NCERT / CBSE Board Exam

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1. Electric Charges and Fields

Coulomb’s Law

$$F=\frac{1}{4\pi\varepsilon_0}\frac{|q_1q_2|}{r^2}$$$$k=\frac{1}{4\pi\varepsilon_0}=9\times10^9\;Nm^2C^{-2}$$

In a medium:$$F=\frac{1}{4\pi\varepsilon}\frac{|q_1q_2|}{r^2}$$$$\varepsilon=K\varepsilon_0$$

Electric Field

$$\vec E=\frac{\vec F}{q_0}$$

For a point charge:$$\vec E=\frac{1}{4\pi\varepsilon_0}\frac{q}{r^2}\hat r$$

Superposition Principle

$$\vec E=\vec E_1+\vec E_2+\vec E_3+\cdots$$

Electric Dipole

Dipole moment:$$\vec p=q(2\vec a)$$

Magnitude:$$p=q(2a)$$

Direction: from $-q$ to $+q$.

Field Due to Dipole

Axial position:$$E_{\rm axial} = \frac{1}{4\pi\varepsilon_0} \frac{2pr}{(r^2-a^2)^2}$$

For $r\gg a$:$$E_{\rm axial} = \frac{1}{4\pi\varepsilon_0}\frac{2p}{r^3}$$

Equatorial position:$$E_{\rm equatorial} = \frac{1}{4\pi\varepsilon_0} \frac{p}{(r^2+a^2)^{3/2}}$$

For $r\gg a$:$$E_{\rm equatorial} = \frac{1}{4\pi\varepsilon_0}\frac{p}{r^3}$$

Torque on an Electric Dipole

$$\vec\tau=\vec p\times\vec E$$$$\tau=pE\sin\theta$$

Electric Flux

$$\Phi_E=\vec E\cdot\vec A$$$$\Phi_E=EA\cos\theta$$

Gauss’s Law

$$\boxed{\oint\vec E\cdot d\vec S=\frac{q_{\rm enclosed}}{\varepsilon_0}}$$

Important Results from Gauss’s Law

Infinite straight wire:$$E=\frac{\lambda}{2\pi\varepsilon_0r}$$

Infinite plane sheet:$$E=\frac{\sigma}{2\varepsilon_0}$$

Two oppositely charged infinite sheets:$$E=\frac{\sigma}{\varepsilon_0}$$

Spherical shell, outside:$$E=\frac{1}{4\pi\varepsilon_0}\frac{Q}{r^2}$$

Inside:$$E=0$$

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2. Electrostatic Potential and Capacitance

Electric Potential

$$V=\frac{W}{q}$$

Point charge:$$V=\frac{1}{4\pi\varepsilon_0}\frac{q}{r}$$

System of charges:$$V=\frac{1}{4\pi\varepsilon_0} \sum_i\frac{q_i}{r_i}$$

Potential Due to Dipole

$$V=\frac{1}{4\pi\varepsilon_0} \frac{p\cos\theta}{r^2} \qquad(r\gg a)$$

Potential Difference

$$V_B-V_A=-\int_A^B\vec E\cdot d\vec l$$

Relation Between E and V

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

Potential Energy

Two charges:$$U=\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{r}$$

System of charges:$$U=\frac{1}{4\pi\varepsilon_0} \sum_{i<j}\frac{q_iq_j}{r_{ij}}$$

Dipole in External Field

$$U=-\vec p\cdot\vec E$$$$U=-pE\cos\theta$$

Capacitance

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

Parallel plate capacitor:$$C=\frac{\varepsilon_0A}{d}$$

With dielectric:$$C=\frac{K\varepsilon_0A}{d}$$

Capacitors in Series

$$\frac1C=\frac1{C_1}+\frac1{C_2}+\cdots$$

Capacitors in Parallel

$$C=C_1+C_2+\cdots$$

Energy Stored

$$U=\frac12CV^2$$$$U=\frac12QV$$$$U=\frac{Q^2}{2C}$$

Energy density:$$u=\frac12\varepsilon_0E^2$$

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3. Current Electricity

Electric Current

$$I=\frac{dQ}{dt}$$

For steady current:$$I=\frac Qt$$

Drift Velocity

$$v_d=\frac{eE\tau}{m}$$$$I=neAv_d$$$$v_d=\frac{I}{neA}$$

Current Density

$$J=\frac IA$$$$J=nev_d$$

Ohm’s Law

$$V=IR$$

Resistance

$$R=\rho\frac{l}{A}$$

Conductivity

$$\sigma=\frac1\rho$$$$\sigma=\frac{ne^2\tau}{m}$$

Temperature Dependence

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

Approximately:$$R_T=R_0[1+\alpha(T-T_0)]$$

Series Combination

$$R=R_1+R_2+\cdots$$

Parallel Combination

$$\frac1R=\frac1{R_1}+\frac1{R_2}+\cdots$$

Electrical Power

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

Electrical Energy

$$W=VIt$$

Cell

$$V=\mathcal E-Ir$$$$I=\frac{\mathcal E}{R+r}$$

Terminal voltage during charging:$$V=\mathcal E+Ir$$

Cells in Series

For identical cells:$$\mathcal E_{\rm eq}=n\mathcal E$$$$r_{\rm eq}=nr$$

Cells in Parallel

For $n$ identical cells:$$\mathcal E_{\rm eq}=\mathcal E$$$$r_{\rm eq}=\frac rn$$

Kirchhoff’s Junction Rule

$$\boxed{\sum I_{\rm in}=\sum I_{\rm out}}$$

Kirchhoff’s Loop Rule

$$\boxed{\sum\Delta V=0}$$

Wheatstone Bridge

At balance:$$\boxed{\frac{P}{Q}=\frac{R}{S}}$$

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4. Moving Charges and Magnetism

Magnetic Force

$$\vec F=q(\vec v\times\vec B)$$$$F=qvB\sin\theta$$

Lorentz Force

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

Force on Current-Carrying Conductor

$$\vec F=I(\vec l\times\vec B)$$$$F=IlB\sin\theta$$

Motion of Charged Particle

For $v\perp B$:$$qvB=\frac{mv^2}{r}$$$$r=\frac{mv}{qB}$$$$\omega=\frac{qB}{m}$$$$T=\frac{2\pi m}{qB}$$$$f=\frac{qB}{2\pi m}$$

Velocity Selector

$$qE=qvB$$$$v=\frac EB$$

Biot-Savart Law

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

Circular Current Loop

At centre:$$B=\frac{\mu_0I}{2R}$$

For $N$ turns:$$B=\frac{\mu_0NI}{2R}$$

On axis:$$B= \frac{\mu_0NIR^2} {2(R^2+x^2)^{3/2}}$$

Ampere’s Circuital Law

$$\oint\vec B\cdot d\vec l=\mu_0I$$

Long Straight Wire

$$B=\frac{\mu_0I}{2\pi r}$$

Long Solenoid

$$B=\mu_0nI$$

where$$n=\frac NL$$

Force Between Parallel Wires

$$\frac Fl=\frac{\mu_0I_1I_2}{2\pi d}$$

Magnetic Dipole Moment of Current Loop

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

Torque on Current Loop

$$\vec\tau=\vec m\times\vec B$$$$\tau=NIAB\sin\theta$$

Moving Coil Galvanometer

$$NIAB=k\theta$$$$I=\frac{k\theta}{NAB}$$

Current sensitivity:$$\frac{\theta}{I}=\frac{NAB}{k}$$

Galvanometer → Ammeter

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

Galvanometer → Voltmeter

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

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5. Magnetism and Matter

Magnetic Dipole Moment of Bar Magnet

$$m=(\text{pole strength})(\text{magnetic length})$$

Torque

$$\vec\tau=\vec m\times\vec B$$$$\tau=mB\sin\theta$$

Potential Energy

$$U=-\vec m\cdot\vec B$$$$U=-mB\cos\theta$$

Magnetisation

$$M=\frac{\text{net magnetic moment}}{\text{volume}}$$

Magnetic Intensity

$$H=\frac{B}{\mu_0}-M$$

Therefore:$$B=\mu_0(H+M)$$

Magnetic Susceptibility

$$M=\chi_mH$$

Relative Permeability

$$\mu_r=1+\chi_m$$$$B=\mu_0\mu_rH$$

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6. Electromagnetic Induction

Magnetic Flux

$$\Phi_B=\vec B\cdot\vec A$$$$\Phi_B=BA\cos\theta$$

Faraday’s Law

$$\boxed{\mathcal E=-\frac{d\Phi_B}{dt}}$$

For $N$ turns:$$\boxed{\mathcal E=-N\frac{d\Phi_B}{dt}}$$

Motional EMF

$$\mathcal E=Blv$$

for mutually perpendicular $B,l,v$.

Induced Current

$$I=\frac{\mathcal E}{R}$$

Self-Inductance

$$N\Phi=LI$$$$\mathcal E=-L\frac{dI}{dt}$$

Mutual Inductance

$$N_2\Phi_2=MI_1$$$$\mathcal E_2=-M\frac{dI_1}{dt}$$

Energy Stored in Inductor

$$U=\frac12LI^2$$

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7. Alternating Current

AC Current and Voltage

$$i=i_0\sin\omega t$$$$v=v_0\sin\omega t$$$$\omega=2\pi f$$

RMS Values

$$I_{\rm rms}=\frac{I_0}{\sqrt2}$$$$V_{\rm rms}=\frac{V_0}{\sqrt2}$$

Inductive Reactance

$$X_L=\omega L$$

Capacitive Reactance

$$X_C=\frac1{\omega C}$$

Series LCR Circuit

$$Z=\sqrt{R^2+(X_L-X_C)^2}$$$$I_{\rm rms}=\frac{V_{\rm rms}}Z$$

Phase Angle

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

Power Factor

$$\cos\phi=\frac RZ$$

Average Power

$$P=V_{\rm rms}I_{\rm rms}\cos\phi$$

Resonance

$$X_L=X_C$$$$\omega_0=\frac1{\sqrt{LC}}$$$$f_0=\frac1{2\pi\sqrt{LC}}$$

At resonance:$$Z=R$$

AC Generator

$$\Phi=NBA\cos\omega t$$$$e=NBA\omega\sin\omega t$$$$e_0=NBA\omega$$

Transformer

$$\boxed{\frac{V_s}{V_p}=\frac{N_s}{N_p}}$$

Ideal transformer:$$V_pI_p=V_sI_s$$$$\frac{I_s}{I_p}=\frac{N_p}{N_s}$$

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8. Electromagnetic Waves

Speed of Electromagnetic Wave

$$\boxed{c=\frac1{\sqrt{\mu_0\varepsilon_0}}}$$

Wave Relation

$$c=\nu\lambda$$

Relation Between Electric and Magnetic Fields

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

or$$E_0=cB_0$$

Displacement Current

$$I_d=\varepsilon_0\frac{d\Phi_E}{dt}$$

Ampere-Maxwell Law

$$\oint\vec B\cdot d\vec l = \mu_0(I_c+I_d)$$

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9. Ray Optics and Optical Instruments

Refractive Index

$$n=\frac cv$$

Snell’s Law

$$n_1\sin i=n_2\sin r$$$$\frac{\sin i}{\sin r}=\frac{n_2}{n_1}$$

Critical Angle

For denser to rarer medium:$$\sin C=\frac{n_{\rm rarer}}{n_{\rm denser}}$$

For air:$$\sin C=\frac1n$$

Spherical Mirror

$$\boxed{\frac1f=\frac1v+\frac1u}$$$$f=\frac R2$$

Magnification:$$m=\frac{h_i}{h_o}=-\frac vu$$

Refraction at Spherical Surface

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

Thin Lens Formula

$$\boxed{\frac1v-\frac1u=\frac1f}$$

Magnification:$$m=\frac{h_i}{h_o}=\frac vu$$

Lens Maker’s Formula

In air:$$\frac1f=(n-1) \left( \frac1{R_1}-\frac1{R_2} \right)$$

Power of Lens

$$P=\frac1f$$

$f$ in metre.

Unit:$$1D=1m^{-1}$$

Lenses in Contact

$$P=P_1+P_2+\cdots$$$$\frac1F=\frac1{f_1}+\frac1{f_2}+\cdots$$

Prism

$$\delta=i+e-A$$

At minimum deviation:$$i=e$$$$r_1=r_2=\frac A2$$$$\boxed{ n= \frac{\sin[(A+\delta_m)/2]} {\sin(A/2)} }$$

For small-angle prism:$$\delta_m=(n-1)A$$

Simple Microscope

Final image at infinity:$$M=\frac Df$$

At near point:$$M=1+\frac Df$$

Compound Microscope

Normal adjustment:$$M=\frac{L}{f_o}\frac{D}{f_e}$$

Astronomical Telescope

Normal adjustment:$$M=-\frac{f_o}{f_e}$$

Magnitude:$$|M|=\frac{f_o}{f_e}$$

Length:$$L=f_o+f_e$$

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10. Wave Optics

Path Difference

$$\Delta=d\sin\theta$$

For small angle:$$\Delta\simeq\frac{dy}{D}$$

Constructive Interference

$$\Delta=n\lambda$$

Destructive Interference

$$\Delta=(2n+1)\frac\lambda2$$

YDSE Fringe Width

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

Bright fringe:$$y_n=n\frac{\lambda D}{d}$$

Dark fringe:$$y_n=(2n+1)\frac{\lambda D}{2d}$$

Resultant Intensity

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

For equal intensities:$$I=4I_0\cos^2\frac\phi2$$

Single-Slit Diffraction

Minima:$$a\sin\theta=n\lambda$$

For small angle:$$y_n=\frac{n\lambda D}{a}$$

Width of central maximum:$$\boxed{W=\frac{2\lambda D}{a}}$$

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11. Dual Nature of Radiation and Matter

Photon Energy

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

Photon Momentum

$$p=\frac{h}{\lambda}$$$$p=\frac Ec$$

Einstein’s Photoelectric Equation

$$\boxed{h\nu=\phi+K_{\max}}$$$$K_{\max}=\frac12mv_{\max}^2$$$$K_{\max}=eV_0$$

Therefore:$$h\nu=\phi+eV_0$$

Work Function

$$\phi=h\nu_0$$$$\phi=\frac{hc}{\lambda_0}$$

Stopping Potential

$$eV_0=h(\nu-\nu_0)$$

de Broglie Wavelength

$$\boxed{\lambda=\frac hp}$$$$\lambda=\frac{h}{mv}$$

For particle of kinetic energy $K$:$$\lambda=\frac{h}{\sqrt{2mK}}$$

For electron accelerated through potential $V$:$$\lambda=\frac{h}{\sqrt{2meV}}$$

Approximately:$$\boxed{\lambda(\mathring A)=\frac{12.27}{\sqrt V}}$$

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12. Atoms

Bohr Quantisation

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

Radius of $n^{th}$ Orbit

For hydrogen:$$r_n=n^2a_0$$

where$$a_0=0.529\,\mathring A$$

For hydrogen-like atom:$$r_n=\frac{n^2a_0}{Z}$$

Velocity

$$v_n=\frac{2.18\times10^6Z}{n}\;m/s$$

Energy of Electron

$$\boxed{E_n=-\frac{13.6Z^2}{n^2}\;eV}$$

For hydrogen:$$E_n=-\frac{13.6}{n^2}\;eV$$

Photon Emission/Absorption

$$h\nu=|E_f-E_i|$$

Rydberg Formula

$$\boxed{ \frac1\lambda = RZ^2 \left( \frac1{n_1^2}-\frac1{n_2^2} \right) }$$

where $n_2>n_1$.

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13. Nuclei

Nuclear Radius

$$\boxed{R=R_0A^{1/3}}$$$$R_0\approx1.2\times10^{-15}m$$

Mass-Energy Equivalence

$$E=mc^2$$

Mass Defect

$$\Delta m= Zm_p+(A-Z)m_n-M_{\rm nucleus}$$

Using atomic masses:$$\Delta m= Zm_H+(A-Z)m_n-M_{\rm atom}$$

Binding Energy

$$\boxed{BE=\Delta mc^2}$$

If $\Delta m$ is in u:$$BE=\Delta m\times931.5\;MeV$$

Binding Energy per Nucleon

$$\frac{BE}{A}$$

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14. Semiconductor Electronics: Materials, Devices and Simple Circuits

Energy Gap

Conductor:$$E_g\approx0$$

Semiconductor:$$E_g\sim1\,eV$$

Insulator:$$E_g\gtrsim3\,eV$$

Current in Semiconductor

$$I=I_e+I_h$$

Conductivity

$$\sigma=e(n_e\mu_e+n_h\mu_h)$$

For intrinsic semiconductor:$$n_e=n_h=n_i$$

Therefore:$$\sigma_i=en_i(\mu_e+\mu_h)$$

Resistivity

$$\rho=\frac1\sigma$$

Diode

Forward bias → low resistance

Reverse bias → high resistance

Rectifier

Half-wave rectifier:$$f_{\rm output}=f_{\rm input}$$

Full-wave rectifier:$$f_{\rm output}=2f_{\rm input}$$

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⭐ Constants to Memorise for CBSE

Constant	Value
$e$	$1.6\times10^{-19}C$
$m_e$	$9.1\times10^{-31}kg$
$m_p$	$1.67\times10^{-27}kg$
$\varepsilon_0$	$8.85\times10^{-12}C^2N^{-1}m^{-2}$
$\mu_0$	$4\pi\times10^{-7}TmA^{-1}$
$k=1/4\pi\varepsilon_0$	$9\times10^9Nm^2C^{-2}$
$h$	$6.626\times10^{-34}Js$
$c$	$3\times10^8m/s$
$R$ (Rydberg)	$1.097\times10^7m^{-1}$
$1\,eV$	$1.6\times10^{-19}J$
$1\,u$	$1.66\times10^{-27}kg$
$1\,u\,c^2$	$931.5\,MeV$

⚠️ Board-exam points to remember

For Ray Optics, always apply the NCERT Cartesian sign convention rather than trying to memorize separate positive/negative versions of the formulas.

For EMI, the negative sign in$$\mathcal E=-\frac{d\Phi}{dt}$$

represents Lenz’s law.

For AC, distinguish carefully between peak and RMS values:$$V_{\rm rms}=\frac{V_0}{\sqrt2}, \qquad I_{\rm rms}=\frac{I_0}{\sqrt2}$$

For Atoms and Nuclei, keep units consistent—especially eV, MeV, joule and atomic mass unit.

For Dual Nature, remember the three equations that solve most CBSE numericals:$$\boxed{h\nu=\phi+eV_0}$$$$\boxed{\lambda=\frac hp}$$$$\boxed{\lambda=\frac{h}{\sqrt{2meV}}}$$

And for Wave Optics, the two especially important board formulas are:$$\boxed{\beta=\frac{\lambda D}{d}} \qquad \boxed{W_{\rm central}=\frac{2\lambda D}{a}}$$

This covers the 14 chapters of the current Class 12 NCERT Physics sequence, with the emphasis on formulas that are directly useful for CBSE board preparation.

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