Modern Physics flash cards

1. 1.State Einstein's photoelectric equation.

   $K_{max}=h\nu-\phi_0$, where $\phi_0$ is the work function. Equivalently $eV_0=h\nu-\phi_0$, with $V_0$ the stopping potential.

   Hint: Photon energy = work function + max KE of ejected electron.

2. 2.What is the work function $\phi_0$ of a metal?

   The minimum energy needed to just liberate an electron from the metal surface. Related to threshold frequency by $\phi_0=h\nu_0$ and to threshold wavelength by $\phi_0=hc/\lambda_0$.

   Hint: Energy to escape the surface, no leftover KE.

3. 3.Define threshold frequency $\nu_0$.

   The minimum frequency of incident light below which no photoemission occurs, regardless of intensity. $\nu_0=\phi_0/h$.

   Hint: Below it, even bright light gives no electrons.

4. 4.What is the stopping potential $V_0$?

   The reverse voltage that just stops the most energetic photoelectrons: $eV_0=K_{max}=h\nu-\phi_0$. It depends on frequency, not intensity.

   Hint: Voltage that makes photocurrent zero.

5. 5.How does photoelectric current depend on light intensity (at fixed frequency $>\nu_0$)?

   Saturation photocurrent is directly proportional to intensity (more photons → more electrons). Stopping potential and $K_{max}$ are unaffected by intensity.

   Hint: Intensity sets number of electrons, not their energy.

6. 6.How does $K_{max}$ of photoelectrons depend on frequency?

   Linearly: $K_{max}=h\nu-\phi_0$. A graph of $K_{max}$ vs $\nu$ is a straight line of slope $h$ and $x$-intercept $\nu_0$.

   Hint: Slope of the line is Planck's constant.

7. 7.On a $V_0$ vs $\nu$ graph, what do the slope and intercepts represent?

   Slope $=h/e$ (same for all metals). $x$-intercept $=\nu_0$ (threshold frequency). $y$-intercept magnitude $=\phi_0/e$.

   Hint: Parallel lines for different metals; slope is universal.

8. 8.Why does the photoelectric effect show no time lag?

   Emission is essentially instantaneous ($<10^{-9}$ s) because a single photon delivers its whole energy to one electron at once, contradicting the wave prediction of gradual energy build-up.

   Hint: One-photon-one-electron, immediate energy transfer.

9. 9.What is a photon and its energy/momentum?

   A quantum of light with energy $E=h\nu=hc/\lambda$ and momentum $p=E/c=h/\lambda$. It has zero rest mass and travels at $c$.

   Hint: Massless packet; $p=h/\lambda$.

10. 10.Give the convenient numerical form for photon energy in eV.

    $E(\text{eV})=\dfrac{1240}{\lambda(\text{nm})}$. So $620$ nm light $\approx 2$ eV.

    Hint: Remember 1240 eV·nm.

11. 11.State the de Broglie wavelength of a matter wave.

    $\lambda=\dfrac{h}{p}=\dfrac{h}{mv}$. Every moving particle has an associated wave; the effect is significant only for small $mv$.

    Hint: Wavelength = h over momentum.

12. 12.Express de Broglie wavelength in terms of kinetic energy $K$.

    $\lambda=\dfrac{h}{\sqrt{2mK}}$. For a charge accelerated through potential $V$: $\lambda=\dfrac{h}{\sqrt{2mqV}}$.

    Hint: $p=\sqrt{2mK}$.

13. 13.de Broglie wavelength of an electron accelerated through $V$ volts (numerical).

    $\lambda=\dfrac{12.27}{\sqrt{V}}\ \text{Å}$ (i.e. $\approx 1.227$ nm$/\sqrt{V}$). For $V=100$ V, $\lambda\approx1.23$ Å.

    Hint: 12.27 Å over root V.

14. 14.What experiment confirmed the wave nature of electrons?

    The Davisson–Germer experiment (1927): electrons scattered off a nickel crystal showed diffraction maxima matching the de Broglie wavelength.

    Hint: Electron diffraction off a crystal.

15. 15.State the postulates of Bohr's model of hydrogen.

    (1) Electrons orbit in stationary states without radiating. (2) Angular momentum is quantized: $mvr=n\dfrac{h}{2\pi}$. (3) Radiation on jumps: $h\nu=E_i-E_f$.

    Hint: Quantized L, stationary orbits, quantum jumps.

16. 16.How does Bohr's quantization link to de Broglie?

    A standing wave fits the orbit: $2\pi r=n\lambda=n\dfrac{h}{mv}$, giving $mvr=n\dfrac{h}{2\pi}$. Only whole numbers of wavelengths survive.

    Hint: Circumference = integer × wavelength.

17. 17.Bohr radius formula for hydrogen-like atom.

    $r_n=\dfrac{n^2}{Z}\,a_0$, with $a_0=0.529$ Å $=\dfrac{\varepsilon_0 h^2}{\pi m e^2}$. Radius grows as $n^2$, shrinks with $Z$.

    Hint: $r\propto n^2/Z$.

18. 18.Bohr energy of the $n$-th level (hydrogen-like).

    $E_n=-13.6\dfrac{Z^2}{n^2}\ \text{eV}$. Negative (bound); magnitude falls as $1/n^2$.

    Hint: $-13.6\,Z^2/n^2$ eV.

19. 19.Bohr orbital velocity of the electron.

    $v_n=\dfrac{Z}{n}\times 2.19\times10^6\ \text{m/s}=\alpha c\dfrac{Z}{n}$, where $\alpha\approx 1/137$.

    Hint: $v\propto Z/n$; ground state $\approx c/137$.

20. 20.Relations among KE, PE and total energy in Bohr atom.

    $K=-E$, $U=2E$, so $U=-2K$ and $E=-K=U/2$. Total energy magnitude equals kinetic energy.

    Hint: Virial theorem: PE = −2·KE.

21. 21.What is the ionization energy of hydrogen?

    $13.6$ eV — energy to remove the ground-state ($n=1$) electron to $n=\infty$. For hydrogen-like: $13.6\,Z^2$ eV.

    Hint: Energy from $n=1$ to free.

Open the interactive deck for the other 73 cards, with self-grading so the ones you keep missing come back.