Formula Sheet
Every key formula of the course on one page, grouped by chapter, with a link back to the section that develops it.
Chapter 1 · Electric Charges and Fields
Quantisation of charge§ 1.1 →
$q=ne,\quad n=0,\pm1,\pm2,\dots,\quad e=1.6\times10^{-19}\ \text{C}$ (any free charge)
Coulomb's law (vacuum)§ 1.1 →
$F=\dfrac{1}{4\pi\varepsilon_0}\dfrac{|q_1q_2|}{r^2},\quad \dfrac{1}{4\pi\varepsilon_0}\approx9.0\times10^{9}\ \text{N m}^2\text{C}^{-2}$ — stationary point charges; like repel, unlike attract
Coulomb's law, vector form§ 1.1 →
$\mathbf F_{21}=\dfrac{1}{4\pi\varepsilon_0}\dfrac{q_1q_2}{r^2}\,\hat{\mathbf r}_{21}=-\mathbf F_{12}$ ($\hat{\mathbf r}_{21}$ points from $q_1$ to $q_2$; signs included)
Force in a medium§ 1.1 →
$F_\text{medium}=\dfrac{1}{4\pi\varepsilon_0\varepsilon_r}\dfrac{|q_1q_2|}{r^2}=\dfrac{F_\text{vacuum}}{\varepsilon_r}$ (charges inside a large uniform insulating medium)
Superposition of forces§ 1.1 →
$\mathbf F_1=\sum_{i\ne1}\dfrac{1}{4\pi\varepsilon_0}\dfrac{q_1q_i}{r_{1i}^2}\,\hat{\mathbf r}_{1i}$ ($\hat{\mathbf r}_{1i}$ from $q_i$ towards $q_1$; point charges)
Electric field§ 1.2 →
$\mathbf E=\dfrac{\mathbf F}{q_0}$ (small positive test charge), $\ \mathbf F=q\mathbf E$; unit N/C
Field of point charges§ 1.2 →
$\mathbf E=\dfrac{1}{4\pi\varepsilon_0}\displaystyle\sum_i\dfrac{q_i}{r_i^2}\,\hat{\mathbf r}_i$ ($\hat{\mathbf r}_i$ from $q_i$ to the point; charges at rest in vacuum)
Dipole moment§ 1.2 →
$p=q\,(2a)$, from $-q$ to $+q$; unit C m
Dipole field on the axial line§ 1.2 →
$\mathbf E=\dfrac{2kpr}{(r^2-a^2)^2}\hat{\mathbf p}\approx\dfrac{2k\mathbf p}{r^3}$ ($r$ from the centre, $r\gt a$; approximation for $r\gg a$)
Dipole field on the equatorial line§ 1.2 →
$\mathbf E=-\dfrac{k\mathbf p}{(r^2+a^2)^{3/2}}\approx-\dfrac{k\mathbf p}{r^3}$ (opposite to $\mathbf p$; approximation for $r\gg a$)
Dipole in a uniform field§ 1.2 →
$\boldsymbol\tau=\mathbf p\times\mathbf E,\ \tau=pE\sin\theta$; net force $0$; work to turn $W=pE(\cos\theta_1-\cos\theta_2)$ (uniform field only)
Charge densities§ 1.3 →
$\lambda=\dfrac{dq}{dl},\ \ \sigma=\dfrac{dq}{dA},\ \ \rho=\dfrac{dq}{dV}$ (uniform: $Q/L$, $Q/A$, $Q/V$)
Electric flux§ 1.3 →
$\Phi_E=\displaystyle\int\mathbf E\cdot d\mathbf A$; flat surface, uniform field: $\Phi_E=EA\cos\theta$
Gauss's law§ 1.3 →
$\displaystyle\oint\mathbf E\cdot d\mathbf A=\dfrac{q_\text{enc}}{\varepsilon_0}$ (any closed surface; $\mathbf E$ = total field, $q_\text{enc}$ = net charge inside)
Infinite line charge§ 1.3 →
$E=\dfrac{\lambda}{2\pi\varepsilon_0r}=\dfrac{2k\lambda}{r}$, radial (long straight uniform line)
Infinite plane sheet§ 1.3 →
$E=\dfrac{\sigma}{2\varepsilon_0}$, normal to the sheet, independent of distance; plates $\pm\sigma$: $E=\dfrac{\sigma}{\varepsilon_0}$ between, $0$ outside
Thin spherical shell§ 1.3 →
$E=\dfrac{kQ}{r^2}$ for $r\gt R$ (just outside $=\sigma/\varepsilon_0$); $E=0$ for $r\lt R$ (uniform thin shell)
Chapter 2 · Electrostatic Potential and Capacitance
Potential and potential difference§ 2.1 →
$V_P=\dfrac{W_{\infty P}}{q},\quad V_B-V_A=\dfrac{W_{AB}}{q}$ (external work, test charge moved slowly; $V=0$ at infinity); $1\ \text{V}=1\ \text{J/C}$
Energy of a charge moving freely§ 2.1 →
$\Delta K=-q\,\Delta V$; through $\Delta V$ from rest: $\tfrac12mv^2=|q\Delta V|$; $1\ \text{eV}=1.6\times10^{-19}$ J
Potential of a point charge§ 2.1 →
$V=\dfrac{1}{4\pi\varepsilon_0}\dfrac{q}{r}$ (vacuum, $V=0$ at infinity; also outside any spherically symmetric charge)
Potential of a short dipole§ 2.1 →
$V=\dfrac{1}{4\pi\varepsilon_0}\dfrac{p\cos\theta}{r^2}=\dfrac{k\,\mathbf p\cdot\hat{\mathbf r}}{r^2}$ ($r\gg a$); axis $\pm kp/r^2$, equatorial plane $0$
Potential of a system of charges§ 2.1 →
$V_P=k\displaystyle\sum_i\frac{q_i}{r_{iP}}$ (algebraic sum, signs included); continuous: $V=k\displaystyle\int\frac{dq}{r}$
Charged thin spherical shell§ 2.1 →
$V=\dfrac{kQ}{r}\ (r\ge R),\qquad V=\dfrac{kQ}{R}\ (r\le R)$ — constant inside, where $E=0$ (also a charged conducting sphere)
Field from potential§ 2.2 →
$E=-\dfrac{dV}{dl}$ (along the normal to the equipotential, $\mathbf E$ towards lower $V$); along an axis $E_x=-\dfrac{dV}{dx}$
Uniform field§ 2.2 →
$V_A-V_B=E\,d$ ($d$ measured along $\mathbf E$ from $A$ to $B$); $1\ \text{N/C}=1\ \text{V/m}$
Energy of a system of charges§ 2.2 →
$U=\dfrac{1}{4\pi\varepsilon_0}\displaystyle\sum_{\text{pairs}}\frac{q_iq_j}{r_{ij}}$ (each pair once, signs kept; $U=0$ at infinite separation)
Charges in an external field§ 2.2 →
$U=qV(\mathbf r)$; two charges: $U=q_1V(\mathbf r_1)+q_2V(\mathbf r_2)+\dfrac{kq_1q_2}{r_{12}}$
Dipole in a uniform field§ 2.2 →
$U=-pE\cos\theta=-\mathbf p\cdot\mathbf E$ ($U=0$ at $90^\circ$); $W_{\theta_1\to\theta_2}=pE(\cos\theta_1-\cos\theta_2)$
Field at a conductor's surface§ 2.3 →
$\mathbf E=\dfrac{\sigma}{\varepsilon_0}\hat{\mathbf n}$ just outside; $E=0$ inside (electrostatic equilibrium)
Dielectric in a field§ 2.3 →
$E=\dfrac{E_0}{K},\quad \mathbf P=\chi_e\mathbf E,\quad K=1+\dfrac{\chi_e}{\varepsilon_0}=\dfrac{\varepsilon}{\varepsilon_0},\quad \sigma_p=P$ (linear dielectric filling the region)
Capacitance§ 2.3 →
$C=\dfrac QV$ ($1\ \text{F}=1\ \text{C/V}$); isolated sphere $C=4\pi\varepsilon_0R$
Parallel-plate capacitor§ 2.3 →
$C=\dfrac{K\varepsilon_0A}{d}$, $\ E=\dfrac{\sigma}{K\varepsilon_0}=\dfrac Vd$ (gap filled; $d$ small compared with the plates)
Slab between the plates§ 2.3 →
$C=\dfrac{\varepsilon_0A}{d-t+t/K}$ (dielectric slab of thickness $t$, parallel to the plates); metal slab: $C=\dfrac{\varepsilon_0A}{d-t}$
Capacitors in series§ 2.4 →
$\dfrac1C=\dfrac1{C_1}+\dfrac1{C_2}+\cdots$ (same $Q$ on each; $V_i=Q/C_i$)
Capacitors in parallel§ 2.4 →
$C=C_1+C_2+\cdots$ (same $V$ across each; $Q_i=C_iV$)
Energy stored in a capacitor§ 2.4 →
$U=\dfrac{Q^2}{2C}=\tfrac12CV^2=\tfrac12QV$ (use $Q^2/2C$ if $Q$ is fixed, $\tfrac12CV^2$ if $V$ is fixed)
Energy density of a field§ 2.4 →
$u=\tfrac12\varepsilon_0E^2$ (vacuum); $u=\tfrac12K\varepsilon_0E^2$ (dielectric)
Sharing charge between capacitors§ 2.4 →
$V=\dfrac{C_1V_1+C_2V_2}{C_1+C_2},\qquad \Delta U_\text{lost}=\dfrac{C_1C_2(V_1-V_2)^2}{2(C_1+C_2)}$ (like plates joined)
Chapter 3 · Current Electricity
Electric current§ 3.1 →
$I=\dfrac{dq}{dt}$; steady current $q=It$; $\ 1\ \text{A}=1\ \text{C/s}$
Current density§ 3.1 →
$j=\dfrac IA$ (uniform flow normal to $A$); unit A/m²
Ohm's law and resistivity§ 3.1 →
$V=IR,\ \ R=\rho\dfrac lA,\ \ \mathbf j=\sigma\mathbf E,\ \ \sigma=\dfrac1\rho$ (metal at constant temperature; uniform wire)
Drift velocity§ 3.1 →
$v_d=\dfrac{eE\tau}{m},\ \ I=neAv_d$ ($n$ free electrons per m³)
Resistivity from the electron model; mobility§ 3.1 →
$\rho=\dfrac{m}{ne^2\tau},\ \ \mu=\dfrac{v_d}{E}=\dfrac{e\tau}{m},\ \ \rho=\dfrac{1}{ne\mu}$
Temperature dependence of resistivity§ 3.2 →
$\rho_T=\rho_0[1+\alpha(T-T_0)]$, $\ R_T=R_0[1+\alpha(T-T_0)]$ (limited temperature range; $\alpha\gt0$ metals, $\alpha\lt0$ semiconductors)
Electrical power§ 3.2 →
$P=VI$ (any device); resistor: $P=I^2R=\dfrac{V^2}{R}$
Electrical energy§ 3.2 →
$W=VIt$ (steady current); $1\ \text{kWh}=3.6\times10^6$ J
Transmission loss§ 3.2 →
$P_\text{loss}=I^2R_c=\dfrac{P^2R_c}{V^2}$ (power $P$ sent at voltage $V$ through lines of resistance $R_c$)
Current from a cell§ 3.3 →
$I=\dfrac{\varepsilon}{R+r}$ (emf $\varepsilon$, internal resistance $r$, external $R$)
Terminal voltage§ 3.3 →
$V=\varepsilon-Ir$ (delivering current); $\ V=\varepsilon+Ir$ (being charged); short-circuit current $\varepsilon/r$
Cells in series§ 3.3 →
$\varepsilon_\text{eq}=\varepsilon_1\pm\varepsilon_2$, $\ r_\text{eq}=r_1+r_2$; $\ n$ identical: $I=\dfrac{n\varepsilon}{R+nr}$ (minus sign for a reversed cell)
Cells in parallel§ 3.3 →
$\varepsilon_\text{eq}=\dfrac{\varepsilon_1r_2+\varepsilon_2r_1}{r_1+r_2}$, $\ r_\text{eq}=\dfrac{r_1r_2}{r_1+r_2}$; $\ m$ identical: $I=\dfrac{\varepsilon}{R+r/m}$ (like terminals joined)
Kirchhoff's junction rule§ 3.4 →
$\sum I_\text{in}=\sum I_\text{out}$ at any junction (steady currents)
Kirchhoff's loop rule§ 3.4 →
$\sum\Delta V=0$ round a closed loop: resistor along assumed current $-IR$, against $+IR$; cell $-\to+$ gives $+\varepsilon$, $+\to-$ gives $-\varepsilon$
Wheatstone bridge balance§ 3.4 →
$\dfrac PQ=\dfrac RS$ (galvanometer across $BD$ reads zero); $\ S=R\dfrac QP$
Chapter 4 · Moving Charges and Magnetism
Lorentz force§ 4.1 →
$\mathbf F=q(\mathbf E+\mathbf v\times\mathbf B)$; magnetic part $F=|q|vB\sin\theta$, perpendicular to $\mathbf v$ and $\mathbf B$, does no work
Unit of magnetic field§ 4.1 →
$1\ \text{T}=1\ \text{N A}^{-1}\text{m}^{-1}$; $\ 1\ \text{G}=10^{-4}\ \text{T}$
Force on a current-carrying conductor§ 4.1 →
$\mathbf F=I\,\mathbf l\times\mathbf B$, $\ F=IlB\sin\theta$ (straight wire, uniform field; $\mathbf l$ along the current)
Circular motion in a uniform field§ 4.1 →
$r=\dfrac{mv}{|q|B}=\dfrac{\sqrt{2mK}}{|q|B}$, $\ T=\dfrac{2\pi m}{|q|B}$, $\ \nu=\dfrac{|q|B}{2\pi m}$ ($\mathbf v\perp\mathbf B$; $T$ independent of $v$)
Helical path§ 4.1 →
$r=\dfrac{mv\sin\theta}{|q|B}$, $\ \text{pitch}=\dfrac{2\pi mv\cos\theta}{|q|B}$ ($\theta$ between $\mathbf v$ and uniform $\mathbf B$)
🚀 Velocity selector, cyclotron§ 4.1 →
$v=\dfrac EB$ (crossed fields); $\ K_\text{max}=\dfrac{q^2B^2R^2}{2m}$ (JEE/NEET only)
Biot–Savart law§ 4.2 →
$d\mathbf B=\dfrac{\mu_0}{4\pi}\dfrac{I\,d\mathbf l\times\hat{\mathbf r}}{r^2}$, $\ dB=\dfrac{\mu_0}{4\pi}\dfrac{I\,dl\sin\theta}{r^2}$ (steady current, vacuum)
Permeability of free space§ 4.2 →
$\mu_0=4\pi\times10^{-7}$ T m/A; $\ c=\dfrac{1}{\sqrt{\mu_0\varepsilon_0}}$
Field on the axis of a circular coil§ 4.2 →
$B=\dfrac{\mu_0NIR^2}{2(x^2+R^2)^{3/2}}$ along the axis ($N$ turns of radius $R$; $x$ from the centre)
Field at the centre of a coil§ 4.2 →
$B=\dfrac{\mu_0NI}{2R}$; far on the axis $B\approx\dfrac{\mu_0NIR^2}{2x^3}$ ($x\gg R$)
Field at the centre of an arc§ 4.2 →
$B=\dfrac{\mu_0I\phi}{4\pi R}$ ($\phi$ in radians; radial leads add nothing)
🚀 Finite straight wire§ 4.2 →
$B=\dfrac{\mu_0I}{4\pi d}(\sin\beta_1+\sin\beta_2)$; centre of a square of side $a$: $\dfrac{2\sqrt2\,\mu_0I}{\pi a}$ (JEE/NEET only)
Ampère's circuital law§ 4.3 →
$\oint\mathbf B\cdot d\mathbf l=\mu_0I_\text{enc}$ (steady currents; signs by the right-hand rule)
Long straight wire§ 4.3 →
$B=\dfrac{\mu_0I}{2\pi r}$ outside; circles round the wire (right-hand grip)
Inside a thick wire§ 4.3 →
$B=\dfrac{\mu_0Ir}{2\pi R^2}$ ($r\le R$, current uniform over the cross-section)
Long solenoid§ 4.3 →
$B=\mu_0nI$ inside, $n=N/L$; $\ \approx\tfrac12\mu_0nI$ at the ends; $\approx0$ outside
Force between parallel currents§ 4.3 →
$\dfrac FL=\dfrac{\mu_0I_aI_b}{2\pi d}$ (long parallel wires); parallel attract, antiparallel repel
🚀 Toroid§ 4.3 →
$B=\dfrac{\mu_0NI}{2\pi r}$ in the core; $0$ in the hole and outside (JEE/NEET only)
Magnetic moment of a loop§ 4.4 →
$\mathbf m=NIA\,\hat{\mathbf n}$ (right-hand thumb rule); unit A m²
Torque on a current loop§ 4.4 →
$\boldsymbol\tau=\mathbf m\times\mathbf B$, $\ \tau=NIAB\sin\theta$ (uniform field; $\theta$ between the normal and $\mathbf B$; net force zero)
Far field of a current loop§ 4.4 →
axis: $\dfrac{\mu_0}{4\pi}\dfrac{2m}{r^3}$; $\ $ plane: $\dfrac{\mu_0}{4\pi}\dfrac{m}{r^3}$ (opposite to $\mathbf m$; $r\gg$ loop size)
Moving-coil galvanometer§ 4.4 →
$k\phi=NIAB$ (radial field); $\ \dfrac\phi I=\dfrac{NAB}{k}$, $\ \dfrac\phi V=\dfrac{NAB}{kG}$
Galvanometer → ammeter§ 4.4 →
shunt $S=\dfrac{I_gG}{I-I_g}$ in parallel; meter resistance $\dfrac{GS}{G+S}$
Galvanometer → voltmeter§ 4.4 →
$R=\dfrac{V}{I_g}-G$ in series; meter resistance $V/I_g$
Chapter 5 · Magnetism and Matter
Magnetic moment§ 5.1 →
bar magnet $m=q_m(2l)$, S → N; solenoid $m=NIA$; unit $\text{A m}^2=\text{J/T}$
Field of a short magnet (far field)§ 5.1 →
$\mathbf B_\text{axial}=\dfrac{\mu_0}{4\pi}\dfrac{2\mathbf m}{r^3}$, $\ \mathbf B_\text{eq}=-\dfrac{\mu_0}{4\pi}\dfrac{\mathbf m}{r^3}$ ($r$ much larger than the magnet; $\mu_0/4\pi=10^{-7}$ T m/A)
Pole model, closer in§ 5.1 →
$B_\text{axial}=\dfrac{\mu_0}{4\pi}\dfrac{2mr}{(r^2-l^2)^2}$, $\ B_\text{eq}=\dfrac{\mu_0}{4\pi}\dfrac{m}{(r^2+l^2)^{3/2}}$ ($2l$ = magnetic length)
Magnetic dipole in a uniform field§ 5.1 →
$\boldsymbol\tau=\mathbf m\times\mathbf B$, $U=-\mathbf m\cdot\mathbf B$, net force $0$; $W=mB(\cos\theta_1-\cos\theta_2)$ (uniform field only)
Oscillating magnetic needle§ 5.1 →
$T=2\pi\sqrt{\dfrac{I}{mB}}$ (small oscillations, uniform field)
Gauss's law for magnetism§ 5.1 →
$\oint\mathbf B\cdot d\mathbf A=0$ (any closed surface; no monopoles)
Magnetisation§ 5.2 →
$\mathbf M=\dfrac{\mathbf m_\text{net}}{V}$; unit A/m
B, H and M§ 5.2 →
$\mathbf B=\mu_0(\mathbf H+\mathbf M)$; long solenoid $H=nI$ (any core)
Susceptibility and permeability§ 5.2 →
$\mathbf M=\chi\mathbf H$, $\mu_r=1+\chi$, $\mu=\mu_0\mu_r$, $\mathbf B=\mu\mathbf H$ (linear materials — not ferromagnets)
Curie's law§ 5.2 →
$M=C\dfrac{B_0}{T}$, $\ \chi=\dfrac{\mu_0C}{T}$ (paramagnets, far from saturation)
Ferromagnet above $T_C$§ 5.2 →
$\chi=\dfrac{C'}{T-T_C}$ ($T\gt T_C$; paramagnetic)
Saturation magnetisation§ 5.2 →
$M_s=nm$ ($n$ atoms per m³, moment $m$ each, all aligned)
Chapter 6 · Electromagnetic Induction
Magnetic flux§ 6.1 →
$\Phi_B=\mathbf B\cdot\mathbf A=BA\cos\theta$ (uniform field, plane coil; $\theta$ from the normal); flux linkage $N\Phi_B$; $1\ \text{Wb}=1\ \text{T m}^2=1\ \text{V s}$
Faraday's law§ 6.1 →
$\varepsilon=-N\dfrac{d\Phi_B}{dt}$; average $\varepsilon_\text{av}=-N\dfrac{\Delta\Phi_B}{\Delta t}$ (same flux through every turn)
Induced current and charge§ 6.1 →
$I=\dfrac{|\varepsilon|}{R}$, $\ Q=\dfrac{N|\Delta\Phi_B|}{R}$ (closed circuit of total resistance $R$; independent of how fast the flux changes)
Lenz's law§ 6.1 →
The induced current's field opposes the change of flux (the minus sign in $\varepsilon=-d\Phi_B/dt$); flux increasing → induced $\mathbf B$ opposite to the applied field, decreasing → along it
Induced electric field (JEE)§ 6.1 →
$\oint\mathbf E\cdot d\mathbf l=-\dfrac{d\Phi_B}{dt}$; long solenoid of radius $R$: $E=\dfrac r2\dfrac{dB}{dt}$ ($r\le R$), $E=\dfrac{R^2}{2r}\dfrac{dB}{dt}$ ($r\ge R$)
Motional emf of a rod§ 6.2 →
$\varepsilon=Blv$ (rod, $\mathbf v$ and $\mathbf B$ mutually perpendicular; uniform field); $\varepsilon=Blv\sin\theta$ if $\mathbf v\perp$ rod makes $\theta$ with $\mathbf B$
Rotating rod or disc§ 6.2 →
$\varepsilon=\tfrac12B\omega l^2$ (rod rotating about one end, plane of rotation $\perp\mathbf B$; disc of radius $l$, centre to rim)
Force and power for a rod on rails§ 6.2 →
$I=\dfrac{Blv}{R}$, $\ F=\dfrac{B^2l^2v}{R}$ (opposing $\mathbf v$), $\ P=Fv=\dfrac{B^2l^2v^2}{R}=I^2R$ (closed circuit, total resistance $R$)
Terminal velocity§ 6.2 →
$v_t=\dfrac{FR}{B^2l^2}$ (constant force $F$, frictionless rails); vertical rails: $v_t=\dfrac{mgR}{B^2l^2}$
Charge and a coasting rod§ 6.2 →
$Q=\dfrac{Bl\,\Delta x}{R}$; launched at $v_0$ and left alone: $Q=\dfrac{mv_0}{Bl}$, distance $=\dfrac{mv_0R}{B^2l^2}$, heat $=\tfrac12mv_0^2$
Mutual inductance§ 6.3 →
$N_2\Phi_2=MI_1$, $\ \varepsilon_2=-M\dfrac{dI_1}{dt}$, $\ M_{12}=M_{21}$; $1\ \text{H}=1\ \text{Wb/A}$
Coaxial solenoids§ 6.3 →
$M=\mu_0n_1n_2\pi r_1^2l=\dfrac{\mu_0N_1N_2\pi r_1^2}{l}$ (long solenoids, common length $l$, $r_1$ = radius of the inner one; $\times\mu_r$ with a core)
Self-inductance§ 6.3 →
$N\Phi_B=LI$, $\ \varepsilon=-L\dfrac{dI}{dt}$; long solenoid $L=\mu_0n^2Al=\dfrac{\mu_0N^2A}{l}$ ($\times\mu_r$ with a core)
Energy in an inductor§ 6.3 →
$U=\tfrac12LI^2$; field energy per unit volume $u_B=\dfrac{B^2}{2\mu_0}$ (vacuum/air)
AC generator§ 6.3 →
$\varepsilon=NBA\omega\sin\omega t$, $\ \varepsilon_0=NBA\omega$, $\ \nu=\dfrac{\omega}{2\pi}$ ($\theta=\omega t$ between the normal and $\mathbf B$; uniform field, constant $\omega$)
LR circuit (JEE)§ 6.3 →
$i=\dfrac{\varepsilon}{R}\left(1-e^{-t/\tau}\right)$, decay $i=I_0e^{-t/\tau}$, $\ \tau=\dfrac LR$; $\ M=k\sqrt{L_1L_2}$, $0\le k\le1$
Chapter 7 · Alternating Current
RMS values§ 7.1 →
$I=\dfrac{i_m}{\sqrt2}$, $\ V=\dfrac{v_m}{\sqrt2}$ (sinusoidal AC only)
AC in a resistor§ 7.1 →
$i=\dfrac{v_m}{R}\sin\omega t$ (in phase); $\ P=I^2R=\dfrac{V^2}{R}=VI$ (rms values)
AC in a pure inductor§ 7.1 →
$X_L=\omega L$; $\ i=\dfrac{v_m}{X_L}\sin\!\left(\omega t-\dfrac\pi2\right)$ — current lags by $\pi/2$; average power zero
AC in a capacitor§ 7.1 →
$X_C=\dfrac{1}{\omega C}$; $\ i=\dfrac{v_m}{X_C}\sin\!\left(\omega t+\dfrac\pi2\right)$ — current leads by $\pi/2$; average power zero
Impedance of a series LCR circuit§ 7.2 →
$Z=\sqrt{R^2+(X_C-X_L)^2}$, $\ I=\dfrac{V}{Z}$ (RL: put $X_C=0$; RC: put $X_L=0$)
Phase angle§ 7.2 →
$i=i_m\sin(\omega t+\phi)$, $\ \tan\phi=\dfrac{X_C-X_L}{R}$ — current lags if $X_L\gt X_C$, leads if $X_C\gt X_L$
Voltages in a series LCR circuit§ 7.2 →
$V=\sqrt{V_R^2+(V_L-V_C)^2}$ (rms values add as phasors, not as numbers)
Series resonance§ 7.2 →
$\omega_0=\dfrac{1}{\sqrt{LC}}$, $\ \nu_0=\dfrac{1}{2\pi\sqrt{LC}}$; at resonance $Z=R$, $I_0=V/R$, $\phi=0$
Bandwidth and quality factor§ 7.2 →
$\omega_2-\omega_1=\dfrac RL$, $\ Q=\dfrac{\omega_0L}{R}=\dfrac{1}{\omega_0CR}=\dfrac1R\sqrt{\dfrac LC}$ (series LCR circuit)
Average power in AC§ 7.3 →
$P=VI\cos\phi=\tfrac12v_mi_m\cos\phi$ (rms $V$, $I$); series LCR: $P=I^2R=\dfrac{V^2R}{Z^2}$
Power factor§ 7.3 →
$\cos\phi=\dfrac{P}{VI}=\dfrac{R}{Z}$ (series LCR)
Wattless current§ 7.3 →
active $I\cos\phi$; $\ $ wattless $I\sin\phi$
Ideal transformer§ 7.3 →
$\dfrac{V_s}{V_p}=\dfrac{N_s}{N_p}$, $\ \dfrac{I_s}{I_p}=\dfrac{N_p}{N_s}$ (ideal transformer: no flux leakage, no losses; AC only)
Line loss and efficiency§ 7.3 →
$P_\text{loss}=I^2R_\text{line}=\dfrac{P^2R_\text{line}}{V^2}$ (power factor 1); $\ \eta=\dfrac{V_sI_s}{V_pI_p}$
Chapter 8 · Electromagnetic Waves
Displacement current§ 8.1 →
$i_d=\varepsilon_0\dfrac{d\Phi_E}{dt}$; for a capacitor $i_d=\dfrac{dq}{dt}=C\dfrac{dV}{dt}=i_c$
Ampère–Maxwell law§ 8.1 →
$\oint\mathbf B\cdot d\mathbf l=\mu_0i_c+\mu_0\varepsilon_0\dfrac{d\Phi_E}{dt}$ (any surface bounded by the loop)
B between circular capacitor plates§ 8.1 →
$B=\dfrac{\mu_0ir}{2\pi R^2}$ ($r\le R$), $\ B=\dfrac{\mu_0i}{2\pi r}$ ($r\ge R$); uniform $\mathbf E$, no fringing
Maxwell's equations (vacuum)§ 8.1 →
$\oint\mathbf E\cdot d\mathbf A=\dfrac{Q}{\varepsilon_0}$, $\ \oint\mathbf B\cdot d\mathbf A=0$, $\ \oint\mathbf E\cdot d\mathbf l=-\dfrac{d\Phi_B}{dt}$, $\ \oint\mathbf B\cdot d\mathbf l=\mu_0i_c+\mu_0\varepsilon_0\dfrac{d\Phi_E}{dt}$
Speed of EM waves in vacuum§ 8.2 →
$c=\dfrac{1}{\sqrt{\mu_0\varepsilon_0}}=\nu\lambda=\dfrac{\omega}{k}\approx3.0\times10^8$ m/s
Plane EM wave along +x§ 8.2 →
$E_y=E_0\sin(kx-\omega t)$, $\ B_z=B_0\sin(kx-\omega t)$, $\ E_0=cB_0$; travels along $\mathbf E\times\mathbf B$ (vacuum)
EM waves in a medium§ 8.2 →
$v=\dfrac{1}{\sqrt{\mu\varepsilon}}=\dfrac{c}{\sqrt{\mu_r\varepsilon_r}}$, $\ n=\sqrt{\mu_r\varepsilon_r}$; $\nu$ unchanged
Energy density and intensity§ 8.2 →
$u_E=u_B$; $\ \langle u\rangle=\tfrac12\varepsilon_0E_0^2=\dfrac{B_0^2}{2\mu_0}$; $\ I=\tfrac12c\varepsilon_0E_0^2$ (sinusoidal wave, vacuum)
Momentum; point source§ 8.2 →
$p=\dfrac{U}{c}$ (complete absorption); $\ I=\dfrac{P}{4\pi r^2}$ (isotropic source)
🚀 Radiation pressure§ 8.2 →
$\dfrac{I}{c}$ (absorbed), $\ \dfrac{2I}{c}$ (reflected), normal incidence (JEE/NEET only)
Chapter 9 · Ray Optics and Optical Instruments
Cartesian sign convention§ 9.1 →
Distances from the pole (optical centre); along the incident light $+$, against it $-$; heights up $+$. Real object: $u\lt0$; concave mirror $f\lt0$, convex mirror $f\gt0$
Focal length of a mirror§ 9.1 →
$f=\dfrac R2$ (paraxial rays)
Mirror equation§ 9.1 →
$\dfrac1v+\dfrac1u=\dfrac1f=\dfrac2R$ (paraxial rays, Cartesian signs)
Magnification by a mirror§ 9.1 →
$m=\dfrac{h'}{h}=-\dfrac vu=\dfrac{f}{f-u}=\dfrac{f-v}{f}$
🚀 Longitudinal magnification§ 9.1 →
$\dfrac{dv}{du}=-m^2$; image velocity $v_I=-m^2v_O$ along the axis (JEE/NEET only)
Snell's law, refractive index§ 9.2 →
$n_1\sin i=n_2\sin r$ (angles from the normal); $\ n=\dfrac cv$, $\ n_{21}=\dfrac{n_2}{n_1}=\dfrac{v_1}{v_2}$, $\ \lambda=\dfrac{\lambda_0}{n}$ (frequency unchanged)
Apparent depth§ 9.2 →
$h'=\dfrac hn$; raised by $h\left(1-\dfrac1n\right)$ (near-normal viewing)
Lateral shift by a slab§ 9.2 →
$d=\dfrac{t\sin(i-r)}{\cos r}$ (parallel-sided slab)
Critical angle§ 9.2 →
$\sin C=\dfrac{n_\text{rarer}}{n_\text{denser}}$ ($=\tfrac1n$ in air); TIR if denser → rarer and $i\gt C$
Optical fibre, light circle§ 9.2 →
$\sin\theta_{a,\max}=\sqrt{n_1^2-n_2^2}$ (fibre in air); source at depth $h$: $r=\dfrac{h}{\sqrt{n^2-1}}$
🚀 Normal shift by a slab§ 9.2 →
$\Delta=t\left(1-\dfrac1n\right)$ along the light; $n\sin\theta$ constant across parallel layers (JEE/NEET only)
Refraction at a spherical surface§ 9.3 →
$\dfrac{n_2}{v}-\dfrac{n_1}{u}=\dfrac{n_2-n_1}{R}$ (light from $n_1$ into $n_2$; paraxial rays; Cartesian signs)
Lens maker's formula§ 9.3 →
$\dfrac1f=(n_{21}-1)\left(\dfrac1{R_1}-\dfrac1{R_2}\right)$, $n_{21}=\dfrac{n_\text{lens}}{n_\text{medium}}$ (thin lens, paraxial rays)
Thin-lens formula§ 9.3 →
$\dfrac1v-\dfrac1u=\dfrac1f$, $\ m=\dfrac vu$ (thin lens, paraxial rays)
Power; lenses in contact§ 9.3 →
$P=\dfrac1{f\,(\text{m})}$ D; in contact $P=P_1+P_2$, $\ m=m_1m_2$
Displacement method§ 9.3 →
$f=\dfrac{D^2-d^2}{4D}$ (object–screen distance $D\ge4f$, lens positions $d$ apart)
🚀 Separated and silvered lenses§ 9.3 →
$P=P_1+P_2-dP_1P_2$; silvered lens $P=2P_\text{lens}+P_\text{mirror}$ (JEE/NEET only)
Prism§ 9.4 →
$r_1+r_2=A$, $\ \delta=i+e-A$
Minimum deviation; thin prism§ 9.4 →
$n=\dfrac{\sin\frac{A+\delta_m}{2}}{\sin\frac A2}$ ($i=e$, $r=\tfrac A2$); thin prism $\delta=(n-1)A$ (small angles)
Simple microscope§ 9.4 →
$m=1+\dfrac Df$ (near point), $\ \dfrac Df$ (infinity); $D=25$ cm
Compound microscope§ 9.4 →
$m=m_om_e$, $m_o=\dfrac{v_o}{u_o}$; normal adjustment $|m|\approx\dfrac{L}{f_o}\cdot\dfrac{D}{f_e}$
Astronomical telescope§ 9.4 →
$m=\dfrac{f_o}{f_e}$, length $f_o+f_e$ (normal adjustment); near point: $m=\dfrac{f_o}{f_e}\left(1+\dfrac{f_e}{D}\right)$
🚀 Dispersion by a thin prism§ 9.4 →
$\delta_v-\delta_r=(n_v-n_r)A$, $\ \omega=\dfrac{n_v-n_r}{n_y-1}$ (JEE only)
Chapter 10 · Wave Optics
Refraction from Huygens' principle§ 10.1 →
$\dfrac{\sin i}{\sin r}=\dfrac{v_1}{v_2}=\dfrac{n_2}{n_1}$ (plane wave, plane boundary; $i,r$ measured from the normal)
Refractive index and speed§ 10.1 →
$n=\dfrac{c}{v}$, $\quad n_1\sin i=n_2\sin r$
Wavelength and frequency in a medium§ 10.1 →
$\nu$ unchanged; $\ \lambda'=\dfrac{\lambda}{n}$ ($\lambda$ = vacuum wavelength); $\ \dfrac{\lambda_1}{\lambda_2}=\dfrac{v_1}{v_2}$
Reflection from Huygens' principle§ 10.1 →
$r=i$; same speed, frequency and wavelength (plane mirror)
Phase difference along a ray§ 10.1 →
$\Delta\phi=\dfrac{2\pi}{\lambda}\,\Delta x$ (same medium); zero between points of one wavefront
Critical angle in terms of wave speeds§ 10.1 →
$\sin i_c=\dfrac{v_1}{v_2}=\dfrac{n_2}{n_1}$ (only when $v_2\gt v_1$)
Two coherent waves§ 10.2 →
$I=I_1+I_2+2\sqrt{I_1I_2}\cos\phi$; $\ I=4I_0\cos^2\dfrac{\phi}{2}$ if $I_1=I_2=I_0$ (same frequency, constant $\phi$)
Maximum and minimum intensity§ 10.2 →
$I_\text{max}=\left(\sqrt{I_1}+\sqrt{I_2}\right)^2$, $\ I_\text{min}=\left(\sqrt{I_1}-\sqrt{I_2}\right)^2$
Incoherent sources§ 10.2 →
$\overline I=I_1+I_2$ (phase difference random in time)
Conditions for interference§ 10.2 →
$\phi=\dfrac{2\pi}{\lambda}\Delta$; bright $\Delta=n\lambda$, dark $\Delta=\left(n+\tfrac12\right)\lambda$ (sources in phase)
Young's double-slit experiment§ 10.2 →
$\Delta\approx\dfrac{xd}{D}$; bright $x_n=\dfrac{n\lambda D}{d}$, dark $x_n'=\left(n+\tfrac12\right)\dfrac{\lambda D}{d}$; $\ \beta=\dfrac{\lambda D}{d}$, angular width $\dfrac{\lambda}{d}$ ($D\gg d$, small angles)
YDSE intensity; apparatus in a liquid§ 10.2 →
$I(x)=4I_0\cos^2\dfrac{\pi x}{\beta}$; in a medium of index $n$: $\beta'=\dfrac{\beta}{n}$
Single slit: minima§ 10.3 →
$a\sin\theta=n\lambda$, $n=\pm1,\pm2,\dots$; small angles: $y_n=\dfrac{n\lambda D}{a}$ (normal incidence, distant screen)
Width of the central maximum§ 10.3 →
angular $\dfrac{2\lambda}{a}$; linear $w=\dfrac{2\lambda D}{a}$ (small angles; $D\to f$ with a lens)
Secondary maxima§ 10.3 →
$a\sin\theta\approx\left(n+\tfrac12\right)\lambda$, $n=1,2,\dots$; $I\approx4.7\%,\ 1.6\%,\dots$ of $I_0$; width $\dfrac{\lambda D}{a}$
Fringes inside the diffraction envelope§ 10.3 →
bright fringes with $|n|\lt\dfrac{d}{a}$; if $\dfrac da=m$ (whole number): $2m-1$ fringes, orders $\pm m,\pm2m,\dots$ missing
Malus's law§ 10.3 →
$I=I_1\cos^2\theta$ (plane-polarised light $I_1$); unpolarised $I_0$ through one polaroid: $\dfrac{I_0}{2}$
Brewster's law (beyond NCERT)§ 10.3 →
$\tan i_B=n$; reflected and refracted rays perpendicular, $i_B+r=90^\circ$
Chapter 11 · Dual Nature of Radiation and Matter
Work function in joules§ 11.1 →
$\phi_0(\text{J})=\phi_0(\text{eV})\times1.6\times10^{-19}$; minimum energy to free an electron from the surface
Stopping potential§ 11.1 →
$K_{\max}=\tfrac12m_ev_{\max}^2=eV_0$ (independent of intensity)
Saturation current§ 11.1 →
$I_\text{sat}=ne$ ($n$ photoelectrons per second; $\propto$ intensity at fixed $\nu\gt\nu_0$)
Threshold condition§ 11.1 →
Emission only if $\nu\gt\nu_0$, i.e. $\lambda\lt\lambda_0=\dfrac{c}{\nu_0}$ — at any intensity
Einstein's photoelectric equation§ 11.2 →
$K_{\max}=eV_0=h\nu-\phi_0=h(\nu-\nu_0)$ (one photon absorbed by one electron; $\nu\ge\nu_0$)
Threshold frequency and wavelength§ 11.2 →
$\nu_0=\dfrac{\phi_0}{h}$, $\ \lambda_0=\dfrac{hc}{\phi_0}$; $\ \lambda_0(\text{nm})=\dfrac{1240}{\phi_0(\text{eV})}$
V₀ against ν§ 11.2 →
$V_0=\dfrac{h}{e}\nu-\dfrac{\phi_0}{e}$: slope $h/e$ (same for all metals), $\nu$-intercept $\nu_0$
Photon energy and momentum§ 11.2 →
$E=h\nu=\dfrac{hc}{\lambda}$, $\ p=\dfrac{h}{\lambda}=\dfrac{E}{c}$; $\ E(\text{eV})=\dfrac{1240}{\lambda(\text{nm})}$
Photons per second§ 11.2 →
$N=\dfrac{P}{h\nu}=\dfrac{P\lambda}{hc}$; flux $\Phi=\dfrac{I}{h\nu}$ (monochromatic beam)
🚀 Force of a photon beam§ 11.2 →
$F=\dfrac{P}{c}$ absorbed, $\dfrac{2P}{c}$ reflected, $(1+r)\dfrac{P}{c}$ in general (normal incidence; JEE/NEET only)
de Broglie relation§ 11.3 →
$\lambda=\dfrac{h}{p}=\dfrac{h}{mv}$ ($v\ll c$; also $\lambda=h/p$ for a photon)
λ from kinetic energy§ 11.3 →
$\lambda=\dfrac{h}{\sqrt{2mK}}$; charge $q$ through $V$: $\lambda=\dfrac{h}{\sqrt{2mqV}}$ (non-relativistic)
Accelerated electron§ 11.3 →
$\lambda=\dfrac{1.227}{\sqrt V}$ nm (electron from rest through $V$ volts; $V\lesssim20$ kV for 1% accuracy)
Comparing particles§ 11.3 →
Same $v$: $\lambda\propto\dfrac1m$; same $K$: $\lambda\propto\dfrac{1}{\sqrt m}$; same $V$: $\lambda\propto\dfrac{1}{\sqrt{mq}}$
🚀 Davisson–Germer check§ 11.3 →
$d\sin\theta=\lambda$: $0.215\sin50^\circ\approx0.165$ nm vs $\dfrac{1.227}{\sqrt{54}}\approx0.167$ nm (JEE/NEET only)
Chapter 12 · Atoms
Distance of closest approach§ 12.1 →
$d_0=\dfrac{1}{4\pi\varepsilon_0}\dfrac{2Ze^2}{K}$ (alpha particle of kinetic energy $K$, head-on, fixed nucleus, Coulomb force only; an upper limit on the nuclear radius)
Handy constant§ 12.1 →
$\dfrac{e^2}{4\pi\varepsilon_0}=1.44$ MeV fm $=1.44$ eV nm
Electron in a circular orbit (hydrogen)§ 12.1 →
$K=\dfrac{e^2}{8\pi\varepsilon_0r}$, $\ U=-\dfrac{e^2}{4\pi\varepsilon_0r}$, $\ E=-\dfrac{e^2}{8\pi\varepsilon_0r}$; $\ K=-E$, $U=2E$
🚀 Impact parameter and angle§ 12.1 →
$b=\dfrac{d_0}{2}\cot\dfrac\theta2$, $\ r_\text{min}=\dfrac{d_0}{2}\left(1+\operatorname{cosec}\dfrac\theta2\right)$, $\ N(\theta)\propto\dfrac{1}{\sin^4(\theta/2)}$ (JEE/NEET only)
Spectral series of hydrogen§ 12.2 →
$\dfrac1\lambda=R\left(\dfrac1{n_f^2}-\dfrac1{n_i^2}\right)$, $R=1.097\times10^7\ \text{m}^{-1}$, $n_i\gt n_f$; Lyman $n_f=1$, Balmer $2$, Paschen $3$, Brackett $4$, Pfund $5$; limit $\lambda_\infty=n_f^2/R$
Bohr's quantum condition§ 12.2 →
$mvr=\dfrac{nh}{2\pi}$, $n=1,2,3,\dots$; $\ h\nu=E_i-E_f$
Bohr orbits of hydrogen§ 12.2 →
$r_n=\dfrac{\varepsilon_0h^2}{\pi me^2}n^2=0.53\,n^2$ Å; $\ v_n=\dfrac{e^2}{2\varepsilon_0nh}=\dfrac{2.18\times10^6}{n}$ m/s; $\ E_n=-\dfrac{me^4}{8\varepsilon_0^2h^2n^2}=-\dfrac{13.6}{n^2}$ eV
Energies in a Bohr orbit§ 12.2 →
$K_n=-E_n=\dfrac{13.6}{n^2}$ eV, $\ U_n=2E_n$; $\ r\propto n^2$, $v\propto\dfrac1n$, $T\propto n^3$
🚀 Hydrogen-like ions§ 12.2 →
$r_n=\dfrac{0.53\,n^2}{Z}$ Å, $\ v_n\propto\dfrac Zn$, $\ E_n=-\dfrac{13.6\,Z^2}{n^2}$ eV, $\ \dfrac1\lambda=RZ^2\left(\dfrac1{n_f^2}-\dfrac1{n_i^2}\right)$ (JEE/NEET only)
Photon emitted or absorbed (hydrogen)§ 12.3 →
$h\nu=E_i-E_f=13.6\left(\dfrac1{n_f^2}-\dfrac1{n_i^2}\right)$ eV; $\ \lambda\,(\text{nm})=\dfrac{1240}{\Delta E\,(\text{eV})}$
Rydberg constant from theory§ 12.3 →
$R=\dfrac{me^4}{8\varepsilon_0^2h^3c}\approx1.097\times10^7\ \text{m}^{-1}$
Ionisation and excitation (hydrogen)§ 12.3 →
Ionisation from $n$: $|E_n|=\dfrac{13.6}{n^2}$ eV; excitation to $n$: $13.6\left(1-\dfrac1{n^2}\right)$ eV ($10.2$, $12.09$, $12.75$ eV)
Number of spectral lines§ 12.3 →
From level $n$ (many atoms): $N=\dfrac{n(n-1)}{2}$; one atom: at most $n-1$ photons
de Broglie's orbit condition§ 12.3 →
$2\pi r_n=n\lambda$, $\ \lambda=\dfrac{h}{mv}$ $\Rightarrow$ $mvr=\dfrac{nh}{2\pi}$
🚀 Recoil of the emitting atom§ 12.3 →
$p=\dfrac Ec$, $\ v_\text{recoil}=\dfrac{E}{m_\text{atom}c}$ (JEE/NEET only)
Chapter 13 · Nuclei
Atomic mass unit§ 13.1 →
$1\ \text{u}=\tfrac{1}{12}\,m\left({}^{12}\text{C atom}\right)=1.66\times10^{-27}\ \text{kg}$
Average atomic mass (natural abundances $f_i$, $\sum f_i=1$)§ 13.1 →
$\bar m=\sum_i f_i\,m_i$
Composition of a nucleus ${}^{A}_{Z}\text{X}$§ 13.1 →
$A=Z+N$
Mass spectrometer (ion of charge $q$, uniform $B$ perpendicular to $v$; accelerated from rest through $V$)§ 13.1 →
$r=\dfrac{mv}{qB}=\dfrac1B\sqrt{\dfrac{2mV}{q}}$
Nuclear radius (approximate; $R_0\approx1.2$ fm)§ 13.1 →
$R=R_0A^{1/3}$
Nuclear density (same for all nuclei; $m$ = nucleon mass)§ 13.1 →
$\rho=\dfrac{3m}{4\pi R_0^3}\approx2.3\times10^{17}\ \text{kg/m}^3$
Mass–energy equivalence§ 13.2 →
$E=mc^2,\qquad \Delta E=\Delta m\,c^2$
Energy of one atomic mass unit§ 13.2 →
$1\ \text{u}\times c^2=931.5\ \text{MeV}$
Mass defect (with atomic masses use $m_\text{H}$ so the electrons cancel)§ 13.2 →
$\Delta M=Zm_\text{H}+(A-Z)m_n-M_\text{atom}$
Binding energy and binding energy per nucleon§ 13.2 →
$E_b=\Delta M\,c^2,\qquad E_{bn}=\dfrac{E_b}{A}$
Energy released in a nuclear change (nucleons conserved)§ 13.2 →
$Q=\sum E_b(\text{products})-\sum E_b(\text{reactants})$
$Q$-value of a nuclear reaction (atomic masses if the electrons balance)§ 13.3 →
$Q=(m_\text{initial}-m_\text{final})\times931.5\ \text{MeV/u}$
$Q$ from binding energies (nucleons conserved)§ 13.3 →
$Q=\sum E_b(\text{products})-\sum E_b(\text{reactants})$
Chain reaction after $n$ generations ($k=1$ critical)§ 13.3 →
$N_n=N_0k^n$
Fission rate for power $P$ (energy $E_f\approx200$ MeV per fission)§ 13.3 →
$\dfrac{dN}{dt}=\dfrac{P}{E_f}$
Key fusion reactions§ 13.3 →
${}^{2}\text{H}+{}^{3}\text{H}\to{}^{4}\text{He}+n+17.6\ \text{MeV}$
Net proton–proton chain (the Sun; includes positron annihilation)§ 13.3 →
$4\,{}^{1}_{1}\text{H}+2e^{-}\to{}^{4}_{2}\text{He}+2\nu+26.7\ \text{MeV}$
Chapter 14 · Semiconductor Electronics: Materials, Devices and Simple Circuits
Classification by band gap§ 14.1 →
metal: bands overlap or partly filled · semiconductor: $E_g\lt3$ eV (Si $1.1$, Ge $0.7$ eV) · insulator: $E_g\gt3$ eV (diamond $5.4$ eV)
Absorption across the gap§ 14.1 →
$h\nu\ge E_g$; $\ \lambda_{\max}=\dfrac{hc}{E_g}$, $\ \lambda_{\max}(\text{nm})\approx\dfrac{1240}{E_g(\text{eV})}$
Thermal energy scale§ 14.1 →
$k_BT$, $\ k_B=8.62\times10^{-5}$ eV/K; $\ k_BT\approx0.026$ eV at $300$ K
Levels in a band§ 14.1 →
$N$ atoms: each atomic level → $N$ levels; Si: $8N$ ($3s,3p$) states → valence band $4N$ (full at $0$ K) + conduction band $4N$
Intrinsic semiconductor§ 14.2 →
$n_e=n_h=n_i$ (pure crystal; $n_i$ depends on material and $T$; Si $1.5\times10^{16}$, Ge $2.4\times10^{19}\ \text{m}^{-3}$ at $300$ K)
Law of mass action§ 14.2 →
$n_en_h=n_i^2$ (thermal equilibrium, intrinsic or doped, same $T$)
Doped semiconductor§ 14.2 →
$n_e+N_A=n_h+N_D$; n-type ($N_D\gg n_i$): $n_e\approx N_D$, $n_h\approx n_i^2/N_D$; p-type ($N_A\gg n_i$): $n_h\approx N_A$, $n_e\approx n_i^2/N_A$
Conductivity of a semiconductor§ 14.2 →
$\sigma=e(n_e\mu_e+n_h\mu_h)$, $\ \rho=1/\sigma$
Logic gates (beyond NCERT; JEE Main/NEET)§ 14.2 →
NOT $\overline A$ · OR $A+B$ · AND $A\cdot B$ · NAND $\overline{A\cdot B}$ · NOR $\overline{A+B}$; $\ \overline{A+B}=\overline A\cdot\overline B$, $\ \overline{A\cdot B}=\overline A+\overline B$
Junction under bias§ 14.3 →
barrier: $V_0-V$ (forward), $V_0+V$ (reverse); $V_0\approx0.7$ V (Si), $0.3$ V (Ge)
Dynamic resistance of a diode§ 14.3 →
$r_d=\dfrac{\Delta V}{\Delta I}$ (small change about an operating point)
Diode in a series circuit§ 14.3 →
$I=\dfrac{E-0.7\ \text{V}}{R}$ (forward-biased Si diode, constant-drop model); reverse biased: $I\approx0$
Rectifier outputs§ 14.3 →
half-wave: $V_\text{dc}=\dfrac{V_m}{\pi}$, $V_\text{rms}=\dfrac{V_m}{2}$, pulses at $f$ · full-wave: $V_\text{dc}=\dfrac{2V_m}{\pi}$, $V_\text{rms}=\dfrac{V_m}{\sqrt2}$, pulses at $2f$ (ideal diodes, resistive load)
Zener regulator (beyond NCERT; JEE Main/NEET)§ 14.3 →
$I_s=\dfrac{V_\text{in}-V_Z}{R_s}$, $\ I_L=\dfrac{V_Z}{R_L}$, $\ I_Z=I_s-I_L$ (valid while $I_Z\gt0$)