AK Vision Zone - NEET Physics Formula Sheet

NEET Physics Formula Sheet

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3. Optics

3.1: Reflection of Light

Laws of reflection: (i) Incident ray, reflected ray, and normal lie in the same plane (ii) $\angle i=\angle r$
Plane mirror: (i) the image and the object are equidistant from mirror (ii) virtual image of real object
Spherical Mirror: 1. Focal length $f=R/2$
2. Mirror equation: $\frac{1}{v}+\frac{1}{u}=\frac{1}{f}$
3. Magnification: $m=-\frac{v}{u}$

3.2: Refraction of Light

Refractive index: $\mu=\frac{speed~of~light~in~vacuum}{speed~of~light~in~medium}=\frac{c}{v}$
Snell's Law: $\frac{sin~i}{sin~r}=\frac{\mu_{2}}{\mu_{1}}$
Apparent depth: $\mu=\frac{real~depth}{apparent~depth}=\frac{d}{d^{\prime}}$
Critical angle: $\theta_{c}=sin^{-1}\frac{1}{\mu}$
Deviation by a prism: $\delta=i+i^{\prime}-A$ general result
$\mu=\frac{sin\frac{A+\delta_{m}}{2}}{sin\frac{A}{2}}, i=i^{\prime}$ for minimum deviation
$\delta_{m}=(\mu-1)A$, for small A
Refraction at spherical surface: $\frac{\mu_{2}}{v}-\frac{\mu_{1}}{u}=\frac{\mu_{2}-\mu_{1}}{R}, m=\frac{\mu_{1}v}{\mu_{2}u}$

3.3: Optical Instruments

Lens maker's formula: $\frac{1}{f}=(\mu-1)[\frac{1}{R_{1}}-\frac{1}{R_{2}}]$
Lens formula: $\frac{1}{v}-\frac{1}{u}=\frac{1}{f}, m=\frac{v}{u}$
Power of the lens: $P=\frac{1}{f}$, P in diopter if f in metre.
Two thin lenses separated by distance d: $\frac{1}{F}=\frac{1}{f_{1}}+\frac{1}{f_{2}}-\frac{d}{f_{1}f_{2}}$
Simple microscope: $m=D/f$ in normal adjustment.
Compound microscope: 1. Magnification in normal adjustment: $m=\frac{v}{u}\frac{D}{f_{e}}$
2. Resolving power: $R=\frac{1}{\Delta d}=\frac{2\mu~sin~\theta}{\lambda}$
Astronomical telescope: 1. In normal adjustment: $m=-\frac{f_{o}}{f_{e}}, L=f_{o}+f_{e}$
2. Resolving power: $R=\frac{1}{\Delta\theta}=\frac{1}{1.22\lambda}$

3.4: Dispersion

Cauchy's equation: $\mu=\mu_{0}+\frac{A}{\lambda^{2}}, A>0$
Dispersion by prism with small A and i:
1. Mean deviation: $\delta_{y}=(\mu_{y}-1)A$
2. Angular dispersion: $\theta=(\mu_{v}-\mu_{r})A$
Dispersive power: $\omega=\frac{\mu_{v}-\mu_{r}}{\mu_{y}-1}\approx\frac{\theta}{\delta_{y}}$ (if A and i small)
Dispersion without deviation: $(\mu_{y}-1)A+(\mu_{y}^{\prime}-1)A^{\prime}=0$
Deviation without dispersion: $(\mu_{v}-\mu_{r})A=(\mu_{v}^{\prime}-\mu_{r}^{\prime})A^{\prime}$

4. Heat & Thermodynamics

4.1: Heat and Temperature

Temp scales: $F=32+\frac{9}{5}C, K=C+273.16$
Ideal gas equation: $pV=nRT$ n: number of moles
van der Waals equation: $(p+\frac{a}{V^{2}})(V-b)=nRT$
Thermal expansion: $L=L_{0}(1+\alpha\Delta T)$
$A=A_{0}(1+\beta\Delta T), V=V_{0}(1+\gamma\Delta T), \gamma=2\beta=3\alpha$
Thermal stress of a material: $\frac{F}{A}=Y\frac{\Delta l}{l}$

4.2: Kinetic Theory of Gases

General: $M=mN_{A}, k=R/N_{A}$
Maxwell distribution of speed:
RMS speed: $v_{rms}=\sqrt{\frac{3kT}{m}}=\sqrt{\frac{3RT}{M}}$
Average speed: $\overline{v}=\sqrt{\frac{8kT}{\pi m}}=\sqrt{\frac{8RT}{\pi M}}$
Most probable speed: $v_{p}=\sqrt{\frac{2kT}{m}}$
Pressure: $p=\frac{1}{3}\rho v_{rms}^{2}$
Equipartition of energy: $K=\frac{1}{2}kT$ for each degree of freedom.
Thus, $K=\frac{f}{2}kT$ for molecule having f degrees of freedoms.
Internal energy of n moles of an ideal gas is $U=\frac{f}{2}nRT$

4.3: Specific Heat

Specific heat: $s=\frac{Q}{m\Delta T}$
Latent heat: $L=Q/m$
Specific heat at const volume: $C_{v}=\frac{\Delta Q}{n\Delta T}|_{V}$
Specific heat at const pressure: $C_{p}=\frac{\Delta Q}{n\Delta T}|_{p}$
Relation between $C_{p}$ and $C_{v}$: $C_{p}-C_{v}=R$
Ratio of specific heats: $\gamma=C_{p}/C_{v}$
Relation between U and $C_{v}$: $\Delta U=nC_{v}\Delta T$
Specific heat of gas mixture: $C_{v}=\frac{n_{1}C_{v1}+n_{2}C_{v2}}{n_{1}+n_{2}}, \gamma=\frac{n_{1}C_{p1}+n_{2}C_{p2}}{n_{1}C_{v1}+n_{2}C_{v2}}$
Molar internal energy of an ideal gas: $U=\frac{f}{2}RT$
$f=3$ for monatomic and $f=5$ for diatomic gas.

4.4: Thermodynamic Processes

First law of thermodynamics: $\Delta Q=\Delta U+\Delta W$
Work done by the gas: $\Delta W=p\Delta V, W=\int_{V_{1}}^{V_{2}}pdV$
$W_{isothermal}=nRT~ln(\frac{V_{2}}{V_{1}})$
$W_{isobaric}=p(V_{2}-V_{1})$
$W_{adiabatic}=\frac{p_{1}V_{1}-p_{2}V_{2}}{\gamma-1}$
$W_{isochoric}=0$
Efficiency of the heat engine: $\eta=\frac{work~done}{heat~supplied}=\frac{Q_{1}-Q_{2}}{Q_{1}}$
$\eta_{carnot}=1-\frac{Q_{2}}{Q_{1}}=1-\frac{T_{2}}{T_{1}}$
Coeff. of performance of refrigerator: $COP=\frac{Q_{2}}{W}=\frac{Q_{2}}{Q_{1}-Q_{2}}$
Entropy: $\Delta S=\frac{\Delta Q}{T}, S_{f}-S_{i}=\int_{i}^{f}\frac{\Delta Q}{T}$
Const. T: $\Delta S=\frac{Q}{T}$, Varying T: $\Delta S=ms~ln\frac{T_{f}}{T_{i}}$
Adiabatic process: $\Delta Q=0, PV^{\gamma} = constant$

4.5: Heat Transfer

Conduction: $\frac{\Delta Q}{\Delta t}=-KA\frac{\Delta T}{x}$
Thermal resistance: $R=\frac{x}{KA}$
$R_{series}=R_{1}+R_{2}=\frac{1}{A}(\frac{x_{1}}{K_{1}}+\frac{x_{2}}{K_{2}})$
$\frac{1}{R_{parallel}}=\frac{1}{R_{1}}+\frac{1}{R_{2}}=\frac{1}{x}(K_{1}A_{1}+K_{2}A_{2})$
Kirchhoff's Law: $\frac{emissive~power}{absorptive~power}=\frac{E_{body}}{a_{body}}=E_{blackbody}$
Wien's displacement law: $\lambda_{m}T=b$
Stefan-Boltzmann law: $\frac{\Delta Q}{\Delta t}=\sigma eAT^{4}$
Newton's law of cooling: $\frac{dT}{dt}=-bA(T-T_{0})$

5. Electricity & Magnetism

5.1: Electrostatics

Coulomb's law: $\vec{F}=\frac{1}{4\pi\epsilon_{0}}\frac{q_{1}q_{2}}{r^{2}}\hat{r}$
Electric field: $\vec{E}(\vec{r})=\frac{1}{4\pi\epsilon_{0}}\frac{q}{r^{2}}\hat{r}$
Electrostatic energy: $U=\frac{1}{4\pi\epsilon_{0}}\frac{q_{1}q_{2}}{r}$
Electrostatic potential: $V=\frac{1}{4\pi\epsilon_{0}}\frac{q}{r}$
$dV=-\vec{E}\cdot\vec{r}, V(\vec{r})=-\int_{\infty}^{\vec{r}}\vec{E}\cdot d\vec{r}$
Electric dipole moment: $\vec{p}=q\vec{d}$
Potential of a dipole: $V=\frac{1}{4\pi\epsilon_{0}}\frac{p~cos~\theta}{r^{2}}$
Field of a dipole: $E_{r}=\frac{1}{4\pi\epsilon_{0}}\frac{2p~cos~\theta}{r^{3}}, E_{\theta}=\frac{1}{4\pi\epsilon_{0}}\frac{p~sin~\theta}{r^{3}}$
Torque on a dipole placed in $\vec{E}$: $\vec{\tau}=\vec{p}\times\vec{E}$
Pot. energy of a dipole placed in $\vec{E}$: $U=-\vec{p}\cdot\vec{E}$

5.2: Gauss's Law and Applications

Electric flux: $\phi=\oint\vec{E}\cdot d\vec{S}$
Gauss's law: $\oint\vec{E}\cdot d\vec{S}=q_{in}/\epsilon_{0}$
Field of a uniformly charged ring on its axis: $E_{P}=\frac{1}{4\pi\epsilon_{0}}\frac{qx}{(a^{2}+x^{2})^{3/2}}$
E and V of a uniformly charged sphere:
$E=\frac{1}{4\pi\epsilon_{0}}\frac{Qr}{R^{3}}$ for $r
$V=\frac{Q}{8\pi\epsilon_{0}R}(3-\frac{r^{2}}{R^{2}})$ for $r
E and V of a uniformly charged spherical shell:
$E=0$ for $r
$V=\frac{1}{4\pi\epsilon_{0}}\frac{Q}{R}$ for $r
Field of a line charge: $E=\frac{\lambda}{2\pi\epsilon_{0}r}$
Field of an infinite sheet: $E=\frac{\sigma}{2\epsilon_{0}}$
Field in the vicinity of conducting surface: $E=\frac{\sigma}{\epsilon_{0}}$

5.3: Capacitors

Capacitance: $C=q/V$
Parallel plate capacitor: $C=\epsilon_{0}A/d$
Spherical capacitor: $C=\frac{4\pi\epsilon_{0}r_{1}r_{2}}{r_{2}-r_{1}}$
Cylindrical capacitor: $C=\frac{2\pi\epsilon_{0}l}{ln(r_{2}/r_{1})}$
Capacitors in parallel: $C_{eq}=C_{1}+C_{2}$
Capacitors in series: $\frac{1}{C_{eq}}=\frac{1}{C_{1}}+\frac{1}{C_{2}}$
Force between plates of a parallel plate capacitor: $F=\frac{Q^{2}}{2A\epsilon_{0}}$
Energy stored in capacitor: $U=\frac{1}{2}CV^{2}=\frac{Q^{2}}{2C}=\frac{1}{2}QV$
Energy density in electric field E: $U/V=\frac{1}{2}\epsilon_{0}E^{2}$
Capacitor with dielectric: $C=\frac{\epsilon_{0}KA}{d}$

5.4: Current electricity

Current density: $j=i/A=\sigma E$
Drift speed: $v_{d}=\frac{1}{2}\frac{eE}{m}\tau=\frac{i}{neA}$
Resistance of a wire: $R=\rho l/A$, where $\rho=1/\sigma$
Temp. dependence of resistance: $R=R_{0}(1+\alpha\Delta T)$
Ohm's law: $V=iR$
Kirchhoff's Laws: (i) Junction Law: $\Sigma_{node}I_{i}=0$ (ii) Loop Law: $\Sigma_{loop}\Delta V_{i}=0$
Resistors in parallel: $\frac{1}{R_{eq}}=\frac{1}{R_{1}}+\frac{1}{R_{2}}$
Resistors in series: $R_{eq}=R_{1}+R_{2}$
Wheatstone bridge: Balanced if $R_{1}/R_{2}=R_{3}/R_{4}$
Electric Power: $P=V^{2}/R=I^{2}R=IV$
Galvanometer as an Ammeter: $i_{g}G=(i-i_{g})S$
Galvanometer as a Voltmeter: $V_{AB}=i_{g}(R+G)$
Charging of capacitors: $q(t)=CV[1-e^{-\frac{t}{RC}}]$
Discharging of capacitors: $q(t)=q_{0}e^{-\frac{t}{RC}}$
Time constant in RC circuit: $\tau=RC$
Peltier effect: emf $e=\frac{\Delta H}{\Delta Q}=\frac{Peltier~heat}{charge~transferred}$
Seeback effect: 1. Thermo-emf: $e=aT+\frac{1}{2}bT^{2}$, 2. Thermoelectric power: $de/dt=a+bT$, 3. Neutral temp.: $T_{n}=-a/b$, 4. Inversion temp.: $T_{i}=-2a/b$
Thomson effect: emf $e=\frac{\Delta H}{\Delta Q}=\frac{Thomson~heat}{charge~transferred}=\sigma\Delta T$
Faraday's law of electrolysis: $m=Zit=\frac{1}{F}Eit$ where $F=96485~C/g$ Faraday constant.

5.5 & 5.6: Magnetism & Magnetic Field due to Current

Lorentz force on a moving charge: $\vec{F}=q\vec{v}\times\vec{B}+q\vec{E}$
Charged particle in uniform magnetic field: $r=\frac{mv}{qB}, T=\frac{2\pi m}{qB}$
Force on a current carrying wire: $\vec{F}=i\vec{l}\times\vec{B}$
Magnetic moment of a current loop: $\vec{\mu}=i\vec{A}$
Torque on a magnetic dipole placed in $\vec{B}$: $\vec{\tau}=\vec{\mu}\times\vec{B}$
Energy of a magnetic dipole placed in $\vec{B}$: $U=-\vec{\mu}\cdot\vec{B}$
Hall effect: $V_{H}=\frac{Bi}{ned}$
Biot-Savart law: $d\vec{B}=\frac{\mu_{0}}{4\pi}\frac{i~d\vec{l}\times\vec{r}}{r^{3}}$
Field due to a straight conductor: $B=\frac{\mu_{0}i}{4\pi d}(cos~\theta_{1}-cos~\theta_{2})$
Field due to an infinite straight wire: $B=\frac{\mu_{0}i}{2\pi d}$
Force between parallel wires: $\frac{dF}{dl}=\frac{\mu_{0}i_{1}i_{2}}{2\pi d}$
Field on the axis of a ring: $B_{P}=\frac{\mu_{0}ia^{2}}{2(a^{2}+d^{2})^{3/2}}$
Field at the centre of an arc: $B=\frac{\mu_{0}i\theta}{4\pi a}$
Field at the centre of a ring: $B=\frac{\mu_{0}i}{2a}$
Ampere's law: $\oint\vec{B}\cdot d\vec{l}=\mu_{0}I_{in}$
Field inside a solenoid: $B=\mu_{0}ni, n=\frac{N}{l}$
Field inside a toroid: $B=\frac{\mu_{0}Ni}{2\pi r}$
Field of a bar magnet: $B_{1}=\frac{\mu_{0}}{4\pi}\frac{2M}{d^{3}}$ (axial), $B_{2}=\frac{\mu_{0}}{4\pi}\frac{M}{d^{3}}$ (equatorial)
Angle of dip: $B_{h}=B~cos~\delta$
Tangent galvanometer: $B_{h}tan~\theta=\frac{\mu_{0}ni}{2r}, i=K~tan~\theta$
Moving coil galvanometer: $niAB=k\theta \Rightarrow i=\frac{k}{nAB}\theta$
Time period of magnetometer: $T=2\pi\sqrt{\frac{I}{MB_{h}}}$
Permeability: $\vec{B}=\mu\vec{H}$

5.7: Electromagnetic Induction

Magnetic flux: $\phi=\oint\vec{B}\cdot d\vec{S}$
Faraday's law: $e=-\frac{d\phi}{dt}$
Lenz's Law: Induced current creates a B-field that opposes the change in magnetic flux.
Motional emf: $e=Blv$
Self inductance: $\phi=Li, e=-L\frac{di}{dt}$
Self inductance of a solenoid: $L=\mu_{0}n^{2}(\pi r^{2}l)$
Growth of current in LR circuit: $i=\frac{e}{R}[1-e^{-\frac{t}{L/R}}]$
Decay of current in LR circuit: $i=i_{0}e^{-\frac{t}{L/R}}$
Time constant of LR circuit: $\tau=L/R$
Energy stored in an inductor: $U=\frac{1}{2}Li^{2}$
Energy density of B field: $u=\frac{U}{V}=\frac{B^{2}}{2\mu_{0}}$
Mutual inductance: $\phi=Mi, e=-M\frac{di}{dt}$
EMF induced in a rotating coil: $e=NAB\omega~sin~\omega t$
Alternating current: $i=i_{0}sin(\omega t+\phi), T=2\pi/\omega$
Average current in AC: $\overline{i}=\frac{1}{T}\int_{0}^{T}i~dt=0$
RMS current: $i_{rms}=[\frac{1}{T}\int_{0}^{T}i^{2}dt]^{1/2}=\frac{i_{0}}{\sqrt{2}}$
Energy: $E=i_{rms}{}^{2}RT$
Capacitive reactance: $X_{c}=\frac{1}{\omega C}$
Inductive reactance: $X_{L}=\omega L$
Imepedance: $Z=e_{0}/i_{0}$
RC circuit: $Z=\sqrt{R^{2}+(1/\omega C)^{2}}, tan~\phi=\frac{1}{\omega CR}$
LR circuit: $Z=\sqrt{R^{2}+\omega^{2}L^{2}}, tan~\phi=\frac{\omega L}{R}$
LCR Circuit: $Z=\sqrt{R^{2}+(\frac{1}{\omega C}-\omega L)^{2}}, tan~\phi=\frac{\frac{1}{\omega C}-\omega L}{R}$
Resonance: $\nu_{resonance}=\frac{1}{2\pi}\sqrt{\frac{1}{LC}}$
Power factor: $P=e_{rms}i_{rms}cos~\phi$
Transformer: $\frac{N_{1}}{N_{2}}=\frac{e_{1}}{e_{2}}, e_{1}i_{1}=e_{2}i_{2}$
Speed of the EM waves in vacuum: $c=1/\sqrt{\mu_{0}\epsilon_{0}}$

6. Modern Physics

6.1: Photo-electric effect

Photon's energy: $E=h\nu=hc/\lambda$
Photon's momentum: $p=h/\lambda=E/c$
Max. KE of ejected photo-electron: $K_{max}=h\nu-\phi$
Threshold freq. in photo-electric effect: $\nu_{0}=\phi/h$
Stopping potential: $V_{o}=\frac{hc}{e}(\frac{1}{\lambda})-\frac{\phi}{e}$
de Broglie wavelength: $\lambda=h/p$

6.2: The Atom

Energy in nth Bohr's orbit: $E_{n}=-\frac{mZ^{2}e^{4}}{8{\epsilon_{0}}^{2}h^{2}n^{2}}$, $E_{n}=-\frac{13.6Z^{2}}{n^{2}}eV$
Radius of the nth Bohr's orbit: $r_{n}=\frac{\epsilon_{0}h^{2}n^{2}}{\pi mZe^{2}}, r_{n}=\frac{n^{2}a_{0}}{Z}, a_{0}=0.529~A$
Quantization of the angular momentum: $l=\frac{nh}{2\pi}$
Photon energy in state transition: $E_{2}-E_{1}=h\nu$
Wavelength of emitted radiation from nth to mth state: $\frac{1}{\lambda}=RZ^{2}[\frac{1}{n^{2}}-\frac{1}{m^{2}}]$

6.3: The Nucleus

Nuclear radius: $R=R_{0}A^{1/3}$, $R_{0}\approx1.1\times10^{-15}m$
Decay rate: $\frac{dN}{dt}=-\lambda N$
Population at time t: $N=N_{0}e^{-\lambda t}$
Half life: $t_{1/2}=0.693/\lambda$
Average life: $t_{av}=1/\lambda$
Population after n half lives: $N=N_{0}/2^{n}$
Mass defect: $\Delta m=[Zm_{p}+(A-Z)m_{n}]-M$
Binding energy: $B=[Zm_{p}+(A-Z)m_{n}-M]c^{2}$
Q-value: $Q=U_{i}-U_{f}$
Energy released in nuclear reaction: $\Delta E=\Delta mc^{2}$ where $\Delta m=m_{reactants}-m_{products}$

6.4: Vacuum tubes and Semiconductors

Plate resistance of a triode: $r_{p}=\frac{\Delta V_{p}}{\Delta i_{p}}|_{\Delta V_{g}=0}$
Transconductance of a triode: $g_{m}=\frac{\Delta i_{p}}{\Delta V_{g}}|_{\Delta V_{p}=0}$
Amplification by a triode: $\mu=-\frac{\Delta V_{p}}{\Delta V_{g}}|_{\Delta i_{p}=0}$
Relation between $r_{p},\mu,$ and $g_{m}:\mu=r_{p}\times g_{m}$
X-ray spectrum: $\lambda_{min}=\frac{hc}{eV}$
Moseley's law: $\sqrt{\nu}=a(Z-b)$
X-ray diffraction: $2d~sin~\theta=n\lambda$
Heisenberg uncertainity principle: $\Delta p\Delta x\ge h/(2\pi)$, $\Delta E\Delta t\ge h/(2\pi)$
Current in a transistor: $I_{e}=I_{b}+I_{c}$
$\alpha$ and $\beta$ parameters of a transistor: $\alpha=\frac{I_{c}}{I_{e}}, \beta=\frac{I_{c}}{I_{b}}, \beta=\frac{\alpha}{1-\alpha}$
Transconductance: $g_{m}=\frac{\Delta I_{c}}{\Delta V_{be}}$
Logic Gates:

A	B	AND ($AB$)	OR ($A+B$)	NAND ($\overline{AB}$)	NOR ($\overline{A+B}$)	XOR ($A\overline{B}+\overline{A}B$)
0	0	0	0	1	1	0
0	1	0	1	1	0	1
1	0	0	1	1	0	1
1	1	1	1	0	0	0

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