CBSE Class 12 Physics Formulas

Cheatsheet Content

### Electrostatics #### Electric Charges and Fields - **Coulomb's Law:** $F = k \frac{|q_1 q_2|}{r^2}$, where $k = \frac{1}{4\pi\epsilon_0}$ - **Electric Field:** $\vec{E} = \frac{\vec{F}}{q_0} = k \frac{q}{r^2} \hat{r}$ - **Electric Dipole Moment:** $\vec{p} = q(2\vec{a})$ - **Torque on Dipole in Uniform E-field:** $\vec{\tau} = \vec{p} \times \vec{E}$ - **Potential Energy of Dipole:** $U = -\vec{p} \cdot \vec{E}$ - **Electric Flux:** $\Phi_E = \oint \vec{E} \cdot d\vec{A}$ - **Gauss's Law:** $\Phi_E = \frac{Q_{enc}}{\epsilon_0}$ #### Electrostatic Potential and Capacitance - **Electric Potential:** $V = \frac{U}{q_0} = k \frac{q}{r}$ - **Potential Difference:** $\Delta V = V_B - V_A = -\int_A^B \vec{E} \cdot d\vec{l}$ - **Relationship between E and V:** $E = -\frac{dV}{dr}$ - **Capacitance:** $C = \frac{Q}{V}$ - **Capacitance of Parallel Plate Capacitor:** $C = \frac{\epsilon_0 A}{d}$ - **Capacitors in Series:** $\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + ...$ - **Capacitors in Parallel:** $C_{eq} = C_1 + C_2 + ...$ - **Energy Stored in Capacitor:** $U = \frac{1}{2}CV^2 = \frac{1}{2}\frac{Q^2}{C} = \frac{1}{2}QV$ - **Energy Density:** $u = \frac{1}{2}\epsilon_0 E^2$ - **Capacitance with Dielectric:** $C' = KC = K \frac{\epsilon_0 A}{d}$ ### Current Electricity - **Electric Current:** $I = \frac{dQ}{dt} = nAv_e e$ - **Drift Velocity:** $v_d = \frac{eE\tau}{m}$ - **Ohm's Law:** $V = IR$ - **Resistance:** $R = \rho \frac{L}{A}$ - **Resistivity:** $\rho_T = \rho_0 [1 + \alpha(T - T_0)]$ - **Resistors in Series:** $R_{eq} = R_1 + R_2 + ...$ - **Resistors in Parallel:** $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + ...$ - **Kirchhoff's Junction Rule:** $\sum I = 0$ (at any junction) - **Kirchhoff's Loop Rule:** $\sum \Delta V = 0$ (around any closed loop) - **Electric Power:** $P = VI = I^2R = \frac{V^2}{R}$ - **Internal Resistance of Cell:** $r = (\frac{E}{V} - 1)R$ - **Cells in Series:** $E_{eq} = E_1 + E_2 + ...$, $r_{eq} = r_1 + r_2 + ...$ - **Cells in Parallel (Identical):** $E_{eq} = E$, $r_{eq} = \frac{r}{n}$ - **Wheatstone Bridge (Balanced):** $\frac{P}{Q} = \frac{R}{S}$ ### Magnetic Effects of Current & Magnetism #### Moving Charges and Magnetism - **Lorentz Force:** $\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})$ - **Magnetic Force on Current Carrying Conductor:** $\vec{F} = I(\vec{l} \times \vec{B})$ - **Magnetic Force between Two Parallel Currents:** $F = \frac{\mu_0 I_1 I_2 L}{2\pi d}$ - **Biot-Savart Law:** $d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{l} \times \hat{r}}{r^2}$ - **Magnetic Field due to Long Straight Wire:** $B = \frac{\mu_0 I}{2\pi r}$ - **Magnetic Field at Center of Circular Loop:** $B = \frac{\mu_0 I}{2R}$ - **Magnetic Field at Axis of Circular Loop:** $B = \frac{\mu_0 nIR^2}{2(R^2+x^2)^{3/2}}$ - **Ampere's Circuital Law:** $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$ - **Magnetic Field inside Solenoid:** $B = \mu_0 nI$ - **Magnetic Field inside Toroid:** $B = \frac{\mu_0 NI}{2\pi r}$ - **Torque on Current Loop:** $\vec{\tau} = \vec{M} \times \vec{B}$, where $\vec{M} = NI\vec{A}$ - **Moving Coil Galvanometer:** $I = k\theta$, where $k = \frac{c}{NBA}$ #### Magnetism and Matter - **Magnetic Dipole Moment of Electron (Orbital):** $M_l = \frac{e v r}{2} = \frac{e}{2m_e} L$ - **Magnetic Field Intensity (H):** $H = \frac{B}{\mu_0} - M$ - **Magnetisation (M):** $M = \frac{m_{net}}{V}$ - **Magnetic Susceptibility:** $\chi_m = \frac{M}{H}$ - **Magnetic Permeability:** $\mu = \mu_0 (1 + \chi_m) = \mu_0 \mu_r$ - **Relative Permeability:** $\mu_r = 1 + \chi_m$ ### Electromagnetic Induction & Alternating Current #### Electromagnetic Induction - **Magnetic Flux:** $\Phi_B = \int \vec{B} \cdot d\vec{A} = BA \cos\theta$ - **Faraday's Law of EMI:** $\mathcal{E} = -\frac{d\Phi_B}{dt}$ - **Motional EMF:** $\mathcal{E} = (B l v)$ (if B, l, v are mutually perpendicular) - **Self-Inductance:** $\Phi_B = LI$, $\mathcal{E} = -L\frac{dI}{dt}$ - **Energy Stored in Inductor:** $U = \frac{1}{2}LI^2$ - **Mutual Inductance:** $\Phi_{21} = M_{21}I_1$, $\mathcal{E}_2 = -M_{21}\frac{dI_1}{dt}$ #### Alternating Current - **AC Voltage/Current:** $V = V_m \sin(\omega t + \phi)$, $I = I_m \sin(\omega t)$ - **RMS Value:** $V_{rms} = \frac{V_m}{\sqrt{2}}$, $I_{rms} = \frac{I_m}{\sqrt{2}}$ - **Inductive Reactance:** $X_L = \omega L = 2\pi f L$ - **Capacitive Reactance:** $X_C = \frac{1}{\omega C} = \frac{1}{2\pi f C}$ - **Impedance (Series LCR Circuit):** $Z = \sqrt{R^2 + (X_L - X_C)^2}$ - **Phase Angle:** $\tan\phi = \frac{X_L - X_C}{R}$ - **Resonant Frequency:** $f_0 = \frac{1}{2\pi\sqrt{LC}}$ - **Power in AC Circuit:** $P_{avg} = V_{rms} I_{rms} \cos\phi$ - **Power Factor:** $\cos\phi = \frac{R}{Z}$ - **Transformer Relation:** $\frac{V_s}{V_p} = \frac{N_s}{N_p} = \frac{I_p}{I_s}$ ### Electromagnetic Waves - **Speed of EM Wave in Vacuum:** $c = \frac{1}{\sqrt{\mu_0\epsilon_0}}$ - **Speed of EM Wave in Medium:** $v = \frac{1}{\sqrt{\mu\epsilon}}$ - **Relation between E and B Field:** $E_0 = c B_0$ - **Energy Density of EM Wave:** $u = u_E + u_B = \frac{1}{2}\epsilon_0 E^2 + \frac{1}{2\mu_0} B^2 = \epsilon_0 E^2 = \frac{B^2}{\mu_0}$ - **Poynting Vector (Intensity):** $S_{avg} = \frac{1}{2}c\epsilon_0 E_0^2 = \frac{1}{2}\frac{E_0 B_0}{\mu_0}$ ### Optics #### Ray Optics and Optical Instruments - **Mirror Formula:** $\frac{1}{f} = \frac{1}{v} + \frac{1}{u}$ (Concave $f 0$) - **Magnification (Mirror):** $m = -\frac{v}{u} = \frac{h_I}{h_O}$ - **Refraction (Snell's Law):** $n_1 \sin i = n_2 \sin r$ - **Refractive Index:** $n = \frac{c}{v}$ - **Critical Angle:** $\sin C = \frac{n_2}{n_1}$ (for $n_1 > n_2$) - **Lens Maker's Formula:** $\frac{1}{f} = (n-1)(\frac{1}{R_1} - \frac{1}{R_2})$ - **Lens Formula:** $\frac{1}{f} = \frac{1}{v} - \frac{1}{u}$ (Concave $f 0$) - **Magnification (Lens):** $m = \frac{v}{u} = \frac{h_I}{h_O}$ - **Power of Lens:** $P = \frac{1}{f}$ (in dioptres, f in meters) - **Lenses in Contact:** $P_{eq} = P_1 + P_2$, $\frac{1}{f_{eq}} = \frac{1}{f_1} + \frac{1}{f_2}$ - **Refraction at Spherical Surface:** $\frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2 - n_1}{R}$ - **Prism Formula:** $n = \frac{\sin((A+\delta_m)/2)}{\sin(A/2)}$ (for minimum deviation) #### Wave Optics - **Wavefront:** Locus of points having the same phase of vibration. - **Huygens' Principle:** Every point on a wavefront is a source of secondary wavelets. - **Path Difference ($\Delta x$):** - Constructive Interference (Bright Fringes): $\Delta x = n\lambda$ - Destructive Interference (Dark Fringes): $\Delta x = (2n+1)\frac{\lambda}{2}$ - **Phase Difference ($\Delta \phi$):** $\Delta \phi = \frac{2\pi}{\lambda} \Delta x$ - **Young's Double Slit Experiment (YDSE):** - **Fringe Width:** $\beta = \frac{\lambda D}{d}$ - **Position of Bright Fringes:** $y_n = \frac{n\lambda D}{d}$ - **Position of Dark Fringes:** $y_n = (n+\frac{1}{2})\frac{\lambda D}{d}$ - **Diffraction (Single Slit):** - **Condition for Minima:** $a \sin\theta = n\lambda$ - **Condition for Maxima:** $a \sin\theta = (2n+1)\frac{\lambda}{2}$ (approx) - **Width of Central Maxima:** $2\theta = \frac{2\lambda}{a}$ or $2y = \frac{2\lambda D}{a}$ - **Brewster's Law (Polarisation):** $\tan i_p = n$ - **Malus's Law:** $I = I_0 \cos^2\theta$ ### Dual Nature of Radiation and Matter - **Planck's Quantum Theory:** $E = h\nu = \frac{hc}{\lambda}$ - **Photon Momentum:** $p = \frac{h}{\lambda} = \frac{E}{c}$ - **Einstein's Photoelectric Equation:** $K_{max} = h\nu - \phi_0 = h\nu - h\nu_0$ - **Work Function:** $\phi_0 = h\nu_0$ - **Stopping Potential:** $eV_0 = K_{max}$ - **de Broglie Wavelength:** $\lambda = \frac{h}{p} = \frac{h}{mv} = \frac{h}{\sqrt{2mK}}$ - **de Broglie Wavelength for Electron:** $\lambda = \frac{1.227}{\sqrt{V}}$ nm (for V in Volts) ### Atoms and Nuclei #### Atoms - **Rutherford's Model:** - **Distance of Closest Approach:** $r_0 = \frac{Z e^2}{4\pi\epsilon_0 K}$ - **Impact Parameter:** $b = \frac{Ze^2 \cot(\theta/2)}{4\pi\epsilon_0 K}$ - **Bohr's Model:** - **Quantization Condition:** $L = mvr = n\frac{h}{2\pi}$ - **Radius of n-th Orbit:** $r_n = \frac{n^2 h^2 \epsilon_0}{\pi m e^2 Z} = 0.529 \frac{n^2}{Z} \text{ Å}$ - **Velocity of Electron in n-th Orbit:** $v_n = \frac{Ze^2}{2\epsilon_0 n h}$ - **Energy of Electron in n-th Orbit:** $E_n = -\frac{m e^4 Z^2}{8\epsilon_0^2 h^2 n^2} = -13.6 \frac{Z^2}{n^2} \text{ eV}$ - **Wavelength of Emitted Photon:** $\frac{1}{\lambda} = RZ^2 (\frac{1}{n_f^2} - \frac{1}{n_i^2})$ (Rydberg Formula) - **Rydberg Constant:** $R = 1.097 \times 10^7 \text{ m}^{-1}$ #### Nuclei - **Nuclear Radius:** $R = R_0 A^{1/3}$, where $R_0 \approx 1.2 \times 10^{-15} \text{ m}$ - **Mass Defect:** $\Delta m = [Z m_p + (A-Z) m_n] - M_{nucleus}$ - **Binding Energy:** $E_b = \Delta m c^2$ - **Binding Energy per Nucleon:** $E_{bn} = \frac{E_b}{A}$ - **Radioactive Decay Law:** $N = N_0 e^{-\lambda t}$ - **Half-Life:** $T_{1/2} = \frac{\ln 2}{\lambda} = \frac{0.693}{\lambda}$ - **Mean Life:** $\tau = \frac{1}{\lambda}$ - **Activity:** $A = -\frac{dN}{dt} = \lambda N = \lambda N_0 e^{-\lambda t} = A_0 e^{-\lambda t}$ ### Semiconductor Electronics - **Conduction in Semiconductors:** - **Current:** $I = I_e + I_h = (n_e v_e + n_h v_h) A e$ - **Intrinsic Carrier Concentration:** $n_i^2 = n_e n_h$ - **Diode (p-n Junction):** - **Forward Bias Current:** $I = I_0 (e^{eV/k_B T} - 1)$ - **Transistor (npn/pnp):** - **Current Relation:** $I_E = I_B + I_C$ - **Current Gain (Common Emitter):** $\beta_{ac} = (\frac{\Delta I_C}{\Delta I_B})_{V_{CE}}$, $\beta_{dc} = \frac{I_C}{I_B}$ - **Current Gain (Common Base):** $\alpha_{ac} = (\frac{\Delta I_C}{\Delta I_E})_{V_{CB}}$, $\alpha_{dc} = \frac{I_C}{I_E}$ - **Relation between $\alpha$ and $\beta$:** $\beta = \frac{\alpha}{1-\alpha}$ - **Logic Gates:** - **NOT Gate:** $Y = \bar{A}$ - **AND Gate:** $Y = A \cdot B$ - **OR Gate:** $Y = A + B$ - **NAND Gate:** $Y = \overline{A \cdot B}$ - **NOR Gate:** $Y = \overline{A + B}$