FAD1022 — EVERYTHING AT A GLANCE

CONSTANTS

$k = 8.99\times10^9$$\varepsilon_0 = 8.85\times10^{-12}$ F/m$\mu_0 = 4\pi\times10^{-7}$ H/m$e = 1.602\times10^{-19}$ C
$m_e = 9.11\times10^{-31}$ kg$m_p = 1.673\times10^{-27}$ kg$m_n = 1.675\times10^{-27}$ kg$h = 6.626\times10^{-34}$ J·s
$\hbar = 1.055\times10^{-34}$ J·s$c = 3.00\times10^8$ m/s$k_B = 1.381\times10^{-23}$ J/K$\sigma = 5.67\times10^{-8}$ W/m²K⁴
$b = 2.90\times10^{-3}$ m·K$N_A = 6.022\times10^{23}$ /mol$1\text{ eV} = 1.602\times10^{-19}$ J$1\text{ u} = 1.6606\times10^{-27}$ kg = 931.5 MeV
$a_0 = 5.29\times10^{-11}$ m$R_H = 1.097\times10^7$ /m$\lambda_C = 2.43\times10^{-12}$ m$R_0 = 1.2$ fm

1. Electrostatics

$Q = ne$|$F = k\frac{Qq}{r^2}$|$k = \frac{1}{4\pi\varepsilon_0}$
$E = \frac{F}{q_0}$|$E = \frac{kQ}{r^2}$|$\vec{F} = q\vec{E}$
$\vec{E}_{\text{net}} = \sum \vec{E}_i$|$E = \sqrt{E_x^2 + E_y^2}$|$\theta = \tan^{-1}(E_y/E_x)$

Neutral pt: $\vec{E}_{\text{net}} = 0$ (between like charges, closer to smaller)

Dyn. eq: $qE = mg$ | Perp. entry: $a_y = qE/m$, $v_y = qEx/(mv_0)$, $s_y = -\frac{1}{2}a_y t^2$

Parallel entry: $a = qE/m$ | SUVAT: $v=u+at$, $s=ut+\frac{1}{2}at^2$, $v^2=u^2+2as$

Field lines: tangent to $\vec{E}$, $\propto$ density, start $+$, end $-$, never cross

2. Capacitors

$C = \frac{Q}{\Delta V}$|$C_0 = \frac{A\varepsilon_0}{d}$|$E = \frac{\sigma}{\varepsilon_0} = \frac{\Delta V}{d}$
$\kappa = \varepsilon/\varepsilon_0$|$C = \kappa C_0$|$\tau = RC$

Isolated (bat. disc.): $Q = Q_0$, $\Delta V = \Delta V_0/\kappa$, $E = E_0/\kappa$

Energy: $U = \frac{1}{2}\frac{Q^2}{C} = \frac{1}{2}C(\Delta V)^2 = \frac{1}{2}Q\,\Delta V$

3. DC Circuits

$V = IR$|$V_{\text{term}} = \varepsilon - Ir$|$I = \frac{\varepsilon}{R_{\text{ext}} + r}$
$P = IV = I^2R = \frac{V^2}{R}$|$P_{\text{src}} = I\varepsilon$|$P_{\text{lost}} = I^2r$
$R_{\text{eq,ser}} = \sum R_i$|$\frac{1}{R_{\text{eq,par}}} = \sum \frac{1}{R_i}$|$R_{\text{eq}} = \frac{R_1R_2}{R_1+R_2}$ (2)
$I_1 = I_T \frac{R_2}{R_1+R_2}$|$I_2 = I_T \frac{R_1}{R_1+R_2}$

KCL: $\sum I_{\text{in}} = \sum I_{\text{out}}$ | KVL: $\sum_{\text{loop}} \Delta V = 0$ | Single loop: $I = \frac{\sum\varepsilon}{\sum R}$

4. AC Analysis

$I(t) = I_0\sin(\omega t)$|$\omega = 2\pi f = \frac{2\pi}{T}$|$Z = \frac{V_{\text{rms}}}{I_{\text{rms}}}$
$I_{\text{rms}} = \frac{I_{\text{max}}}{\sqrt{2}}$|$V_{\text{rms}} = \frac{V_{\text{max}}}{\sqrt{2}}$|$P = V_{\text{rms}}I_{\text{rms}}$

$A(t) = A_m\sin(\omega t + \phi)$; +$\phi$ = lead, -$\phi$ = lag

Circuit$Z$PhaseLead/Lag
Pure R$R$$0°$In phase
Pure C$X_C = \frac{1}{\omega C}$$-90°$I leads V
Pure L$X_L = \omega L$$+90°$V leads I

CIVIL: C → I leads V | I(nductor) → V leads I | R → in phase

5. AC Series Circuits

$X_L = \omega L$|$X_C = \frac{1}{\omega C}$|$X = X_L - X_C$
RL: $Z = \sqrt{R^2 + X_L^2}$|RC: $Z = \sqrt{R^2 + X_C^2}$
RLC: $Z = \sqrt{R^2 + (X_L - X_C)^2}$|$\theta = \tan^{-1}\!\frac{X_L - X_C}{R}$
$V_T = \sqrt{V_R^2 + (V_L - V_C)^2}$|$I_{\text{rms}} = \frac{V_{\text{rms}}}{Z}$

Resonance: $X_L = X_C$, $f_0 = \frac{1}{2\pi\sqrt{LC}}$, $Z = R$, $I_{\text{max}}$, $\theta = 0°$, PF = 1

$\frac{X_C}{X_L} = \left(\frac{f_0}{f}\right)^2$ | From phase: $C = \frac{1}{2\pi f R \tan|\theta|}$

PowerFormulaUnit
Real ($P_{\text{ave}}$)$V_{\text{rms}}I_{\text{rms}}\cos\phi = I^2R$W
Reactive ($P_R$)$V_{\text{rms}}I_{\text{rms}}\sin\phi = I^2|X_L-X_C|$VAr
Apparent ($P_A$)$V_{\text{rms}}I_{\text{rms}} = I^2Z$VA

$P_A^2 = P_{\text{ave}}^2 + P_R^2$ | PF = $\cos\phi = \frac{R}{Z} = \frac{P_{\text{ave}}}{P_A}$

6. Magnetism

$1\text{ T} = 1\text{ N/(A·m)}$, $1\text{ G} = 10^{-4}\text{ T}$

Wire: $B = \frac{\mu_0 I}{2\pi r}$|Loop: $B = \frac{\mu_0 NI}{2r}$|Solenoid: $B = \mu_0 nI$
$\vec{F}_B = q\vec{v} \times \vec{B}$|$|F_B| = |q|vB\sin\theta$
$r = \frac{mv}{qB}$|$T = \frac{2\pi m}{qB}$|$\omega = \frac{qB}{m}$
Vel. selector: $v = E/B$|Mass spec: $m = \frac{qB'B^2r}{E}$
$F = \frac{\mu_0 I_1 I_2 L}{2\pi d}$|$f = \frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}$

Ampère: $\oint \vec{B}\cdot d\vec{l} = \mu_0 I_{\text{enc}}$ | Cyl. wire: $B_{\text{in}} = \frac{\mu_0 Ir}{2\pi R^2}$, $B_{\text{out}} = \frac{\mu_0 I}{2\pi r}$

$\tau = NIAB\sin\theta$ | Same-dir currents → attractive | RHR grip = field, RHR = force on +q

7. EM Induction

$\Phi_B = BA\cos\theta$|$\varepsilon = -N\frac{d\Phi_B}{dt}$|$\varepsilon_{\text{avg}} = -N\frac{\Delta\Phi_B}{\Delta t}$
Motional: $\varepsilon = Blv$|$I = \frac{\varepsilon}{R}$|$P = I^2R = \frac{\varepsilon^2}{R}$
$F = IlB = \frac{B^2l^2v}{R}$|$\varepsilon_{\text{max}} = NBA\omega$

Lenz: induced current opposes flux change | $P_{\text{mech}} = Fv$

8. Inductance & Transformers

$\mathcal{E} = -L\frac{dI}{dt}$|$L = \frac{\mu_0 N^2 A}{\ell}$|$L = \frac{N\Phi_B}{I}$
$U = \frac{1}{2}LI^2$|$M = \frac{\mu_0 N_p N_s A}{l}$|$\varepsilon_2 = M\frac{di_1}{dt}$
$\frac{V_s}{V_p} = \frac{N_s}{N_p}$|$\frac{I_s}{I_p} = \frac{N_p}{N_s}$|$P_p = P_s$ (ideal)

Step-up: $N_s > N_p$ | Losses: Cu ($I^2R$), eddy ($B^2f^2t^2$), hysteresis, flux leakage

Transmission: $P_{\text{loss}} = I^2R_{\text{line}}$ | $\% = \frac{I^2R_{\text{line}}}{P_{\text{total}}}\times100$

CapacitorInductor
Geometry$C = \varepsilon_0 A/d$$L = \mu_0 N^2 A/\ell$
Energy$U = \frac{1}{2}CV^2$$U = \frac{1}{2}LI^2$
Definition$C = Q/V$$L = N\Phi/I$

9. Semiconductors & Op-Amps

$V_D$: Ge = 0.3V, Si = 0.7V, GaAs = 1.5V

FB (ON): $E - V_R - V_D = 0$|RB (OFF): $I_D = 0$, $V_D = E$
$I = \frac{V_{\text{supply}} - V_D}{R}$|$V_{DC,HW} = 0.318(V_m - V_D)$|$V_{DC,FW} = \frac{2V_O}{\pi}$
$I_E = I_B + I_C$|$\beta = I_C/I_B$|$\alpha = I_C/I_E$
$I_E = (\beta+1)I_B$|$I_B = \frac{V_{CC}-V_{BE}}{R_B}$ (fixed)

Fixed: $V_{CE} = V_{CC} - I_CR_C$, $I_{C(\text{sat})} = V_{CC}/R_C$

Emitter-stabilized: $I_B = \frac{V_{CC}-V_{BE}}{R_B+(\beta+1)R_E}$, $V_{CE} = V_{CC}-I_C(R_C+R_E)$

Voltage divider: $V_B = \frac{R_{B2}}{R_{B1}+R_{B2}}V_{CC}$, $V_E = V_B - V_{BE}$, $I_C \approx V_E/R_E$

Stability: $(\beta+1)R_E \geq 10R_B$ → $I_C \approx \frac{V_{CC}-V_{BE}}{R_E}$ | Approx: $\beta R_E \geq 10R_{B2}$

Region$V_{BE}$$V_{CE}$$I_C$
Cutoff$< 0.7$V$= V_{CC}$$\approx 0$
Active$\approx 0.7$V$> V_{CE(\text{sat})}$$\beta I_B$
Saturation$\approx 0.7$V$\approx 0.2$V$I_{C(\text{sat})}$
Op-AmpGainPhase
Inverting$V_{\text{out}} = -\frac{R_f}{R_1}V_{\text{in}}$$180°$
Non-inverting$V_{\text{out}} = \left(1+\frac{R_f}{R_1}\right)V_{\text{in}}$$0°$

10. Atomic Physics

$L = n\hbar$|$r_n = a_0 n^2$|$E_n = -\frac{13.6}{n^2}\text{ eV}$
$E = K+U = -\frac{ke^2}{2r}$|$\Delta E = hf = E_i - E_f$
$\frac{1}{\lambda} = R_H\left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)$|$E_{\text{ion}} = 13.6$ eV

Lyman $n_f=1$ (UV) | Balmer $n_f=2$ (vis) | Paschen $n_f=3$ (IR)

$n$1234
$E_n$ (eV)$-13.6$$-3.4$$-1.51$$-0.85$
Q.#NameValues
$n$Principal$1,2,3,\dots$
$l$Orbital$0,1,\dots,n-1$
$m_l$Magnetic$-l,\dots,+l$
$m_s$Spin$\pm\frac{1}{2}$

Absorption: $E_{\text{photon}} = E_{\text{high}} - E_{\text{low}}$ | LASER: pop. inversion + optical feedback

Reduced mass: $\mu = \frac{m_e M}{m_e + M}$

11. Nuclear Physics

$^A_Z X$|$N = A - Z$|$R = R_0 A^{1/3}$
$E = mc^2$|$1\text{ u} = 931.5$ MeV|$\rho \approx 2.3\times10^{17}$ kg/m³
$\Delta m = Zm_p + Nm_n - m_N$|$E_B = \Delta m \times 931.5$ MeV
$\frac{E_B}{A}$|Peak: Fe-56 $\approx 8.8$ MeV/nucleon

Decay: $N(t) = N_0 e^{-\lambda t}$ | $A = \lambda N = A_0 e^{-\lambda t}$ | $T_{1/2} = \frac{0.693}{\lambda}$

$\frac{N}{N_0} = \left(\frac{1}{2}\right)^{t/T_{1/2}}$ | $N_0 = \frac{N_A}{M}m_{\text{sample}}$ | $t = \frac{T_{1/2}\ln(N_0/N)}{\ln 2}$

$1\text{ Bq} = 1$ decay/s | $1\text{ Ci} = 3.70\times10^{10}$ Bq | C-14: $\lambda = 1.21\times10^{-4}$ yr⁻¹

DecayParticle$\Delta Z$$\Delta A$Trigger
$\alpha$$^4_2$He$-2$$-4$Too heavy
$\beta^-$$^0_{-1}e$$+1$$0$Too many n
$\beta^+$$^0_{+1}e$$-1$$0$Too many p
$\gamma$Photon$0$$0$Excited

Q-value: $Q = \Delta m \times 931.5$ MeV; $Q>0$ exo, $Q<0$ endo | Fusion: $E = E_B(\text{prod}) - E_B(\text{react})$

Fission: $\Delta m = (m_{\text{parent}}+m_n) - (m_1+m_2+\text{neutrons})$, $Q = \Delta m \times 931.5$

12. Modern Physics & QM

$E = hf = \frac{hc}{\lambda}$|$\lambda_{\text{max}} = \frac{b}{T}$|$\frac{P}{A} = \sigma T^4$
$\phi = hf_0$|$\lambda_c = \frac{hc}{\phi}$|$KE_{\text{max}} = hf - \phi$
$KE_{\text{max}} = eV_s$|$KE_{\text{max}} = \frac{1}{2}m_e v_{\text{max}}^2$
$\lambda' - \lambda = \lambda_C(1-\cos\theta)$|$\lambda_C = \frac{h}{m_e c}$
$\lambda = \frac{h}{p} = \frac{h}{mv}$|$\lambda = \frac{h}{\sqrt{2m\cdot KE}}$|$\lambda = \frac{h}{\sqrt{2meV}}$
$\Delta x\,\Delta p \geq \frac{\hbar}{2}$|$\Delta E\,\Delta t \geq \frac{\hbar}{2}$

TISE: $-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + V\psi = E\psi$ | $\hat{H}\psi = E\psi$ | $P = |\Psi|^2$ | $\int_{-\infty}^{\infty}|\Psi|^2 dx = 1$

1D Box: $E_n = \frac{n^2h^2}{8mL^2}$|$\psi_n = \sqrt{\frac{2}{L}}\sin\left(\frac{n\pi x}{L}\right)$

$E_{n+1}-E_n = \frac{(2n+1)h^2}{8mL^2}$ | Nodes = $n-1$ | $E_1 = \frac{h^2}{8mL^2}$ (ZPE)

Photon from box: $\Delta E = \frac{h^2}{8mL^2}(n_i^2 - n_f^2)$, $\lambda = \frac{hc}{\Delta E}$

Black body: $\alpha_\nu + \rho_\nu + \tau_\nu = 1$ | $u(\lambda,T) = \frac{8\pi hc}{\lambda^5}\frac{1}{e^{hc/\lambda k_BT}-1}$