All Class 12 Physics formulas

Charge and force

Quantisation of charge

$\displaystyle q = ne$

e = 1.6 × 10⁻¹⁹ C.

Coulomb's law

$\displaystyle F = \frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{r^2}$

1/(4πε₀) ≈ 9 × 10⁹ N m² C⁻².

Electric field of a point charge

$\displaystyle E = \frac{1}{4\pi\varepsilon_0}\frac{q}{r^2}$

Force on a charge in a field

$\displaystyle \vec F = q\vec E$

Electric dipole

Dipole moment

$\displaystyle \vec p = q(2\vec a)$

Points from negative to positive charge.

Field on the axis

$\displaystyle E = \frac{1}{4\pi\varepsilon_0}\frac{2p}{r^3}$

For r much larger than a.

Field on the equatorial line

$\displaystyle E = \frac{1}{4\pi\varepsilon_0}\frac{p}{r^3}$

Opposite in direction to p.

Torque in a uniform field

$\displaystyle \vec\tau = \vec p\times\vec E$

Gauss's law

Electric flux

$\displaystyle \Phi = \oint\vec E\cdot d\vec A = \frac{q_{\text{enclosed}}}{\varepsilon_0}$

Infinite line of charge

$\displaystyle E = \frac{\lambda}{2\pi\varepsilon_0 r}$

Infinite plane sheet

$\displaystyle E = \frac{\sigma}{2\varepsilon_0}$

Outside a charged spherical shell

$\displaystyle E = \frac{1}{4\pi\varepsilon_0}\frac{q}{r^2}$

Inside the shell, E = 0.

Potential

Potential of a point charge

$\displaystyle V = \frac{1}{4\pi\varepsilon_0}\frac{q}{r}$

Potential of a dipole

$\displaystyle V = \frac{1}{4\pi\varepsilon_0}\frac{p\cos\theta}{r^2}$

Zero on the equatorial plane.

Field from potential

$\displaystyle E = -\frac{dV}{dr}$

Potential energy of two charges

$\displaystyle U = \frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{r}$

Dipole energy in a uniform field

$\displaystyle U = -\vec p\cdot\vec E$

Capacitors

Capacitance

$\displaystyle C = \frac{Q}{V}$

Parallel plate capacitor

$\displaystyle C = \frac{\varepsilon_0 A}{d}$

With a dielectric filling the gap

$\displaystyle C = \frac{K\varepsilon_0 A}{d}$

Series combination

$\displaystyle \frac{1}{C} = \frac{1}{C_1} + \frac{1}{C_2}$

Same charge on each.

Parallel combination

$\displaystyle C = C_1 + C_2$

Same voltage across each.

Energy stored

$\displaystyle U = \frac{Q^2}{2C} = \tfrac12 CV^2 = \tfrac12 QV$

Energy density

$\displaystyle u = \tfrac12\varepsilon_0 E^2$

Current and resistance

Current

$\displaystyle I = \frac{dq}{dt} = neAv_d$

Drift velocity

$\displaystyle v_d = \frac{eE\tau}{m}$

Ohm's law

$\displaystyle V = IR$

Resistance

$\displaystyle R = \frac{\rho L}{A}$

Conductivity

$\displaystyle \sigma = \frac{1}{\rho} = \frac{ne^2\tau}{m}$

Current density

$\displaystyle J = \frac{I}{A} = \sigma E$

Temperature dependence

$\displaystyle \rho_T = \rho_0[1 + \alpha(T - T_0)]$

Combinations and power

Series

$\displaystyle R = R_1 + R_2$

Parallel

$\displaystyle \frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2}$

Power

$\displaystyle P = VI = I^2R = \frac{V^2}{R}$

Cells and networks

Terminal voltage

$\displaystyle V = \varepsilon - Ir$

Cells in series

$\displaystyle \varepsilon_{\text{eq}} = \varepsilon_1 + \varepsilon_2, \quad r_{\text{eq}} = r_1 + r_2$

Cells in parallel

$\displaystyle \frac{\varepsilon_{\text{eq}}}{r_{\text{eq}}} = \frac{\varepsilon_1}{r_1} + \frac{\varepsilon_2}{r_2}, \quad \frac{1}{r_{\text{eq}}} = \frac{1}{r_1} + \frac{1}{r_2}$

Kirchhoff's junction rule

$\displaystyle \sum I = 0$

Kirchhoff's loop rule

$\displaystyle \sum V = 0$

Balanced Wheatstone bridge

$\displaystyle \frac{P}{Q} = \frac{R}{S}$

Force

Lorentz force

$\displaystyle \vec F = q(\vec E + \vec v\times\vec B)$

Force on a wire

$\displaystyle \vec F = I\vec L\times\vec B$

Radius in a magnetic field

$\displaystyle r = \frac{mv}{qB}$

Cyclotron frequency

$\displaystyle \nu = \frac{qB}{2\pi m}$

It doesn't depend on speed.

Force between parallel wires

$\displaystyle \frac{F}{L} = \frac{\mu_0 I_1I_2}{2\pi d}$

Like currents attract.

Magnetic field

Biot-Savart law

$\displaystyle d\vec B = \frac{\mu_0}{4\pi}\frac{I\,d\vec l\times\hat r}{r^2}$

Long straight wire

$\displaystyle B = \frac{\mu_0 I}{2\pi r}$

Centre of a circular loop

$\displaystyle B = \frac{\mu_0 I}{2R}$

Axis of a circular loop

$\displaystyle B = \frac{\mu_0 IR^2}{2(R^2 + x^2)^{3/2}}$

Ampere's law

$\displaystyle \oint\vec B\cdot d\vec l = \mu_0 I_{\text{enclosed}}$

Long solenoid

$\displaystyle B = \mu_0 nI$

n is turns per unit length.

Current loops

Magnetic moment

$\displaystyle \vec m = NI\vec A$

Torque on a loop

$\displaystyle \vec\tau = \vec m\times\vec B$

Galvanometer to ammeter

$\displaystyle S = \frac{I_gG}{I - I_g}$

Shunt in parallel.

Galvanometer to voltmeter

$\displaystyle R = \frac{V}{I_g} - G$

High resistance in series.

Bar magnet

Field on the axis

$\displaystyle B = \frac{\mu_0}{4\pi}\frac{2m}{r^3}$

Field on the equator

$\displaystyle B = \frac{\mu_0}{4\pi}\frac{m}{r^3}$

Torque in a uniform field

$\displaystyle \vec\tau = \vec m\times\vec B$

Potential energy

$\displaystyle U = -\vec m\cdot\vec B$

Time period in a field

$\displaystyle T = 2\pi\sqrt{\frac{I}{mB}}$

Gauss's law for magnetism

$\displaystyle \oint\vec B\cdot d\vec A = 0$

There are no isolated magnetic poles.

Magnetic materials

Magnetisation

$\displaystyle M = \frac{m_{\text{net}}}{V}$

Total field

$\displaystyle B = \mu_0(H + M)$

Susceptibility

$\displaystyle \chi = \frac{M}{H}$

Permeability

$\displaystyle \mu = \mu_0\mu_r, \quad \mu_r = 1 + \chi$

Curie's law

$\displaystyle \chi = \frac{C\mu_0}{T}$

For paramagnetic materials.

Induced EMF

Magnetic flux

$\displaystyle \Phi = \vec B\cdot\vec A = BA\cos\theta$

Faraday's law

$\displaystyle \varepsilon = -N\frac{d\Phi}{dt}$

The minus sign is Lenz's law.

Motional EMF

$\displaystyle \varepsilon = Blv$

Rotating coil

$\displaystyle \varepsilon = NBA\omega\sin\omega t$

Inductance

Self-inductance

$\displaystyle \Phi = LI, \quad \varepsilon = -L\frac{dI}{dt}$

Long solenoid

$\displaystyle L = \mu_0 n^2 Al$

Mutual inductance

$\displaystyle \Phi_2 = MI_1, \quad \varepsilon_2 = -M\frac{dI_1}{dt}$

Energy in an inductor

$\displaystyle U = \tfrac12 LI^2$

AC values

Instantaneous voltage

$\displaystyle v = v_m\sin\omega t$

RMS current

$\displaystyle I = \frac{i_m}{\sqrt2} \approx 0.707\,i_m$

RMS voltage

$\displaystyle V = \frac{v_m}{\sqrt2}$

Reactance and impedance

Inductive reactance

$\displaystyle X_L = \omega L$

Current lags voltage by π/2.

Capacitive reactance

$\displaystyle X_C = \frac{1}{\omega C}$

Current leads voltage by π/2.

Impedance of an LCR circuit

$\displaystyle Z = \sqrt{R^2 + (X_L - X_C)^2}$

Phase angle

$\displaystyle \tan\phi = \frac{X_L - X_C}{R}$

Resonance and power

Resonant frequency

$\displaystyle \omega_0 = \frac{1}{\sqrt{LC}}$

Here X_L = X_C and Z = R.

Quality factor

$\displaystyle Q = \frac{\omega_0 L}{R} = \frac{1}{R}\sqrt{\frac{L}{C}}$

Average power

$\displaystyle P = VI\cos\phi$

Power factor

$\displaystyle \cos\phi = \frac{R}{Z}$

Transformer

Voltage and turns

$\displaystyle \frac{V_s}{V_p} = \frac{N_s}{N_p} = \frac{I_p}{I_s}$

For an ideal transformer.

Formulas

Displacement current

$\displaystyle i_d = \varepsilon_0\frac{d\Phi_E}{dt}$

Speed in vacuum

$\displaystyle c = \frac{1}{\sqrt{\mu_0\varepsilon_0}} = 3\times10^8\ \text{m/s}$

Speed in a medium

$\displaystyle v = \frac{1}{\sqrt{\mu\varepsilon}}$

Ratio of the fields

$\displaystyle \frac{E_0}{B_0} = c$

Wave relation

$\displaystyle c = \nu\lambda$

Energy density

$\displaystyle u_{\text{av}} = \tfrac12\varepsilon_0E_0^2$

Spectrum, longest to shortest wavelength

• Radio, microwaves, infrared, visible, ultraviolet, X-rays, gamma rays.
• Frequency and photon energy increase in the same order.
• Visible light runs from about 400 nm (violet) to 700 nm (red).
• E, B and the direction of travel are all perpendicular to each other.

Mirrors

Focal length

$\displaystyle f = \frac{R}{2}$

Mirror equation

$\displaystyle \frac{1}{v} + \frac{1}{u} = \frac{1}{f}$

Magnification

$\displaystyle m = -\frac{v}{u}$

Refraction

Snell's law

$\displaystyle n_1\sin i = n_2\sin r$

Apparent depth

$\displaystyle \text{apparent depth} = \frac{\text{real depth}}{n}$

Critical angle

$\displaystyle \sin C = \frac{1}{n}$

For light going from the denser medium into air.

Refraction at a spherical surface

$\displaystyle \frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2 - n_1}{R}$

Lenses

Lens equation

$\displaystyle \frac{1}{v} - \frac{1}{u} = \frac{1}{f}$

Lens maker's formula

$\displaystyle \frac{1}{f} = (n - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$

Magnification

$\displaystyle m = \frac{v}{u}$

Power

$\displaystyle P = \frac{1}{f}$

f in metres gives P in dioptres.

Lenses in contact

$\displaystyle \frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2}, \quad P = P_1 + P_2$

Prism and instruments

Prism angles

$\displaystyle A = r_1 + r_2, \quad \delta = i + e - A$

Refractive index of a prism

$\displaystyle n = \frac{\sin\frac{A + \delta_m}{2}}{\sin\frac{A}{2}}$

Thin prism

$\displaystyle \delta = (n - 1)A$

Simple microscope

$\displaystyle m = 1 + \frac{D}{f}$

Image at the near point D.

Compound microscope

$\displaystyle m = \frac{L}{f_o}\cdot\frac{D}{f_e}$

Telescope

$\displaystyle m = \frac{f_o}{f_e}$

Normal adjustment. Tube length is f_o + f_e.

Interference

Bright fringes

$\displaystyle \Delta x = n\lambda$

Constructive, n = 0, 1, 2, ...

Dark fringes

$\displaystyle \Delta x = \left(n + \tfrac12\right)\lambda$

Destructive.

Path difference in YDSE

$\displaystyle \Delta x = \frac{yd}{D}$

Fringe width

$\displaystyle \beta = \frac{\lambda D}{d}$

Resultant intensity

$\displaystyle I = I_1 + I_2 + 2\sqrt{I_1I_2}\cos\phi$

Phase from path

$\displaystyle \phi = \frac{2\pi}{\lambda}\Delta x$

Diffraction

Minima of a single slit

$\displaystyle a\sin\theta = n\lambda$

n = 1, 2, 3, ...

Width of the central maximum

$\displaystyle w = \frac{2\lambda D}{a}$

Polarisation

Malus's law

$\displaystyle I = I_0\cos^2\theta$

Brewster's law

$\displaystyle n = \tan i_B$

Photoelectric effect

Photon energy

$\displaystyle E = h\nu = \frac{hc}{\lambda}$

hc ≈ 1240 eV nm.

Photon momentum

$\displaystyle p = \frac{h\nu}{c} = \frac{h}{\lambda}$

Einstein's equation

$\displaystyle K_{\max} = h\nu - \phi_0$

Stopping potential

$\displaystyle eV_0 = K_{\max}$

Work function

$\displaystyle \phi_0 = h\nu_0$

ν₀ is the threshold frequency.

Matter waves

de Broglie wavelength

$\displaystyle \lambda = \frac{h}{p} = \frac{h}{mv}$

From kinetic energy

$\displaystyle \lambda = \frac{h}{\sqrt{2mK}}$

Electron through potential V

$\displaystyle \lambda = \frac{1.227}{\sqrt V}\ \text{nm}$

Bohr model

Quantised angular momentum

$\displaystyle mvr = \frac{nh}{2\pi}$

Radius of the nth orbit

$\displaystyle r_n = 0.529\,\frac{n^2}{Z}\ \text{Å}$

Speed in the nth orbit

$\displaystyle v_n = 2.19\times10^6\,\frac{Z}{n}\ \text{m/s}$

Energy of the nth orbit

$\displaystyle E_n = -13.6\,\frac{Z^2}{n^2}\ \text{eV}$

Energy of a photon

$\displaystyle h\nu = E_i - E_f$

Hydrogen spectrum

Rydberg formula

$\displaystyle \frac{1}{\lambda} = R\left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)$

R ≈ 1.097 × 10⁷ m⁻¹.

Number of spectral lines

$\displaystyle N = \frac{n(n - 1)}{2}$

From level n down to level 1.

Series names

• Lyman ends at n = 1 (ultraviolet).
• Balmer ends at n = 2 (visible).
• Paschen ends at n = 3, Brackett at n = 4 and Pfund at n = 5 (all infrared).

Nucleus

Nuclear radius

$\displaystyle R = R_0A^{1/3}$

R₀ ≈ 1.2 fm. Nuclear density is about the same for all nuclei.

Mass-energy relation

$\displaystyle E = mc^2$

1 u ≈ 931.5 MeV.

Mass defect

$\displaystyle \Delta m = [Zm_p + (A - Z)m_n] - M$

Binding energy

$\displaystyle E_b = \Delta m\,c^2$

Radioactive decay

Decay law

$\displaystyle N = N_0e^{-\lambda t}$

Activity

$\displaystyle R = \lambda N = R_0e^{-\lambda t}$

Half-life

$\displaystyle T_{1/2} = \frac{\ln 2}{\lambda} = \frac{0.693}{\lambda}$

Mean life

$\displaystyle \tau = \frac{1}{\lambda}$

Left after n half-lives

$\displaystyle N = \frac{N_0}{2^n}$

Decays

• Alpha decay: A falls by 4 and Z by 2.
• Beta-minus decay: Z rises by 1, and A stays the same.
• Gamma decay: A and Z both stay the same, and only energy is released.
• Fission splits heavy nuclei and fusion joins light ones. Both release energy because binding energy per nucleon rises.

Formulas

Intrinsic semiconductor

$\displaystyle n_e = n_h = n_i$

Mass action law

$\displaystyle n_en_h = n_i^2$

Current in a semiconductor

$\displaystyle I = I_e + I_h$

Half-wave rectifier output

$\displaystyle \nu_{\text{out}} = \nu_{\text{in}}$

Full-wave rectifier output

$\displaystyle \nu_{\text{out}} = 2\nu_{\text{in}}$

Key facts

• n-type: doped with pentavalent atoms such as phosphorus, and electrons are the majority carriers.
• p-type: doped with trivalent atoms such as boron, and holes are the majority carriers.
• Both types stay electrically neutral overall.
• Forward bias narrows the depletion region and current flows. Reverse bias widens it, and only a tiny current flows.
• An insulator has a large band gap, a semiconductor a small one, and a conductor none or overlapping bands.