohr's hydrogen: $E_n = -\dfrac{13.6\ \text{eV}}{n^2}$ ($Z^2$ times that for an ion with one electron). A jump gives a photon of the energy difference.
• Series: Lyman ends on $n = 1$ (ultraviolet), Balmer on $n = 2$ (visible), Paschen on $n = 3$ (infrared).
• Matter waves: $\lambda = \dfrac{h}{mv}$; for an electron through $V$ volts, $\lambda = \dfrac{1.227}{\sqrt{V}}\ \text{nm}$.

The facts at a glance

• Photon: $E = \dfrac{1240\ \text{eV nm}}{\lambda}$
• Einstein: $KE_{\max} = h\nu - \phi$, $eV_0 = KE_{\max}$
• Graph of $V_0$ against $\nu$: slope $\dfrac{h}{e}$, the same for every metal
• Hydrogen level: $E_n = -\dfrac{13.6\ \text{eV}}{n^2}$
• Lines from level $n$: up to $\dfrac{n(n - 1)}{2}$
• De Broglie: $\lambda = \dfrac{h}{p}$

A worked example

Light of $300\ \text{nm}$ falls on a metal whose work function is $3.4\ \text{eV}$. Find the largest kinetic energy and the stopping potential. Then find the de Broglie wavelength of an electron speeded up by $81\ \text{V}$.

1. The photon. $\dfrac{1240\ \text{eV nm}}{300\ \text{nm}} = 4.13\ \text{eV}$.
2. Einstein. $KE_{\max} = 4.13\ \text{eV} - 3.4\ \text{eV} = 0.73\ \text{eV}$, so $V_0 = 0.73\ \text{V}$.
3. The electron. $\lambda = \dfrac{1.227}{\sqrt{81}}\ \text{nm} = 0.136\ \text{nm}$.

Practice with new numbers

A program to try

Mistakes to avoid

• Thinking brighter light gives faster electrons. Only the frequency sets $KE_{\max}$.
• Dividing by $n$ instead of $n^2$ in $E_n = -\dfrac{13.6\ \text{eV}}{n^2}$.
• Forgetting $Z^2$ for $\text{He}^+$ and $\text{Li}^{2+}$.
• Forgetting the square root in $\lambda = \dfrac{1.227}{\sqrt{V}}\ \text{nm}$.

Check yourself

1. What is the energy of a photon of $207\ \text{nm}$?

2. What is the energy of the $n = 3$ level of $\text{He}^+$?

3. Why does a dim violet light eject electrons from a metal that a bright red light cannot?

4. What is the de Broglie wavelength of an electron speeded up from rest by $400\ \text{V}$?