Class 12 Physics Formula Sheet 2026-27 (Free PDF), All 14 Chapters

Every formula from all 14 NCERT Class 12 Physics chapters in one place, checked against the NCERT textbook (Reprint 2026-27) and the official CBSE 2026-27 syllabus. Qualitative-only topics in the 2026-27 syllabus are marked separately.

How to use this sheet. Plain boxes cover the CBSE 2026-27 board-exam syllabus. Amber boxes marked ★ Qualitative only are topics the 2026-27 syllabus lists for qualitative treatment, so expect conceptual questions on them rather than derivations or numericals. Grey boxes marked ◇ Not in 2026-27 syllabus are in the NCERT book but not in the current syllabus. They are kept for reference and JEE/NEET. Constants: $e = 1.6 \times 10^{-19}\,\text{C}$, $\frac{1}{4\pi\varepsilon_0} = 9 \times 10^{9}\,\text{N m}^2\,\text{C}^{-2}$, $\varepsilon_0 = 8.854 \times 10^{-12}\,\text{C}^2\,\text{N}^{-1}\,\text{m}^{-2}$, $\mu_0 = 4\pi \times 10^{-7}\,\text{T m A}^{-1}$, $h = 6.63 \times 10^{-34}\,\text{J s}$, $c = 3 \times 10^{8}\,\text{m s}^{-1}$, $m_e = 9.1 \times 10^{-31}\,\text{kg}$.

Electric Charges and Fields
Electrostatic Potential and Capacitance
Current Electricity
Moving Charges and Magnetism
Magnetism and Matter
Electromagnetic Induction
Alternating Current
Electromagnetic Waves
Ray Optics and Optical Instruments
Wave Optics
Dual Nature of Radiation and Matter
Atoms
Nuclei
Semiconductor Electronics

1Electric Charges and Fields

Charge and Coulomb's law

Quantisation: $q = ne$, $n = 0, \pm 1, \pm 2, \dots$
Charge is additive and conserved.
Coulomb's law: $$F = \frac{1}{4\pi\varepsilon_0}\,\frac{q_1 q_2}{r^2}$$
Vector form: $\vec{F}_{21} = \dfrac{1}{4\pi\varepsilon_0}\dfrac{q_1 q_2}{r_{21}^2}\,\hat{r}_{21}$ (force on $q_2$ due to $q_1$)
Superposition: $\vec{F}_1 = \vec{F}_{12} + \vec{F}_{13} + \dots + \vec{F}_{1n}$

Electric field

$\vec{E} = \vec{F}/q$, SI unit N C$^{-1}$ (= V m$^{-1}$)
Point charge: $$\vec{E} = \frac{1}{4\pi\varepsilon_0}\,\frac{q}{r^2}\,\hat{r}$$
System of charges: $\vec{E} = \dfrac{1}{4\pi\varepsilon_0}\sum_i \dfrac{q_i}{r_{iP}^2}\,\hat{r}_{iP}$
Charge densities: $\lambda = \frac{\Delta Q}{\Delta l}$, $\sigma = \frac{\Delta Q}{\Delta S}$, $\rho = \frac{\Delta Q}{\Delta V}$

Electric dipole

Dipole moment $\vec{p} = q \times 2a\,\hat{p}$, from $-q$ to $+q$
Axial point ($r \gg a$): $\vec{E} = \dfrac{1}{4\pi\varepsilon_0}\dfrac{2\vec{p}}{r^3}$
Equatorial point ($r \gg a$): $\vec{E} = -\dfrac{1}{4\pi\varepsilon_0}\dfrac{\vec{p}}{r^3}$
Dipole field falls as $1/r^3$; point-charge field as $1/r^2$.

Dipole in a uniform field

Torque: $\vec{\tau} = \vec{p} \times \vec{E}$, $\ |\tau| = pE\sin\theta$
Net force = 0 in a uniform field
Non-uniform field: net force is non-zero

Flux and Gauss's law

Flux: $\Delta\phi = \vec{E}\cdot\Delta\vec{S} = E\,\Delta S\cos\theta$
Gauss's law: $$\oint \vec{E}\cdot d\vec{S} = \frac{q_{\text{enclosed}}}{\varepsilon_0}$$
Charges outside the surface contribute zero net flux.

Applications of Gauss's law

Infinite straight wire: $\vec{E} = \dfrac{\lambda}{2\pi\varepsilon_0 r}\,\hat{n}$
Infinite plane sheet: $\vec{E} = \dfrac{\sigma}{2\varepsilon_0}\,\hat{n}$ (independent of distance)
Thin spherical shell, outside ($r \ge R$): $\vec{E} = \dfrac{q}{4\pi\varepsilon_0 r^2}\,\hat{r}$
Thin spherical shell, inside ($r < R$): $E = 0$

2Electrostatic Potential and Capacitance

Potential

$V = W/q$: work done by an external force to bring a unit positive charge from infinity
Point charge: $V = \dfrac{1}{4\pi\varepsilon_0}\dfrac{q}{r}$
Dipole: $V = \dfrac{1}{4\pi\varepsilon_0}\dfrac{p\cos\theta}{r^2}$ (zero on the equatorial plane)
System of charges: $V = \dfrac{1}{4\pi\varepsilon_0}\sum_i \dfrac{q_i}{r_{iP}}$
Shell: $V = \dfrac{q}{4\pi\varepsilon_0 r}$ outside, $V = \dfrac{q}{4\pi\varepsilon_0 R}$ inside (constant)

Field and potential

$E = -\dfrac{dV}{dr}$, i.e. $|\vec{E}| = \dfrac{|\delta V|}{\delta l}$ along the normal to the equipotential
$\vec{E}$ points in the direction of steepest decrease of $V$
$\vec{E}$ is perpendicular to every equipotential surface
No work is done moving a charge along an equipotential surface

Potential energy

Two charges: $U = \dfrac{1}{4\pi\varepsilon_0}\dfrac{q_1 q_2}{r_{12}}$
Three charges: $U = \dfrac{1}{4\pi\varepsilon_0}\left(\dfrac{q_1q_2}{r_{12}} + \dfrac{q_1q_3}{r_{13}} + \dfrac{q_2q_3}{r_{23}}\right)$
Single charge in external field: $qV(\vec{r})$
Two charges in external field: $q_1V(\vec{r}_1) + q_2V(\vec{r}_2) + \dfrac{q_1q_2}{4\pi\varepsilon_0 r_{12}}$
Dipole in uniform field: $U(\theta) = -pE\cos\theta = -\vec{p}\cdot\vec{E}$

Conductors and dielectrics

Inside a conductor: $E = 0$; $V$ is constant throughout
Just outside the surface: $\vec{E} = \dfrac{\sigma}{\varepsilon_0}\,\hat{n}$
Excess charge resides only on the surface
Polarisation: $\vec{P} = \chi_e\vec{E}$ ($\chi_e$ = electric susceptibility)
Dielectric constant: $K = \varepsilon_r = \varepsilon/\varepsilon_0$

Capacitance

$C = Q/V$, SI unit farad (F)
Parallel plate (vacuum): $$C_0 = \frac{\varepsilon_0 A}{d}$$
Filled with dielectric: $C = \dfrac{K\varepsilon_0 A}{d} = KC_0$
Series: $\dfrac{1}{C} = \dfrac{1}{C_1} + \dfrac{1}{C_2} + \dots$ (same $Q$ on each)
Parallel: $C = C_1 + C_2 + \dots$ (same $V$ across each)

Energy stored (formulae only)

$$U = \frac{Q^2}{2C} = \frac{1}{2}CV^2 = \frac{1}{2}QV$$
Energy density of the field: $u = \dfrac{1}{2}\varepsilon_0 E^2$

3Current Electricity

Current, Ohm's law, resistivity

$I = \dfrac{q}{t}$; instantaneous $I = \dfrac{dq}{dt}$
Ohm's law: $V = IR$
$R = \rho\dfrac{l}{A}$; conductivity $\sigma = 1/\rho$
Current density $j = I/A$; $\ \vec{j} = \sigma\vec{E}$

Drift velocity and mobility

$\vec{v}_d = -\dfrac{e\vec{E}}{m}\,\tau$ ($\tau$ = relaxation time)
$I = neA\,v_d$
$\rho = \dfrac{m}{ne^2\tau}$
Mobility: $\mu = \dfrac{|v_d|}{E} = \dfrac{e\tau}{m}$

Temperature, energy, power

$\rho_T = \rho_0\left[1 + \alpha(T - T_0)\right]$ ($\alpha$ = temperature coefficient of resistivity)
$R_2 = R_1\left[1 + \alpha(T_2 - T_1)\right]$
Power: $P = VI = I^2R = \dfrac{V^2}{R}$
Energy: $W = VIt$

Cell: emf and internal resistance

Terminal voltage: $V = \varepsilon - Ir$
Current through external $R$: $I = \dfrac{\varepsilon}{R + r}$
Series: $\varepsilon_{eq} = \varepsilon_1 + \varepsilon_2$, $\ r_{eq} = r_1 + r_2$
Parallel: $\varepsilon_{eq} = \dfrac{\varepsilon_1 r_2 + \varepsilon_2 r_1}{r_1 + r_2}$, $\ r_{eq} = \dfrac{r_1 r_2}{r_1 + r_2}$

Resistors in combination

Series: $R = R_1 + R_2 + \dots$
Parallel: $\dfrac{1}{R} = \dfrac{1}{R_1} + \dfrac{1}{R_2} + \dots$

Kirchhoff's rules and Wheatstone bridge

Junction rule: $\sum I_{\text{in}} = \sum I_{\text{out}}$ (charge conservation)
Loop rule: $\sum \Delta V = 0$ round any closed loop (energy conservation)
Balanced bridge ($I_g = 0$): $$\frac{R_2}{R_1} = \frac{R_4}{R_3}$$

4Moving Charges and Magnetism

Lorentz force

$\vec{F} = q\left[\vec{E} + \vec{v}\times\vec{B}\right]$
Magnetic part: $F = qvB\sin\theta$; zero when $\vec{v} \parallel \vec{B}$
Magnetic force does no work (it is always $\perp$ to $\vec{v}$)
Current-carrying conductor: $\vec{F} = I\,\vec{l}\times\vec{B}$

Charge moving in a uniform $\vec{B}$

Radius: $r = \dfrac{mv}{qB}$ (for $\vec{v} \perp \vec{B}$)
Angular frequency $\omega = \dfrac{qB}{m}$; frequency $\nu = \dfrac{qB}{2\pi m}$
Period $T = \dfrac{2\pi m}{qB}$ (independent of speed)
Helix pitch: $p = v_{\parallel}\,T = \dfrac{2\pi m\,v_{\parallel}}{qB}$

Biot-Savart law

$$d\vec{B} = \frac{\mu_0}{4\pi}\,\frac{I\,d\vec{l}\times\vec{r}}{r^3}$$
$|dB| = \dfrac{\mu_0}{4\pi}\dfrac{I\,dl\sin\theta}{r^2}$
Circular loop, on axis: $B = \dfrac{\mu_0 I R^2}{2(x^2 + R^2)^{3/2}}$
At the centre: $B = \dfrac{\mu_0 I}{2R}$; for $N$ turns multiply by $N$

Ampere's circuital law

$$\oint \vec{B}\cdot d\vec{l} = \mu_0 I_e$$
Infinitely long straight wire: $B = \dfrac{\mu_0 I}{2\pi r}$
Field lines are circles round the wire (right-hand rule).

Solenoid (qualitative only)

Inside a long solenoid: $B = \mu_0 n I$ ($n$ = turns per unit length)
Field inside is uniform; outside it is nearly zero

Parallel currents and the ampere

Force per unit length: $f = \dfrac{\mu_0 I_a I_b}{2\pi d}$
Like currents attract; opposite currents repel
1 A: the current that gives $2 \times 10^{-7}\,\text{N m}^{-1}$ between two wires 1 m apart

Current loop as a magnetic dipole

Magnetic moment: $\vec{m} = NI\vec{A}$
Torque: $\vec{\tau} = \vec{m}\times\vec{B}$, $\ \tau = NIAB\sin\theta$
Field on the axis far away: $B = \dfrac{\mu_0}{4\pi}\dfrac{2m}{x^3}$

Moving coil galvanometer

Equilibrium: $NIAB = k\phi$ ($k$ = torsional constant)
Current sensitivity: $\dfrac{\phi}{I} = \dfrac{NAB}{k}$
Voltage sensitivity: $\dfrac{\phi}{V} = \dfrac{NAB}{kR}$
Ammeter: small shunt $r_s$ in parallel, $\ r_s = \dfrac{I_g R_G}{I - I_g}$
Voltmeter: large $R$ in series, $\ R = \dfrac{V}{I_g} - R_G$

5Magnetism and Matter

Bar magnet (qualitative only)

Behaves like a solenoid; dipole moment $\vec{m}$ points S to N
Axial field: $B = \dfrac{\mu_0}{4\pi}\dfrac{2m}{r^3}$; equatorial: $B = \dfrac{\mu_0}{4\pi}\dfrac{m}{r^3}$ (opposite to $\vec{m}$)
Torque: $\vec{\tau} = \vec{m}\times\vec{B}$; energy $U = -\vec{m}\cdot\vec{B}$
Field lines are closed loops; they never intersect

Magnetisation

$\vec{M} = \dfrac{\vec{m}_{net}}{V}$ (A m$^{-1}$)
$\vec{B} = \mu_0(\vec{H} + \vec{M})$
$\vec{M} = \chi\vec{H}$ ($\chi$ = magnetic susceptibility)
$\vec{B} = \mu_0\mu_r\vec{H} = \mu\vec{H}$, $\ \mu_r = 1 + \chi$

Magnetic materials

Property	Diamagnetic	Paramagnetic	Ferromagnetic
$\chi$	$-1 \le \chi < 0$	$0 < \chi < \varepsilon$ (small)	$\chi \gg 1$
$\mu_r$	$0 \le \mu_r < 1$	$1 < \mu_r < 1 + \varepsilon$	$\mu_r \gg 1$
In a field	moves to weaker region	moves to stronger region	strongly attracted
Examples	Bi, Cu, Pb, Si, water, NaCl	Al, Na, Ca, O$_2$, CuCl$_2$	Fe, Ni, Co, Gd

Superconductors are perfect diamagnets ($\chi = -1$, Meissner effect). Above the Curie temperature a ferromagnet becomes paramagnetic.

Gauss's law for magnetism

$\oint \vec{B}\cdot d\vec{S} = 0$: net magnetic flux through any closed surface is zero (no isolated poles)

6Electromagnetic Induction

Flux, Faraday and Lenz

$\Phi_B = \vec{B}\cdot\vec{A} = BA\cos\theta$ (weber, Wb)
Faraday's law: $$\varepsilon = -N\,\frac{d\Phi_B}{dt}$$
Lenz's law: the induced current opposes the change in flux that produces it (the minus sign)

Motional emf

Rod of length $l$ moving with speed $v$ $\perp$ to $B$: $\varepsilon = Blv$
The work done to move the rod is dissipated as Joule heat in the circuit (Lenz's law and energy conservation).

Mutual inductance

$N_1\Phi_1 = M_{12}I_2$; $\ \varepsilon_1 = -M\dfrac{dI_2}{dt}$
Coaxial solenoids: $M_{12} = M_{21} = \mu_0 n_1 n_2 \pi r_1^2 l$
Unit: henry (H)

Self-inductance

$N\Phi_B = LI$; $\ \varepsilon = -L\dfrac{dI}{dt}$
Long solenoid: $L = \mu_0 n^2 A l$; with core: $L = \mu_r\mu_0 n^2 A l$
Energy stored: $W = \dfrac{1}{2}LI^2$

AC generator

Flux $\Phi_B = BA\cos\omega t$; induced emf $\varepsilon = NBA\omega\sin\omega t = \varepsilon_0\sin\omega t$
Peak emf: $\varepsilon_0 = NBA\omega$, where $\omega = 2\pi\nu$

7Alternating Current

AC basics and rms values

$v = v_m\sin\omega t$, $\ i = i_m\sin\omega t$
$I = \dfrac{i_m}{\sqrt{2}} = 0.707\,i_m$, $\ V = \dfrac{v_m}{\sqrt{2}}$
Average power in $R$: $\bar{P} = \dfrac{1}{2}i_m^2R = I^2R$
Mean of $i$ over a full cycle = 0

Pure $R$, $L$, $C$

Element	Reactance	Phase of $i$	Avg power
$R$	$R$	in phase	$I^2R$
$L$	$X_L = \omega L$	lags $v$ by $\pi/2$	0
$C$	$X_C = \dfrac{1}{\omega C}$	leads $v$ by $\pi/2$	0

Series LCR circuit

Impedance: $$Z = \sqrt{R^2 + (X_C - X_L)^2}$$
$i_m = \dfrac{v_m}{Z}$; phase $\tan\phi = \dfrac{X_C - X_L}{R}$
Resonance: $X_L = X_C$, $\ \omega_0 = \dfrac{1}{\sqrt{LC}}$, $Z = R$ (current is maximum)

Power and power factor

$P = VI\cos\phi$; power factor $\cos\phi = \dfrac{R}{Z}$
Pure $L$ or $C$: $\phi = \pi/2$, $P = 0$ (wattless current)
Resonance or pure $R$: $\cos\phi = 1$

Transformer (ideal)

$$\frac{V_s}{V_p} = \frac{N_s}{N_p} = \frac{I_p}{I_s}$$
Step-up: $N_s > N_p$; step-down: $N_s < N_p$. Losses: flux leakage, winding resistance, eddy currents, hysteresis.

8Electromagnetic Waves

Displacement current

$i_d = \varepsilon_0\dfrac{d\Phi_E}{dt}$
Ampere-Maxwell law: $\oint \vec{B}\cdot d\vec{l} = \mu_0 i_c + \mu_0\varepsilon_0\dfrac{d\Phi_E}{dt}$

EM wave properties

$E_x = E_0\sin(kz - \omega t)$, $\ B_y = B_0\sin(kz - \omega t)$
$k = 2\pi/\lambda$, $\ \omega = 2\pi\nu$, $\ c = \nu\lambda$
$c = \dfrac{1}{\sqrt{\mu_0\varepsilon_0}}$; in a medium $v = \dfrac{1}{\sqrt{\mu\varepsilon}}$
$\dfrac{E_0}{B_0} = c$; $\vec{E}$, $\vec{B}$ and the direction of travel are mutually $\perp$ (transverse)

Electromagnetic spectrum

Type	Wavelength range	Production / use
Radio	> 0.1 m	Oscillating charges in antennas; radio, TV, cellphones
Microwave	0.1 m to 1 mm	Klystron, magnetron; radar, microwave ovens
Infrared	1 mm to 700 nm	Hot bodies; remote controls, physical therapy, greenhouse effect
Visible	700 nm to 400 nm	Electron transitions in atoms; vision
Ultraviolet	400 nm to 1 nm	Sun, special lamps; LASIK, water purification
X-rays	1 nm to $10^{-3}$ nm	X-ray tubes; medical diagnosis
Gamma rays	< $10^{-3}$ nm	Radioactive decay of nuclei; cancer treatment

9Ray Optics and Optical Instruments

Spherical mirrors

Focal length $f = R/2$
Mirror equation: $$\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$$
Magnification $m = \dfrac{h'}{h} = -\dfrac{v}{u}$
Cartesian sign convention: distances measured from the pole; along incident light positive.

Refraction and TIR

Snell's law: $n_1\sin i = n_2\sin r$; $\ n_{21} = \dfrac{n_2}{n_1} = \dfrac{v_1}{v_2}$
Apparent depth: $\dfrac{\text{real depth}}{\text{apparent depth}} = n$
Critical angle (denser to rarer): $\sin i_c = \dfrac{1}{n_{21}}$, with $n_{21}$ of the denser w.r.t. rarer medium
TIR needs: light going denser to rarer, and $i > i_c$

Spherical surface and lenses

Single surface: $\dfrac{n_2}{v} - \dfrac{n_1}{u} = \dfrac{n_2 - n_1}{R}$
Lens maker's formula: $$\frac{1}{f} = (n_{21} - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$$
Thin lens: $\dfrac{1}{v} - \dfrac{1}{u} = \dfrac{1}{f}$; $\ m = \dfrac{h'}{h} = \dfrac{v}{u}$

Power and lens combinations

$P = \dfrac{1}{f}$ ($f$ in metres), unit dioptre (D)
Lenses in contact: $\dfrac{1}{f} = \dfrac{1}{f_1} + \dfrac{1}{f_2} + \dots$, $\ P = P_1 + P_2 + \dots$
Total magnification: $m = m_1 m_2 m_3 \dots$

Prism

$r_1 + r_2 = A$; $\ A + \delta = i + e$
Minimum deviation ($i = e$, $r_1 = r_2 = A/2$): $$n_{21} = \frac{\sin\left[(A + D_m)/2\right]}{\sin(A/2)}$$
Thin prism: $D_m = (n_{21} - 1)A$

Optical instruments ($D = 25$ cm)

Simple microscope: $m = 1 + \dfrac{D}{f}$ (image at $D$); $\ m = \dfrac{D}{f}$ (image at infinity)
Compound microscope (final image at infinity): $m = m_o m_e = \dfrac{L}{f_o}\cdot\dfrac{D}{f_e}$, $L$ = tube length
Telescope (normal adjustment): $m = \dfrac{f_o}{f_e}$; tube length $f_o + f_e$
Reflecting telescopes (Cassegrain) avoid chromatic aberration and need no huge lens.

10Wave Optics

Huygens principle and refraction

$n_{21} = \dfrac{v_1}{v_2} = \dfrac{\sin i}{\sin r}$
In a medium of index $n$: $\lambda' = \lambda/n$, speed $c/n$; frequency unchanged
Reflection: angle of incidence = angle of reflection

Interference

Constructive: path difference $= n\lambda$; destructive: $= (n + \tfrac{1}{2})\lambda$
Resultant intensity: $I = 4I_0\cos^2\dfrac{\phi}{2}$
Sustained interference needs coherent sources (constant phase difference)

Young's double slit (final expressions only)

Bright fringes: $x_n = \dfrac{n\lambda D}{d}$; dark fringes: $x_n = \left(n + \dfrac{1}{2}\right)\dfrac{\lambda D}{d}$
Fringe width: $$\beta = \frac{\lambda D}{d}$$
$d$ = slit separation, $D$ = slit-to-screen distance. Fringes are equally spaced.

Single-slit diffraction (qualitative only)

Minima: $\theta \approx \dfrac{n\lambda}{a}$, $n = \pm 1, \pm 2, \dots$
Central maximum: angular width $\dfrac{2\lambda}{a}$; it is twice as wide as other maxima

Polarisation

Malus' law: $I = I_0\cos^2\theta$
Unpolarised light through a polaroid: intensity halves

11Dual Nature of Radiation and Matter

Photon

Energy $E = h\nu = \dfrac{hc}{\lambda}$
Momentum $p = \dfrac{h\nu}{c} = \dfrac{h}{\lambda}$
$1\,\text{eV} = 1.6 \times 10^{-19}\,\text{J}$

Einstein's photoelectric equation

$$K_{max} = h\nu - \phi_0 = eV_0$$
Work function $\phi_0 = h\nu_0$; threshold wavelength $\lambda_0 = c/\nu_0$
$V_0 = \dfrac{h}{e}\nu - \dfrac{\phi_0}{e}$: straight-line graph of $V_0$ vs $\nu$, slope $h/e$

Photoelectric observations

Photocurrent $\propto$ intensity (for $\nu > \nu_0$)
Stopping potential depends on $\nu$, not on intensity
No emission below $\nu_0$, however intense the light; emission is instantaneous

de Broglie relation

$$\lambda = \frac{h}{p} = \frac{h}{mv} = \frac{h}{\sqrt{2mK}}$$
Electron accelerated through $V$: $\lambda = \dfrac{h}{\sqrt{2meV}} = \dfrac{1.227}{\sqrt{V}}\,\text{nm}$

12Atoms

Rutherford scattering

Distance of closest approach ($K$ = kinetic energy of the alpha particle): $$d = \frac{1}{4\pi\varepsilon_0}\,\frac{2Ze^2}{K}$$
Most alpha particles pass undeflected: the atom is mostly empty, with a tiny massive nucleus

Bohr's postulates

Stationary orbits: $L = mvr = \dfrac{nh}{2\pi}$
Transition: $h\nu = E_i - E_f$

Bohr model of hydrogen

Radius: $r_n = \dfrac{n^2 h^2 \varepsilon_0}{\pi m e^2} = n^2 a_0$, Bohr radius $a_0 = 5.29 \times 10^{-11}\,\text{m}$
Speed: $v_n = \dfrac{e^2}{2\varepsilon_0 n h} = \dfrac{1}{n}\,\dfrac{e^2}{4\pi\varepsilon_0\,(h/2\pi)}$
Energy: $$E_n = -\frac{me^4}{8n^2\varepsilon_0^2h^2} = -\frac{13.6}{n^2}\,\text{eV}$$
KE $= -E_n$, PE $= 2E_n$; ionisation energy of H = 13.6 eV

Line spectra (qualitative only)

Photon emitted when the electron drops from $n_i$ to $n_f$: $h\nu_{if} = E_{n_i} - E_{n_f}$
Each allowed transition gives one line, so hydrogen emits a discrete line spectrum

13Nuclei

Composition and size

$A = Z + N$; $\ 1\,\text{u} = 1.660539 \times 10^{-27}\,\text{kg}$
Radius: $R = R_0A^{1/3}$, $R_0 = 1.2 \times 10^{-15}\,\text{m}$
Nuclear density is the same for all nuclei (about $2.3 \times 10^{17}\,\text{kg m}^{-3}$)

Mass-energy and binding energy

$E = mc^2$; $\ 1\,\text{u} \equiv 931.5\,\text{MeV}$
Mass defect: $\Delta M = \left[Zm_p + (A - Z)m_n\right] - M$
Binding energy $E_b = \Delta M\,c^2$; per nucleon $E_{bn} = E_b/A$
$E_{bn}$ peaks at about 8.75 MeV near $A = 56$; roughly constant (about 8 MeV) for $30 < A < 170$

Nuclear force

Much stronger than Coulomb force; short range (a few fm)
Nearly independent of charge (n-n, p-p and n-p similar)

Fission and fusion

${}^{1}_{0}\text{n} + {}^{235}_{92}\text{U} \to {}^{236}_{92}\text{U} \to {}^{144}_{56}\text{Ba} + {}^{89}_{36}\text{Kr} + 3\,{}^{1}_{0}\text{n}$ (about 200 MeV)
${}^{1}_{1}\text{H} + {}^{1}_{1}\text{H} \to {}^{2}_{1}\text{H} + e^+ + \nu + 0.42\,\text{MeV}$
${}^{2}_{1}\text{H} + {}^{2}_{1}\text{H} \to {}^{3}_{2}\text{He} + \text{n} + 3.27\,\text{MeV}$

14Semiconductor Electronics

Energy bands (qualitative)

Conductor: bands overlap, or the conduction band is partly filled
Insulator: $E_g > 3$ eV
Semiconductor: $E_g < 3$ eV (Si about 1.1 eV, Ge about 0.7 eV)

Carrier concentration

Intrinsic: $n_e = n_h = n_i$
In equilibrium: $$n_e\,n_h = n_i^2$$
n-type (pentavalent dopant, e.g. As, P): $n_e \gg n_h$
p-type (trivalent dopant, e.g. B, Al, In): $n_h \gg n_e$

p-n junction diode

Diffusion and drift form a depletion layer and barrier potential
Forward bias: barrier lowered, current rises sharply past the threshold (about 0.7 V for Si, 0.2 V for Ge)
Reverse bias: barrier raised, very small current until breakdown

Diode as a rectifier

Half-wave: one diode, output frequency = input frequency $\nu$
Full-wave (centre-tap, two diodes): output frequency = $2\nu$
A capacitor filter smooths the pulsating dc

Element	Reactance	Phase of \(i\)	Avg power
\(R\)	\(R\)	in phase	\(I^2R\)
\(L\)	\(X_L = \omega L\)	lags \(v\) by \(\pi/2\)	0
\(C\)	\(X_C = \dfrac{1}{\omega C}\)	leads \(v\) by \(\pi/2\)	0