Centripetal force = constant $F$:
$$F = \frac{mv^2}{r} \implies v^2 = \frac{Fr}{m} \quad \cdots (1)$$Bohr quantization: $mvr = \dfrac{nh}{2\pi} \implies v = \dfrac{nh}{2\pi mr} \quad \cdots (2)$
Square (2) and equate with (1):
$$\frac{n^2h^2}{4\pi^2 m^2 r^2} = \frac{Fr}{m} \implies r^3 = \frac{n^2h^2}{4\pi^2 mF} \implies \boxed{r \propto n^{2/3}}$$$$v = \frac{nh}{2\pi mr} \propto \frac{n}{r} \propto \frac{n}{n^{2/3}} = \boxed{n^{1/3}}$$Bohr (1913) postulated: (1) Electrons move in circular orbits without radiating. (2) Angular momentum quantized: $L = mvr = nh/(2\pi) = n\hbar$. (3) Radiation emitted/absorbed when electron transitions between orbits: $h\nu = E_i - E_f$. Results for hydrogen: $r_n = a_0 n^2$ where $a_0 = 0.529$ Å (Bohr radius). $v_n = v_1/n$ where $v_1 = 2.18\times 10^6$ m/s. $E_n = -13.6/n^2$ eV.
For different force laws: Force $F = mv^2/r$ (centripetal condition). Bohr: $mvr = n\hbar$. Two equations, two unknowns ($r$ and $v$). For $F = $ constant: $r \propto n^{2/3}$, $v \propto n^{1/3}$. For $F = kr/r^2 = k/r^2$ (Coulomb): $r \propto n^2$, $v \propto n^{-1}$. For $F = kr$ (spring): $r \propto n^{1/2}$, $v \propto n^{1/2}$. General force $F \propto r^{-s}$: $r \propto n^{2/(s+1)}$.
Energy levels: $E_n = -13.6/n^2$ eV. Spectral series (transitions to $n_f$): Lyman ($n_f=1$): UV. Balmer ($n_f=2$): visible ($H_\alpha = 656$ nm red, $H_\beta = 486$ nm blue-green). Paschen ($n_f=3$): IR. Brackett ($n_f=4$): IR. Pfund ($n_f=5$): far IR. Rydberg formula: $1/\lambda = R_H(1/n_f^2 - 1/n_i^2)$ where $R_H = 1.097\times 10^7$ m$^{-1}$.
$\lambda = h/(mv) = h/p$. Bohr condition $mvr = n\hbar$ is equivalent to $2\pi r = n\lambda$ (standing wave condition: circumference = integer number of wavelengths). This gives the wave interpretation of Bohr orbits. For electron in $n$th orbit of H: $\lambda_n = 2\pi r_n / n = 2\pi a_0 n$. Davisson-Germer experiment (1927) confirmed wave nature of electrons.
Nucleus contains protons ($Z$) and neutrons ($N$). Mass number $A = Z + N$. Nuclear radius: $R = R_0 A^{1/3}$ where $R_0 = 1.2\times 10^{-15}$ m. Nuclear density $\approx 2.3\times 10^{17}$ kg/m$^3$ (same for all nuclei!). Binding energy per nucleon: peaks near Fe-56 (most stable). Fission: heavy nuclei split (U-235, Pu-239). Fusion: light nuclei combine (H isotopes). Both release energy via $E = mc^2$.
$N = N_0 e^{-\lambda t}$. Decay constant $\lambda$, half-life $t_{1/2} = \ln 2/\lambda = 0.693/\lambda$. Activity $A = \lambda N = A_0 e^{-\lambda t}$. Types: Alpha ($^4_2$He nucleus), Beta$^-$ (electron), Beta$^+$ (positron), Gamma (photon). Alpha: stopped by paper. Beta: stopped by Al foil. Gamma: reduced by thick Pb/concrete. Units: Becquerel (Bq) = 1 decay/s. Curie (Ci) = $3.7\times 10^{10}$ Bq.
Produced when fast electrons hit a metal target. Two types: Characteristic X-rays: electron knocks out inner shell electron; outer electron falls in, emits X-ray photon. $K_\alpha$: L→K transition. $K_\beta$: M→K. Moseley law: $\sqrt{\nu} \propto (Z - b)$ (used to determine atomic number). Continuous (Bremsstrahlung): electron decelerates, emits photon. Minimum wavelength (maximum energy): $\lambda_{min} = hc/(eV)$. Applications: medical imaging, crystallography (Bragg diffraction), security scanning.
Einstein (1905): light consists of photons, energy $E = h\nu$. Photoelectric equation: $KE_{max} = h\nu - \phi$ where $\phi = h\nu_0$ = work function. Threshold frequency $\nu_0$: minimum frequency for emission. Stopping potential $V_0$: $eV_0 = KE_{max}$. Key observations: (1) Instantaneous emission (no time lag). (2) $KE_{max}$ depends on $\nu$, not intensity. (3) Intensity affects number of electrons, not their energy. Classical wave theory failed to explain these — needed quantum (photon) theory.