physics_formula
 Kinematics and Laws of Motion Motion in straight linev=u+at$v=u+at$s=ut+12at2$s=ut+\frac{1}{2}a{t}^{2}$v2=u2+2as${v}^{2}={u}^{2}+2as$Sn=u+12 a (2n-1) ${S}_{n}=u+\frac{1}{2}\phantom{\rule{0.22em}{0ex}}a\phantom{\rule{0.22em}{0ex}}\left(2n-1\right)\phantom{\rule{0.22em}{0ex}}$ Relative VelocityVelocity of A with respect to B=VA-VB$=\stackrel{\to }{{V}_{A}}-\stackrel{\to }{{V}_{B}}$ Linear Momentum=p=m v $=\stackrel{\to }{p}=m\stackrel{\to }{\phantom{\rule{0.22em}{0ex}}v\phantom{\rule{0.22em}{0ex}}}$ Force = Rate of change of momentum (here mass is considered to be constant)F=dpdt=d (mv)dt=mdvdt=ma$\stackrel{\to }{F}=\frac{d\stackrel{\to }{p}}{dt}=\frac{d\phantom{\rule{0.22em}{0ex}}\left(m\stackrel{\to }{v}\right)}{dt}=m\frac{d\stackrel{\to }{v}}{dt}=m\stackrel{\to }{a}$ (if mass is not constant)F=dpdt=d (mv)dt=mdvdt+v dmdt$\stackrel{\to }{F}=\frac{d\stackrel{\to }{p}}{dt}=\frac{d\phantom{\rule{0.22em}{0ex}}\left(m\stackrel{\to }{v}\right)}{dt}=m\frac{d\stackrel{\to }{v}}{dt}+\stackrel{\to }{v}\phantom{\rule{0.22em}{0ex}}\frac{dm}{dt}$ Elastic CollisionIn elastic collision total kinetic energy of the two bodies remains the same. m1u1+m2u2=m1v1+m2v2 ${m}_{1}{u}_{1}+{m}_{2}{u}_{2}={m}_{1}{v}_{1}+{m}_{2}{v}_{2}\phantom{\rule{0.22em}{0ex}}$12m1u21+12m2u22=12m1v21+12m2v22 $\frac{1}{2}{m}_{1}{u}_{1}^{2}+\frac{1}{2}{m}_{2}{u}_{2}^{2}=\frac{1}{2}{m}_{1}{v}_{1}^{2}+\frac{1}{2}{m}_{2}{v}_{2}^{2}\phantom{\rule{0.22em}{0ex}}$ v1=m1-m2m1+m2 u1+2m2m1+m2 u2${v}_{1}=\frac{{m}_{1}-{m}_{2}}{{m}_{1}+{m}_{2}}\phantom{\rule{0.22em}{0ex}}{u}_{1}+\frac{2{m}_{2}}{{m}_{1}+{m}_{2}}\phantom{\rule{0.22em}{0ex}}{u}_{2}$ v2=m2-m1m1+m2 u2+2m1m1+m2 u1${v}_{2}=\frac{{m}_{2}-{m}_{1}}{{m}_{1}+{m}_{2}}\phantom{\rule{0.22em}{0ex}}{u}_{2}+\frac{2{m}_{1}}{{m}_{1}+{m}_{2}}\phantom{\rule{0.22em}{0ex}}{u}_{1}$ Coefficient of restitution (e)e=v2-v1u1-u2$e=\frac{{v}_{2}-{v}_{1}}{{u}_{1}-{u}_{2}}$ Newton’s law of restitution applies to the component of the velocity of the sphere perpendicular to the surface. F=-dUdx$F=-\frac{dU}{dx}$ U=Potential energyF=Force Condition for stable equilibriumi) dUdx=0$\frac{dU}{dx}=0$ ii) d2Udx2>0 $\frac{{d}^{2}U}{d{x}^{2}}>0\phantom{\rule{0.22em}{0ex}}$ Condition for unstable equilibriumi) dUdx=0$\frac{dU}{dx}=0$ ii) d2Udx2<0 $\frac{{d}^{2}U}{d{x}^{2}}<0\phantom{\rule{0.22em}{0ex}}$ Condition for neutral equilibriumi) dUdx=0$\frac{dU}{dx}=0$ ii) d2Udx2=0 $\frac{{d}^{2}U}{d{x}^{2}}=0\phantom{\rule{0.22em}{0ex}}$ Tangential velocity vv=𝜔r$v=𝜔r$ Moment of inertia : Ring or thin wall hollow cylinderAxis: passing thru center along cylinder axisI=mr2$I=m{r}^{2}$ Moment of inertia : Disc or solid cylinderAxis: passing thru center along cylinder axis I=12mr2$I=\frac{1}{2}m{r}^{2}$ Moment of inertia : Spherical ShellAxis: passing thru centerI=23mr2$I=\frac{2}{3}m{r}^{2}$ Moment of inertia : solid sphereAxis: passing thru centerI=25mr2$I=\frac{2}{5}m{r}^{2}$ Moment of inertia : rodAxis: passing thru center (and perpendicular to axis)I=mL212$I=\frac{m{L}^{2}}{12}$ Gravitation Gravitational ForceF=Gm1 m2r2$F=\frac{G{m}_{1}\phantom{\rule{0.22em}{0ex}}{m}_{2}}{{r}^{2}}$ Potential Energy = U =- Gm1 m2r $Potential\phantom{\rule{0.22em}{0ex}}Energy\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}U\phantom{\rule{0.22em}{0ex}}=-\phantom{\rule{0.22em}{0ex}}\frac{G{m}_{1}\phantom{\rule{0.22em}{0ex}}{m}_{2}}{r}\phantom{\rule{0.22em}{0ex}}$ Gravitational field intensity (E)E(r) =GMr2 $E\left(r\right)\phantom{\rule{0.22em}{0ex}}=\frac{GM}{{r}^{2}}\phantom{\rule{0.22em}{0ex}}$ Acceleration due to gravity on surface of earth=g = GMeR2e$=g\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}\frac{G{M}_{e}}{{R}_{e}^{2}}$ Variation of Acceleration due to gravity with depth= gd=g(1-dRe)$=\phantom{\rule{0.22em}{0ex}}{g}_{d}=g\left(1-\frac{d}{{R}_{e}}\right)$ Variation of Acceleration due to gravity with height (h)gh=GMe(Re+h)2 = gR2e(Re+h)2=g(1+hRe)-2 ${g}_{h}=\frac{G{M}_{e}}{\left({R}_{e}+h{\right)}^{2}}\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}\frac{g{R}_{e}^{2}}{\left({R}_{e}+h{\right)}^{2}}=g\left(1+\frac{h}{{R}_{e}}{\right)}^{-2}\phantom{\rule{0.22em}{0ex}}$gh=g(1+hRe)-2 ≈g(1-2hRe)${g}_{h}=g\left(1+\frac{h}{{R}_{e}}{\right)}^{-2}\phantom{\rule{0.22em}{0ex}}\approx g\left(1-\frac{2h}{{R}_{e}}\right)$ Gravitational potential (V)V(r) =-GMr $V\left(r\right)\phantom{\rule{0.22em}{0ex}}=-\frac{GM}{r}\phantom{\rule{0.44em}{0ex}}$ Gravitational potential Energy (U) U=- GMmr $\phantom{\rule{0.22em}{0ex}}U=-\phantom{\rule{0.22em}{0ex}}\frac{GMm}{r}\phantom{\rule{0.44em}{0ex}}$ Gravitational Self energy of uniform SphereU=35GM2R$U=\frac{3}{5}\frac{G{M}^{2}}{R}$ Escape Velocity: The minimum velocity with which a body must be projected up so as to enable it to just overcome the gravitational pull, is known as escape velocity. vescape = 2gRe =(2 )vorbital${v}_{escape}\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}\sqrt{2g{R}_{e}}\phantom{\rule{0.22em}{0ex}}=\left(\sqrt{2}\phantom{\rule{0.22em}{0ex}}\right){v}_{orbital}$ Binding energy of satellite: The minimum enery required for satellite to escape from earth gravitational influence is called binding energy of satellite. Binding energy of satellite revolving around circular orbit round the earth = 12GMmr$\frac{1}{2}\frac{GMm}{r}$ Kepler’s first law ( law of elliptical orbit) :A planet moves round the sun in an elliptical orbit with sun situated at one of its foci. Kepler’s second law ( law of areal velocities) :A planet moves round the sun in such a way that its areal velocity is constant. Kepler’s third law ( law of time period) :A planet moves round the sun in such a way that the square of its period is proportional to the cube of semi major axis of its elliptical orbit.
 Simple Harmonic Motion d2xdt2+𝜔2x=0$\phantom{\rule{0.22em}{0ex}}\frac{{d}^{2}x}{d{t}^{2}}+{𝜔}^{2}x=0$ v2=𝜔2(a2-x2) ${v}^{2}={𝜔}^{2}\left({a}^{2}-{x}^{2}\right)\phantom{\rule{0.22em}{0ex}}$ x=a sin 𝜔t$x=a\phantom{\rule{0.22em}{0ex}}sin\phantom{\rule{0.22em}{0ex}}𝜔t$ Kinetic Energy in SHM=12mv2=12m𝜔2(a2-x2 )$Kinetic\phantom{\rule{0.22em}{0ex}}Energy\phantom{\rule{0.22em}{0ex}}in\phantom{\rule{0.22em}{0ex}}SHM=\frac{1}{2}m{v}^{2}=\frac{1}{2}m{𝜔}^{2}\left({a}^{2}-{x}^{2}\phantom{\rule{0.22em}{0ex}}\right)$ Potential Energy in SHM=$Potential\phantom{\rule{0.22em}{0ex}}Energy\phantom{\rule{0.22em}{0ex}}in\phantom{\rule{0.22em}{0ex}}SHM=$ 12kx2=12m𝜔2x2 $\phantom{\rule{0.22em}{0ex}}\frac{1}{2}k{x}^{2}=\frac{1}{2}m{𝜔}^{2}{x}^{2}\phantom{\rule{0.22em}{0ex}}$ Total Energy in SHM=12m𝜔2a2 $Total\phantom{\rule{0.22em}{0ex}}Energy\phantom{\rule{0.22em}{0ex}}in\phantom{\rule{0.22em}{0ex}}SHM=\frac{1}{2}m{𝜔}^{2}{a}^{2}\phantom{\rule{0.22em}{0ex}}$
 Damped Periodic Motion md2xdt2+bdxdt+kx=0$m\frac{{d}^{2}x}{d{t}^{2}}+b\frac{dx}{dt}+kx=0$ x=Ae-bt/2m cos(𝜔t+𝜙)$x=A{e}^{-bt/2m}\phantom{\rule{0.22em}{0ex}}cos\left(𝜔t+𝜙\right)$ 𝜔=km-(b2m)2$𝜔=\sqrt{\frac{k}{m}-\left(\frac{b}{2m}{\right)}^{2}}$
 Waves Speed of waves in a string v=T𝜇$v=\sqrt{\frac{T}{𝜇}}$where v=speed of waveT=Tension in string𝜇=$𝜇=$mass per unit length Energy associated with 1 wavelength of string waveE𝜆=12 𝜇 𝜔2A2 𝜆${E}_{𝜆}=\frac{1}{2}\phantom{\rule{0.22em}{0ex}}𝜇\phantom{\rule{0.22em}{0ex}}{𝜔}^{2}{A}^{2}\phantom{\rule{0.22em}{0ex}}𝜆$ where 𝜆=$𝜆=$wavelength of waveA = Wave amplitude𝜇=$𝜇=$mass per unit length𝜔=$𝜔=$angular wave frequency Power transported by a string waveP=12 𝜇 𝜔2A2 v$P=\frac{1}{2}\phantom{\rule{0.22em}{0ex}}𝜇\phantom{\rule{0.22em}{0ex}}{𝜔}^{2}{A}^{2}\phantom{\rule{0.22em}{0ex}}v$ where v=$v=$speed of waveA = Wave amplitude𝜇=$𝜇=$mass per unit length𝜔=$𝜔=$angular wave frequency Due to interference of two waves redistribution of energy takes place in medium y1=A1sin(kx-𝜔t)${y}_{1}={A}_{1}sin\left(kx-𝜔t\right)$y2=A2sin(kx-𝜔t+𝛿)${y}_{2}={A}_{2}sin\left(kx-𝜔t+𝛿\right)$ y=y1+y2$y={y}_{1}+{y}_{2}$y=Asin(kx-𝜔t+𝜖)$y=Asin\left(kx-𝜔t+𝜖\right)$whereA=A21+A22+2A1A2 cos𝛿 $A=\sqrt{{A}_{1}^{2}+{A}_{2}^{2}+2{A}_{1}{A}_{2}\phantom{\rule{0.22em}{0ex}}cos𝛿\phantom{\rule{0.22em}{0ex}}}$tan 𝜖=A2sin𝛿A1+A2cos𝛿 $tan\phantom{\rule{0.22em}{0ex}}𝜖=\frac{{A}_{2}sin𝛿}{{A}_{1}+{A}_{2}cos𝛿\phantom{\rule{0.22em}{0ex}}}$ Simple Harmonic Progressive Wavey=A sin(𝜔t-2𝜋x𝜆)$y=A\phantom{\rule{0.22em}{0ex}}sin\left(𝜔t-\frac{2𝜋x}{𝜆}\right)$ Speed of longitudinal waveSound wave travel in compressions and rarefactionsv=B𝜌$v=\sqrt{\frac{B}{𝜌}}$where v = speed of waveB = Bulk Modulus𝜌=$𝜌=$ mass density For linear medium (like solid bar), speed of soundv=Y𝜌$v=\sqrt{\frac{Y}{𝜌}}$ where v = speed of waveY = Young's Modulus𝜌=$𝜌=$ mass density Speed of sound in gasv=𝛾P𝜌$v=\sqrt{\frac{𝛾P}{𝜌}}$ Waves in String : String fixed at both endsSpeed of waves : v=T𝜇$v=\sqrt{\frac{T}{𝜇}}$where𝜇=mass per unit length $𝜇=mass\phantom{\rule{0.22em}{0ex}}per\phantom{\rule{0.22em}{0ex}}unit\phantom{\rule{0.22em}{0ex}}length\phantom{\rule{0.22em}{0ex}}$T=Tension inString $T=Tension\phantom{\rule{0.22em}{0ex}}inString\phantom{\rule{0.22em}{0ex}}$
 Kinetic Theory The average distance a molecule can travel without colliding is called mean free path.Mean free path=𝜆=12 n 𝜋 d2$Mean\phantom{\rule{0.22em}{0ex}}free\phantom{\rule{0.22em}{0ex}}path=𝜆=\frac{1}{\sqrt{2}\phantom{\rule{0.22em}{0ex}}n\phantom{\rule{0.22em}{0ex}}𝜋\phantom{\rule{0.22em}{0ex}}{d}^{2}}$ where d = diameter of each moleculen=number of molecule per unit volumem-mass of one molecule𝜌= density of gas$𝜌=\phantom{\rule{0.22em}{0ex}}density\phantom{\rule{0.22em}{0ex}}of\phantom{\rule{0.22em}{0ex}}gas$ Boltzmann constant=KB=PVNT $Boltzmann\phantom{\rule{0.22em}{0ex}}constant={K}_{B}=\frac{PV}{NT}\phantom{\rule{0.22em}{0ex}}$where N=number of molecules (count of molecules)P=PressureV=VolumeT=Temperature aPV=KB N T⟹PV=NAKB N NA T $\begin{array}{l}PV={K}_{B}\phantom{\rule{0.22em}{0ex}}N\phantom{\rule{0.22em}{0ex}}T\\ ⟹PV={N}_{A}{K}_{B}\phantom{\rule{0.22em}{0ex}}\frac{N\phantom{\rule{0.22em}{0ex}}}{{N}_{A}}\phantom{\rule{0.22em}{0ex}}T\phantom{\rule{4.4em}{0ex}}\\ \end{array}$a𝜇=number of moles=N NANAKB=R ⟹PV=𝜇RT$\begin{array}{l}𝜇=number\phantom{\rule{0.22em}{0ex}}of\phantom{\rule{0.22em}{0ex}}moles=\frac{N\phantom{\rule{0.22em}{0ex}}}{{N}_{A}}\\ {N}_{A}{K}_{B}=R\\ \\ ⟹PV=𝜇RT\end{array}$ Avogadro’s hypothesis : The number of molecules per unit volume is samefor all gases. KB=Bolzmann constant = PVNT${K}_{B}=Bolzmann\phantom{\rule{0.22em}{0ex}}constant\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}\frac{PV}{NT}$ where P=PressureV=VolumeN- Total Number of molecule (count of molecule)T=Temnperaturea PV=KB N T⟹PV=NAKB N NA T $\begin{array}{l}\phantom{\rule{1.98em}{0ex}}PV={K}_{B}\phantom{\rule{0.22em}{0ex}}N\phantom{\rule{0.22em}{0ex}}T\\ ⟹PV={N}_{A}{K}_{B}\phantom{\rule{0.22em}{0ex}}\frac{N\phantom{\rule{0.22em}{0ex}}}{{N}_{A}}\phantom{\rule{0.22em}{0ex}}T\phantom{\rule{4.18em}{0ex}}\\ \end{array}$ 𝜇=number of moles=N NA$𝜇=number\phantom{\rule{0.22em}{0ex}}of\phantom{\rule{0.22em}{0ex}}moles=\frac{N\phantom{\rule{0.22em}{0ex}}}{{N}_{A}}$ NAKB=R${N}_{A}{K}_{B}=R$ ⟹PV=𝜇RT$⟹PV=𝜇RT$ Gas equation: PV=𝜇RT $\mathbit{G}\mathbit{a}\mathbit{s}\phantom{\rule{0.22em}{0ex}}\mathbit{e}\mathbit{q}\mathbit{u}\mathbit{a}\mathbit{t}\mathbit{i}\mathbit{o}\mathbit{n}:\phantom{\rule{0.22em}{0ex}}PV=𝜇RT\phantom{\rule{1.32em}{0ex}}$where𝜇=number of moles$𝜇=number\phantom{\rule{0.22em}{0ex}}of\phantom{\rule{0.22em}{0ex}}moles$R=NAKB$R={N}_{A}{K}_{B}$ Van der Waals equation(p+an2V2)(V-nb)=n R T$\left(p+\frac{a{n}^{2}}{{V}^{2}}\right)\left(V-nb\right)=n\phantom{\rule{0.22em}{0ex}}R\phantom{\rule{0.22em}{0ex}}T$where an2V2=Pressure correction$\phantom{\rule{0.22em}{0ex}}\frac{a{n}^{2}}{{V}^{2}}=Pressure\phantom{\rule{0.22em}{0ex}}correction$ nb=Volume correction$nb=Volume\phantom{\rule{0.22em}{0ex}}correction$ Specific Heat Capacity : Monoatomic Gas1. Monoatomic Gas has three degree of freedom.2. Total internal energy of 1 mole of gas= U U=32KB NA T =32 R T$\phantom{\rule{0.22em}{0ex}}U=\frac{3}{2}{K}_{B}\phantom{\rule{0.44em}{0ex}}{N}_{A}\phantom{\rule{0.22em}{0ex}}T\phantom{\rule{0.22em}{0ex}}=\frac{3}{2}\phantom{\rule{0.22em}{0ex}}R\phantom{\rule{0.22em}{0ex}}T$3. Cv (monoatomic gas)=dUdT=32 R${C}_{v}\phantom{\rule{0.22em}{0ex}}\left(monoatomic\phantom{\rule{0.22em}{0ex}}gas\right)=\frac{dU}{dT}=\frac{3}{2}\phantom{\rule{0.22em}{0ex}}R$4. Cp (monoatomic gas)=Cv+R=52 R${C}_{p}\phantom{\rule{0.22em}{0ex}}\left(monoatomic\phantom{\rule{0.22em}{0ex}}gas\right)={C}_{v}+R=\frac{5}{2}\phantom{\rule{0.22em}{0ex}}R$5. Ratio of specific heat=𝛾=CpCv=53$\phantom{\rule{0.22em}{0ex}}Ratio\phantom{\rule{0.22em}{0ex}}of\phantom{\rule{0.22em}{0ex}}specific\phantom{\rule{0.22em}{0ex}}heat=𝛾=\frac{{C}_{p}}{{C}_{v}}=\frac{5}{3}$ Specific Heat Capacity : Diatomic Gas (Rigid Diatomic gas) 1. Rigid Diatomic Gas has five degree of freedom.2. Total internal energy of 1 mole of gas= UU=52KB T NA=52 R T$U=\frac{5}{2}{K}_{B}\phantom{\rule{0.22em}{0ex}}T\phantom{\rule{0.22em}{0ex}}{N}_{A}=\frac{5}{2}\phantom{\rule{0.22em}{0ex}}R\phantom{\rule{0.22em}{0ex}}T$3. Cv (Rigid Diatomic gas)=dUdT=52 R${C}_{v}\phantom{\rule{0.22em}{0ex}}\left(Rigid\phantom{\rule{0.22em}{0ex}}Diatomic\phantom{\rule{0.22em}{0ex}}gas\right)=\frac{dU}{dT}=\frac{5}{2}\phantom{\rule{0.22em}{0ex}}R$4. Cp (Rigid Diatomic gas)=Cv+R=72 R${C}_{p}\phantom{\rule{0.22em}{0ex}}\left(Rigid\phantom{\rule{0.22em}{0ex}}Diatomic\phantom{\rule{0.22em}{0ex}}gas\right)={C}_{v}+R=\frac{7}{2}\phantom{\rule{0.22em}{0ex}}R$5. Ratio of specific heat=𝛾=CpCv=75$Ratio\phantom{\rule{0.22em}{0ex}}of\phantom{\rule{0.22em}{0ex}}specific\phantom{\rule{0.22em}{0ex}}heat=𝛾=\frac{{C}_{p}}{{C}_{v}}=\frac{7}{5}$ Specific Heat Capacity : Diatomic Gas which is not rigid1. Non rigid Diatomic Gas has five degree of freedom and additional vibrational mode2. Total internal energy of 1 mole of gas= UU=(52KB T+KB T) NA=72 R T$U=\left(\frac{5}{2}{K}_{B}\phantom{\rule{0.22em}{0ex}}T+{K}_{B}\phantom{\rule{0.22em}{0ex}}T\right)\phantom{\rule{0.22em}{0ex}}{N}_{A}=\frac{7}{2}\phantom{\rule{0.22em}{0ex}}R\phantom{\rule{0.22em}{0ex}}T$3. Cv (Non rigid Diatomic gas)=dUdT=72 R${C}_{v}\phantom{\rule{0.22em}{0ex}}\left(Non\phantom{\rule{0.22em}{0ex}}rigid\phantom{\rule{0.22em}{0ex}}Diatomic\phantom{\rule{0.22em}{0ex}}gas\right)=\frac{dU}{dT}=\frac{7}{2}\phantom{\rule{0.22em}{0ex}}R$4. Cp (Non rigid Diatomic gas)=Cv+R=92 R${C}_{p}\phantom{\rule{0.22em}{0ex}}\left(Non\phantom{\rule{0.22em}{0ex}}rigid\phantom{\rule{0.22em}{0ex}}Diatomic\phantom{\rule{0.22em}{0ex}}gas\right)={C}_{v}+R=\frac{9}{2}\phantom{\rule{0.22em}{0ex}}R$ 5. Ratio of specific heat=𝛾=CpCv=97$Ratio\phantom{\rule{0.22em}{0ex}}of\phantom{\rule{0.22em}{0ex}}specific\phantom{\rule{0.22em}{0ex}}heat=𝛾=\frac{{C}_{p}}{{C}_{v}}=\frac{9}{7}$
 Polyatomic Gas Non rigid Diatomic Gas has six degrees of freedom and certain (f) vibrational mode Total internal energy of 1 mole of gas U=(62KB T+f KB T) NA$U=\left(\frac{6}{2}{K}_{B}\phantom{\rule{0.22em}{0ex}}T+f\phantom{\rule{0.22em}{0ex}}{K}_{B}\phantom{\rule{0.22em}{0ex}}T\right)\phantom{\rule{0.22em}{0ex}}{N}_{A}$ Cv ${C}_{v}\phantom{\rule{0.22em}{0ex}}$(Polyatomic gas)=dUdT=(3+f)R$=\frac{dU}{dT}=\left(3+f\right)R$ 4. Cp (Non rigid Diatomic gas)=Cv+R=(4+f)R$4.\phantom{\rule{0.22em}{0ex}}{C}_{p}\phantom{\rule{0.22em}{0ex}}\left(Non\phantom{\rule{0.22em}{0ex}}rigid\phantom{\rule{0.22em}{0ex}}Diatomic\phantom{\rule{0.22em}{0ex}}gas\right)={C}_{v}+R=\left(4+f\right)R$ Ratio of specific heat 𝛾=CpCv=4+f3+f$𝛾=\frac{{C}_{p}}{{C}_{v}}=\frac{4+f}{3+f}$ Cp-Cv=R ${C}_{p}-{C}_{v}=R\phantom{\rule{0.44em}{0ex}}$ ( this is true for any ideal gas, monoatomic, diatomic or polyatomic) Speed of sound=c=𝛾P𝜌$=c=\sqrt{\frac{𝛾P}{𝜌}}$ Pressure = P = 13 n m v2rms$Pressure\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}P\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}\frac{1}{3}\phantom{\rule{0.22em}{0ex}}n\phantom{\rule{0.22em}{0ex}}m\phantom{\rule{0.22em}{0ex}}{v}_{rms}^{2}$ where n=number of molecules per unit volumem=mass of moleculevrms${v}_{rms}$ =Root mean square speed PV = 23×(12nV m v2rms )$PV\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}\frac{2}{3}×\left(\frac{1}{2}nV\phantom{\rule{0.22em}{0ex}}m\phantom{\rule{0.22em}{0ex}}{v}_{rms}^{2}\phantom{\rule{0.22em}{0ex}}\right)$ wheren=number of molecules per unit volume Internal energy = E =12nV m v2rms $Internal\phantom{\rule{0.22em}{0ex}}energy\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.44em}{0ex}}E\phantom{\rule{0.22em}{0ex}}=\frac{1}{2}nV\phantom{\rule{0.22em}{0ex}}m\phantom{\rule{0.22em}{0ex}}{v}_{rms}^{2}\phantom{\rule{0.22em}{0ex}}$( This is total Internal energy in volume V gas)wheren=number of molecules per unit volume PV=23E$PV=\frac{2}{3}E$ E=32PV $E=\frac{3}{2}PV\phantom{\rule{0.22em}{0ex}}$( This is total Internal energy in volume V gas) E=32KBNT $E=\frac{3}{2}{K}_{B}NT\phantom{\rule{0.22em}{0ex}}$( N=total number of molecules) Average Kinetic Energy ( of one molecule)=EN=32KBT$=\frac{E}{N}=\frac{3}{2}{K}_{B}T$ Average Kinetic Energy=12 m v2rms $=\frac{1}{2}\phantom{\rule{0.22em}{0ex}}m\phantom{\rule{0.22em}{0ex}}{v}_{rms}^{2}\phantom{\rule{0.44em}{0ex}}$where,m=mass of one molecule v2rms=3KBTm=3RTM ${v}_{rms}^{2}=\frac{3{K}_{B}T}{m}=\frac{3RT}{M}\phantom{\rule{0.66em}{0ex}}$where,M=Molar mass of molecule Root mean square velocity vrms=3RTM${v}_{rms}^{}=\sqrt{\frac{3RT}{M}}$ where,M=Molar mass of molecule Average velocity vaverage=8RT𝜋M$\phantom{\rule{0.22em}{0ex}}{v}_{average}^{}=\sqrt{\frac{8RT}{𝜋M}}$ Most probable velocity vp=2RTM${v}_{p}^{}=\sqrt{\frac{2RT}{M}}$ Mean Free Path: Average distance covered by molecule between 2 successive collision⏨l=12 n 𝜋 d2 $\overline{l}=\frac{1}{\sqrt{2}\phantom{\rule{0.22em}{0ex}}n\phantom{\rule{0.22em}{0ex}}𝜋\phantom{\rule{0.22em}{0ex}}{d}^{2}\phantom{\rule{0.22em}{0ex}}}$wheren = number density of gasd = diameter of molecule Root Mean Square Velocity (Vrms${V}_{rms}$)V2rms=v21+v22+v23+ . . . +v2nN${V}_{rms}^{2}=\frac{{v}_{1}^{2}+{v}_{2}^{2}+{v}_{3}^{2}+\phantom{\rule{0.22em}{0ex}}.\phantom{\rule{0.22em}{0ex}}.\phantom{\rule{0.22em}{0ex}}.\phantom{\rule{0.22em}{0ex}}+{v}_{n}^{2}}{N}$where N is number of collision Average Velocity (⏨V$\overline{V}$)V=v1+v2+v3+ . . . +vnN$V=\frac{{v}_{1}^{}+{v}_{2}^{}+{v}_{3}^{}+\phantom{\rule{0.22em}{0ex}}.\phantom{\rule{0.22em}{0ex}}.\phantom{\rule{0.22em}{0ex}}.\phantom{\rule{0.22em}{0ex}}+{v}_{n}^{}}{N}$where N is number of collision
 Elasticity Youngs Modulus Y=Longitudinal Stresslogitudinal strain = 𝜎𝜖 $Y=\frac{Longitudinal\phantom{\rule{0.22em}{0ex}}Stress}{logitudinal\phantom{\rule{0.22em}{0ex}}strain}\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}\frac{\phantom{\rule{0.22em}{0ex}}𝜎}{𝜖}\phantom{\rule{0.22em}{0ex}}$ Shear modulus or Modulus of Rigidity 𝜂 = Shear StressShear strain $\phantom{\rule{0.22em}{0ex}}𝜂\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}\frac{Shear\phantom{\rule{0.22em}{0ex}}Stress}{Shear\phantom{\rule{0.22em}{0ex}}strain}\phantom{\rule{0.22em}{0ex}}$ Bulk modulusK = B = Volumetric StressVolumetric strain$K\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}B\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}\frac{Volumetric\phantom{\rule{0.22em}{0ex}}Stress}{Volumetric\phantom{\rule{0.22em}{0ex}}strain}$ compressibility = 1Bulk modulus$compressibility\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}\frac{1}{Bulk\phantom{\rule{0.22em}{0ex}}modulus}$ Force Constant: F vs ∆L curve (dFdx) $F\phantom{\rule{0.22em}{0ex}}vs\phantom{\rule{0.22em}{0ex}}∆L\phantom{\rule{0.22em}{0ex}}curve\phantom{\rule{0.22em}{0ex}}\left(\frac{dF}{dx}\right)\phantom{\rule{0.22em}{0ex}}$ is the effective value of the force constant at any point. Poisson ratio is the ratio of transverse contraction ( or expansion) strain to longitudinal extension strain in the direction of stretching force.Poisson ratio = 𝜎 =- 𝜖transverse𝜖longitudinal$Poisson\phantom{\rule{0.22em}{0ex}}ratio\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}𝜎\phantom{\rule{0.22em}{0ex}}=-\phantom{\rule{0.22em}{0ex}}\frac{{𝜖}_{transverse}}{{𝜖}_{longitudinal}}$ Relation between Young Modulus, Modulus of rigidity, Bulk modulus and Poisson Raioa𝜂=Y2(1+𝜎) B=Y3(1-2𝜎) 9Y=1B+1𝜂 𝜂=3B-2𝜂2𝜂+6B $\begin{array}{l}𝜂=\frac{Y}{2\left(1+𝜎\right)}\phantom{\rule{0.22em}{0ex}}\\ \\ B=\frac{Y}{3\left(1-2𝜎\right)}\\ \\ \frac{9}{Y}=\frac{1}{B}+\frac{1}{𝜂}\\ \\ 𝜂=\frac{3B-2𝜂}{2𝜂+6B}\phantom{\rule{0.22em}{0ex}}\\ \end{array}$
 Strain Energy: The work done by the external applied force during stretching is stored as potential energy ( U) in the wire and is called as strain energy in the wire. U=12 ⨯ Load ⨯Elongation = 12 ⨯Y ⨯(Strain)2⨯Volume$U=\frac{1}{2}\phantom{\rule{0.22em}{0ex}}⨯\phantom{\rule{0.22em}{0ex}}Load\phantom{\rule{0.22em}{0ex}}⨯Elongation\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}\frac{1}{2}\phantom{\rule{0.22em}{0ex}}⨯Y\phantom{\rule{0.22em}{0ex}}⨯\left(Strain{\right)}^{2}⨯Volume$ aThe ratio of interatomic force to that of change in interatomic distance (∆R) is defined as the interatomic force constantForce Constant = K = F∆RK = Y⨯r0 where r0 = Normal distance between the atoms of wire$\begin{array}{l}The\phantom{\rule{0.22em}{0ex}}ratio\phantom{\rule{0.22em}{0ex}}of\phantom{\rule{0.22em}{0ex}}interatomic\phantom{\rule{0.22em}{0ex}}force\phantom{\rule{0.22em}{0ex}}to\phantom{\rule{0.22em}{0ex}}that\phantom{\rule{0.22em}{0ex}}of\phantom{\rule{0.22em}{0ex}}change\phantom{\rule{0.22em}{0ex}}in\phantom{\rule{0.22em}{0ex}}interatomic\phantom{\rule{0.22em}{0ex}}distance\phantom{\rule{0.22em}{0ex}}\left(∆R\right)\phantom{\rule{0.22em}{0ex}}\\ is\phantom{\rule{0.22em}{0ex}}defined\phantom{\rule{0.22em}{0ex}}as\phantom{\rule{0.22em}{0ex}}the\phantom{\rule{0.22em}{0ex}}interatomic\phantom{\rule{0.22em}{0ex}}force\phantom{\rule{0.22em}{0ex}}constant\\ Force\phantom{\rule{0.22em}{0ex}}Constant\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}K\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}\frac{F}{∆R}\\ K\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}Y⨯{r}_{0}\phantom{\rule{0.66em}{0ex}}where\phantom{\rule{0.22em}{0ex}}{r}_{0}\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.44em}{0ex}}Normal\phantom{\rule{0.22em}{0ex}}distance\phantom{\rule{0.22em}{0ex}}between\phantom{\rule{0.22em}{0ex}}the\phantom{\rule{0.22em}{0ex}}atoms\phantom{\rule{0.22em}{0ex}}of\phantom{\rule{0.22em}{0ex}}wire\end{array}$ BrittleA material that has a tendency to break easily or suddenly without any extension first. It has small range of plastic extension.
 Cohesive Force: Force of attarction between two molecules of same substance. Adhesive Force: Force of attarction between two molecules of different substance Surface Tension: Surface tension is the attractive force in liquids that pulls surface moleculesinto the rest of the liquid, minimizing the surface area.The surface tension T = force along a line of unit length. = surfece energy per unit area. Surface Tension and Bubbles :Pressure is more on concave side For bubble with two surfaces,Pressure difference = 𝛥p=4TR$Pressure\phantom{\rule{0.22em}{0ex}}difference\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}𝛥p=\frac{4T}{R}$ Pressure Difference for droplet which has one surface, 𝛥p=2TR$𝛥p=\frac{2T}{R}$ Total Surface energy in bubble with two surfaces = 8T𝜋R2$\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}8T𝜋{R}^{2}$ Total Surface energy in drop ( one surfaces )= 4T𝜋R2$=\phantom{\rule{0.22em}{0ex}}4T𝜋{R}^{2}$ Bernoulli's equationp+12𝜌v2+𝜌gh=constant$p+\frac{1}{2}𝜌{v}^{2}+𝜌gh=constant$ Pressue Head=𝜌gh$Pressue\phantom{\rule{0.22em}{0ex}}Head=\frac{𝜌}{gh}$ Gravitationa; Head=h$Gravitationa;\phantom{\rule{0.22em}{0ex}}Head=h$ Velocity Head=v22g$Velocity\phantom{\rule{0.22em}{0ex}}Head=\frac{{v}^{2}}{2g}$ Capillary ActionHeight to which capillary action lift liquid h = 2Tcos𝜃𝜌 r g$\phantom{\rule{0.44em}{0ex}}h\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}\frac{2Tcos𝜃}{𝜌\phantom{\rule{0.22em}{0ex}}r\phantom{\rule{0.22em}{0ex}}g}$a if 𝜃 is close to 0°, then cos𝜃 = 1 , then, h = 2 T𝜌 r g$\begin{array}{l}\\ if\phantom{\rule{0.22em}{0ex}}𝜃\phantom{\rule{0.22em}{0ex}}is\phantom{\rule{0.22em}{0ex}}close\phantom{\rule{0.22em}{0ex}}to\phantom{\rule{0.22em}{0ex}}0°,\phantom{\rule{0.22em}{0ex}}then\phantom{\rule{0.22em}{0ex}}cos𝜃\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}1\phantom{\rule{0.22em}{0ex}},\phantom{\rule{0.22em}{0ex}}then,\phantom{\rule{0.44em}{0ex}}h\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}\frac{2\phantom{\rule{0.22em}{0ex}}T}{𝜌\phantom{\rule{0.22em}{0ex}}r\phantom{\rule{0.22em}{0ex}}g}\end{array}$ Viscosity : Viscosity is the resistance of a fluid to flow due to internal friction. When liquid flows over flat surface, a backward viscous force acts tangentially to every layer.This force depends upon the area of the layer, velocity of the layer, and the distance of the layer from the surface.aF∝AdvdxF=𝜂Advdx 𝜂=StressStrain Rate$\begin{array}{l}F\propto A\frac{dv}{dx}\\ F=𝜂A\frac{dv}{dx}\phantom{\rule{1.1em}{0ex}}\\ \\ 𝜂=\frac{Stress}{Strain\phantom{\rule{0.22em}{0ex}}Rate}\end{array}$ where 𝜂$𝜂$ = coefficient of viscosity Laminar Flow Confined to Tubes - Poiseuille’s Law:It is relationship between horizontal flow and pressure.Q=p2- p1R$Q=\frac{{p}_{2}-\phantom{\rule{0.22em}{0ex}}{p}_{1}}{R}$ where Q=Flow Ratep1${p}_{1}$ and p2${p}_{2}$ are pressure difference between two points.R is the resistance to flow ( everything except pressure) The resistance R ( or X) to laminar flow of an incompressible fluid having viscosity 𝜂$𝜂$ through a horizontal tube of uniform radius r and length l , is given byR=8 𝜂 l𝜋 r4$R=\frac{8\phantom{\rule{0.22em}{0ex}}𝜂\phantom{\rule{0.22em}{0ex}}l}{𝜋\phantom{\rule{0.22em}{0ex}}{r}^{4}}$ where r = radius of the tubel = length of the tube Flow of a liquid thru cylindrical pipe: v=(p2- p1)4𝜂L (R2-r2)$v=\frac{\left({p}_{2}-\phantom{\rule{0.22em}{0ex}}{p}_{1}{\right)}^{}}{4𝜂L}\phantom{\rule{0.22em}{0ex}}\left({R}^{2}-{r}^{2}\right)$where L = length of pipeR = radius of tuber = distance r from center where velocity needs to be determined Poiseuille found that if a pressure difference ( P) is maintained across the two ends of a capillary tube of length 'l' and radius r, then the volume of liquid coming out of the tube per second is( i) Directly proportional to the pressure difference ( P) .( ii) Directly proportional to the fourth power of radius ( r) of the capillary tube( iii) Inversely proportional to the coefficient of viscosity ( 𝜂$𝜂$ ) of the liquid.( iv) Inversely proportional to the length ( l) of the capillary tube. According to Stokes’ law, the backward viscous force acting on a small spherical body of radius r moving with uniform velocity v through fluid of viscosity η is given byF=6 𝜋 𝜂 r v$F=6\phantom{\rule{0.22em}{0ex}}𝜋\phantom{\rule{0.22em}{0ex}}𝜂\phantom{\rule{0.22em}{0ex}}r\phantom{\rule{0.22em}{0ex}}v$ where r=radius of sphere𝜂$𝜂$ = cooeficient of viscosityF =Retarding forcev=velocity of sphere When viscous force plus buoyant force becomes equal to force due to gravity, the net force becomes zero. The sphere then descends with a constant terminal velocity (vt${v}_{t}$) .6 𝜋 𝜂 r vt=43𝜋r3(𝜌-𝜎) g$6\phantom{\rule{0.22em}{0ex}}𝜋\phantom{\rule{0.22em}{0ex}}𝜂\phantom{\rule{0.22em}{0ex}}r\phantom{\rule{0.22em}{0ex}}{v}_{t}=\frac{4}{3}𝜋{r}^{3}\left(𝜌-𝜎\right)\phantom{\rule{0.22em}{0ex}}g$ awhere 𝜌= density of sphere 𝜎=density of liquid$\begin{array}{l}where\phantom{\rule{0.22em}{0ex}}𝜌=\phantom{\rule{0.22em}{0ex}}density\phantom{\rule{0.22em}{0ex}}of\phantom{\rule{0.22em}{0ex}}sphere\\ \phantom{\rule{2.42em}{0ex}}𝜎=density\phantom{\rule{0.22em}{0ex}}of\phantom{\rule{0.22em}{0ex}}liquid\end{array}$ Laminar flow occurs when the fluid flows in infinitesimal parallel layers with no disruption between them. Steady Flow: A flow that is not a function of time is called steady flow. Steady state flow refers to the condition where the fluid properties at a point in the system do not change over time. Critical velocity: Velocity of liquiid flow below which its flow is streamline and above which its flow becomes turbulent is called Critical Velocity. The Venturimeter is a device to measure theflow speed of incompressible fluid. REYNOLDS NUMBERRe=𝜌vd𝜂 ${R}_{e}=\frac{𝜌vd}{𝜂}\phantom{\rule{1.54em}{0ex}}$where a d=internal diameter of pipev=speed of fluid𝜌=density of fluid$\begin{array}{l}\phantom{\rule{0.44em}{0ex}}d=internal\phantom{\rule{0.22em}{0ex}}diameter\phantom{\rule{0.22em}{0ex}}of\phantom{\rule{0.22em}{0ex}}pipe\\ v=speed\phantom{\rule{0.22em}{0ex}}of\phantom{\rule{0.22em}{0ex}}fluid\\ 𝜌=density\phantom{\rule{0.22em}{0ex}}of\phantom{\rule{0.22em}{0ex}}fluid\end{array}$
 Heat Transfer Heat transfe rate (heat current) thru cross sectiondQdt=-kAdTdx$\frac{dQ}{dt}=-kA\frac{dT}{dx}$where K=thermal conductivity of material Amount of heat flowing per second Heat Current =i=𝛥Q𝛥t=KAT1-T2x $=i=\frac{𝛥Q}{𝛥t}=KA\frac{{T}_{1}-{T}_{2}}{x}\phantom{\rule{0.22em}{0ex}}$ Thermal Resistance=RR=xKA$R=\frac{x}{KA}$ Heat Current =i=𝛥Q𝛥t=T1-T2R $=i=\frac{𝛥Q}{𝛥t}=\frac{{T}_{1}-{T}_{2}}{R}\phantom{\rule{0.22em}{0ex}}$ where R is thermal resistance Thermal Resistance in seriesR=R1+R2+R3+ . . . $R={R}_{1}+{R}_{2}+{R}_{3}+\phantom{\rule{0.22em}{0ex}}.\phantom{\rule{0.22em}{0ex}}.\phantom{\rule{0.22em}{0ex}}.\phantom{\rule{0.22em}{0ex}}$ Thermal Resistance in parallel1R=1R1+1R2+1R3+ . . . $\frac{1}{R}=\frac{1}{{R}_{1}}+\frac{1}{{R}_{2}}+\frac{1}{{R}_{3}}+\phantom{\rule{0.22em}{0ex}}.\phantom{\rule{0.22em}{0ex}}.\phantom{\rule{0.22em}{0ex}}.\phantom{\rule{0.22em}{0ex}}$ Emmisive Power: It is energy emmited per unit area, per unit time, per unit soldit angle along normal to area.E=𝛥U(𝛥A)(𝛥𝜔)(𝛥t)$E=\frac{𝛥U}{\left(𝛥A\right)\left(𝛥𝜔\right)\left(𝛥t\right)}$ Absorptive Power: It is defined as fraction of the incident radiation that is absorbed by the body.Absorptive Power=𝛼=energy absorbedenergy incident $Absorptive\phantom{\rule{0.22em}{0ex}}Power=𝛼=\frac{energy\phantom{\rule{0.22em}{0ex}}absorbed}{energy\phantom{\rule{0.22em}{0ex}}incident}\phantom{\rule{0.22em}{0ex}}$ Kirchhoff's Law: The ratio of emmisive power to absorptive power is the same for all bodies at a given temperature and is equal to the emmisive power of a black body at that temperature.E(body)𝛼(body)=E (blackbody)$\frac{E\left(body\right)}{𝛼\left(body\right)}=E\phantom{\rule{0.22em}{0ex}}\left(blackbody\right)$ Thermal RadiationAs the temperature is increased, the wavelength corresponding to the highest intensity decreases. Wien's displacement law𝜆maxT=b${𝜆}_{max}T=b$for black body, Wien constant= b=0.288cmK𝜆max${𝜆}_{max}$= wavelength at which Intensity is maximum. Stepan Bolzmann LawThe energy of the thermal radiation emitted per unit time by a black body Thermal Expansion Lt=L0(1+𝛼𝛥T)${L}_{t}={L}_{0}\left(1+𝛼𝛥T\right)$At=A0(1+𝛽𝛥T)${A}_{t}={A}_{0}\left(1+𝛽𝛥T\right)$Vt=V0(1+𝛾𝛥T)${V}_{t}={V}_{0}\left(1+𝛾𝛥T\right)$ where𝛼$𝛼$ =linear coefficient of thermal expansion𝛽$𝛽$ =superficial coefficient of thermal expansion𝛾$𝛾$=volumetric coefficient of thermal expansion a𝛽=2𝛼𝛾=3𝛼𝛼:𝛽:𝛾 = 1:2:3$\begin{array}{l}𝛽=2𝛼\\ 𝛾=3𝛼\\ 𝛼:𝛽:𝛾\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}1:2:3\end{array}$ First Law of thermodynamics𝛥Q=𝛥U+𝛥W$𝛥Q=𝛥U+𝛥W$where 𝛥Q$𝛥Q$ = heat supplied to the system.𝛥W$𝛥W$ = work done by the system.𝛥U$𝛥U$ = change in internal energy in the system. Work done by the system against constant P is𝛥W=P𝛥V$𝛥W=P𝛥V$ Heat Capacity S=𝛥Q𝛥T$Heat\phantom{\rule{0.22em}{0ex}}Capacity\phantom{\rule{0.22em}{0ex}}S=\frac{𝛥Q}{𝛥T}$ Specific Heat Capacity (s) s=Sm =1 𝛥Qm 𝛥T$s=\frac{S}{m}\phantom{\rule{0.22em}{0ex}}=\frac{1\phantom{\rule{0.22em}{0ex}}𝛥Q}{m\phantom{\rule{0.22em}{0ex}}𝛥T}$ Work done by system in isothermal process 𝛥W=𝜇RT lnV2V1$𝛥W=𝜇RT\phantom{\rule{0.22em}{0ex}}ln\frac{{V}_{2}}{{V}_{1}}$ Work done by system in adiabetic process𝛥W=𝜇R(T1-T2)𝛾-1$𝛥W=\frac{𝜇R\left({T}_{1}-{T}_{2}\right)}{𝛾-1}$ 𝛥Wadiabetic=(P2V2-P1V1)1-𝛾=𝜇R(T1-T2)𝛾-1$𝛥{W}_{adiabetic}=\frac{\left({P}_{2}{V}_{2}-{P}_{1}{V}_{1}\right)}{1-𝛾}=\frac{𝜇R\left({T}_{1}-{T}_{2}\right)}{𝛾-1}$ Work done in isochoric process𝛥W=0$𝛥W=0$( as volume is constant) Critical TemperatureCritical temperature ( Tc${T}_{c}$ ) of a substance is the temperature at and above which vapor of the substance cannot be liquified, no matter how much pressure is applied. Critical Pressure:The pressure required to liquify a gas at its critical temperature is called Critical Pressure (Pc${P}_{c}$ ) . Critical Volume:The volume of he gas at critical temperature and critical pressure is called Critical Volume ( Vc${V}_{c}$ ) aVc=3bPc=a27b2Tc=8a27Rb$\begin{array}{l}{V}_{c}=3b\\ {P}_{c}=\frac{a}{27{b}^{2}}\\ {T}_{c}=\frac{8a}{27Rb}\end{array}$ RTcPcVc=83$\frac{R{T}_{c}}{{P}_{c}{V}_{c}}=\frac{8}{3}$This is called critical coefficient. It is same for all gases.
Electrostatics
 Force between two charges = Fe=14𝜋𝜖0q1q2r2 $Force\phantom{\rule{0.22em}{0ex}}between\phantom{\rule{0.22em}{0ex}}two\phantom{\rule{0.22em}{0ex}}charges\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}{F}_{e}=\frac{1}{4𝜋{𝜖}_{0}}\frac{{q}_{1}{q}_{2}}{{r}^{2}}\phantom{\rule{0.44em}{0ex}}$ Polarization : dipole moment per unit volume P=𝜒e E$P={𝜒}_{e}\phantom{\rule{0.22em}{0ex}}E$ where 𝜒e=$\phantom{\rule{0.22em}{0ex}}where\phantom{\rule{0.22em}{0ex}}{𝜒}_{e}=$Electric susceptibility of dielectric material. Polarisation is also defined as amount of induced surface charge per unit area So, P=QinducedA$So,\phantom{\rule{0.22em}{0ex}}P=\frac{{Q}_{induced}}{A}$ aCapacitance=C=k𝜖0Ad where k=dielectric constant in air, dielectric constant=k=1Cair=𝜖0Ad Cd=k Cair$\begin{array}{l}Capacitance=C=\frac{k{𝜖}_{0}A}{d}\phantom{\rule{0.88em}{0ex}}where\phantom{\rule{0.22em}{0ex}}k=dielectric\phantom{\rule{0.22em}{0ex}}constant\phantom{\rule{0.22em}{0ex}}\\ in\phantom{\rule{0.22em}{0ex}}air,\phantom{\rule{0.22em}{0ex}}dielectric\phantom{\rule{0.22em}{0ex}}constant=k=1\\ {C}_{air}=\frac{{𝜖}_{0}A}{d}\\ \\ {C}_{d}=k\phantom{\rule{0.22em}{0ex}}{C}_{air}\end{array}$ Capacitance=C=QV $Capacitance=C=\frac{Q}{V}\phantom{\rule{0.44em}{0ex}}$ Energy stored in capacitor=E=12CV2=12QV=Q22C$=E=\frac{1}{2}C{V}^{2}=\frac{1}{2}QV=\frac{{Q}^{2}}{2C}$ Electric field intensity at a point near and outside the surface of a charged conductor of any shape:E=𝜎𝜖0k where 𝜎=surface charge density $E=\frac{𝜎}{{𝜖}_{0}k}\phantom{\rule{1.76em}{0ex}}where\phantom{\rule{0.44em}{0ex}}𝜎=surface\phantom{\rule{0.22em}{0ex}}charge\phantom{\rule{0.22em}{0ex}}density\phantom{\rule{0.22em}{0ex}}$ Electric field due to uniformly charged infinite plane sheet:E=𝜎2𝜖0 where 𝜎=surface charge density $E=\frac{𝜎}{2{𝜖}_{0}}\phantom{\rule{1.76em}{0ex}}where\phantom{\rule{0.44em}{0ex}}𝜎=surface\phantom{\rule{0.22em}{0ex}}charge\phantom{\rule{0.22em}{0ex}}density\phantom{\rule{0.22em}{0ex}}$ Mechanical force acting on unit area of a charged conductor =f=𝜎22 𝜖0 $=f=\frac{{𝜎}^{2}}{2\phantom{\rule{0.22em}{0ex}}{𝜖}_{0}}\phantom{\rule{0.22em}{0ex}}$ The electrostatic energy per unit volume is called energy density=12 𝜖0 E2 $=\frac{1}{2}\phantom{\rule{0.22em}{0ex}}{𝜖}_{0}\phantom{\rule{0.22em}{0ex}}{E}^{2}\phantom{\rule{0.44em}{0ex}}$
 𝛼2${𝛼}_{2}$𝛼1${𝛼}_{1}$Ex${E}_{x}$Ey${E}_{y}$Electric field due to uniformly charged rod𝜆$𝜆$=charge densityABEx=𝜆4𝜋𝜖0r(sin𝛼1+sin𝛼2)${E}_{x}=\frac{𝜆}{4𝜋{𝜖}_{0}r}\left(sin{𝛼}_{1}+sin{𝛼}_{2}\right)$Ey=𝜆4𝜋𝜖0r(cos𝛼1-cos𝛼2)${E}_{y}=\frac{𝜆}{4𝜋{𝜖}_{0}r}\left(cos{𝛼}_{1}-cos{𝛼}_{2}\right)$
 Electric field due to electric dipole making angle 𝜃$𝜃$ with dipole E=p4𝜋𝜖0r33cos2𝜃+1$E=\frac{p}{4𝜋{𝜖}_{0}{r}^{3}}\sqrt{3co{s}^{2}𝜃+1}$
 𝛼2${𝛼}_{2}$𝛼1${𝛼}_{1}$Electric Potential due to uniformly charged rod𝜆$𝜆$=charge densityABVp=𝜆4𝜋𝜖0ln{(sec𝛼1+tan𝛼1)(sec𝛼2-tan𝛼2)}${V}_{p}=\frac{𝜆}{4𝜋{𝜖}_{0}}ln\left\{\frac{\left(sec{𝛼}_{1}+tan{𝛼}_{1}\right)}{\left(sec{𝛼}_{2}-tan{𝛼}_{2}\right)}\right\}$P
 Current Electricity CurrentI=QT$I=\frac{Q}{T}$ V = I R ResistanceR=𝜌lA$R=𝜌\frac{l}{A}$where 𝜌=resistivity$𝜌=resistivity$ Current Densityj=IA$j=\frac{I}{A}$ Conductivity=𝜎=1𝜌$Conductivity=𝜎=\frac{1}{𝜌}$ I=n e A Vd$I=n\phantom{\rule{0.22em}{0ex}}e\phantom{\rule{0.22em}{0ex}}A\phantom{\rule{0.22em}{0ex}}{V}_{d}$I = currentn = number of electrons per unit volumee = charge of electronA = Cross sectional AreaVd${V}_{d}$ = drift velocity Mobility is drift velocity per unit volume𝜇=|vd|E$𝜇=\frac{|{v}_{d}|}{E}$ Temperature dependance of resistivity𝜌T=𝜌0 [1+𝛼(T-T0) ]${𝜌}_{T}={𝜌}_{0}\phantom{\rule{0.22em}{0ex}}\left[1+𝛼\left(T-{T}_{0}\right)\phantom{\rule{0.22em}{0ex}}\right]$
 Magnetism Magnetic force on current carrying conductor (F)F=(nAl) q Vd⨯B $F=\left(nAl\right)\phantom{\rule{0.22em}{0ex}}q\phantom{\rule{0.22em}{0ex}}{V}_{d}⨯B\phantom{\rule{0.44em}{0ex}}$ where n = number of charge per unit volumeA=cross-sectional area of conductorl = length of conductor\Vd${V}_{d}$ = drift\ velocityB = magnetic field F=I l⨯B $F=I\phantom{\rule{0.22em}{0ex}}\stackrel{\to }{l}⨯\stackrel{\to }{B}\phantom{\rule{0.22em}{0ex}}$wherea I =current flowing thru conductor l = vector magnitude of length of conductor B =magnetic field$\begin{array}{l}\phantom{\rule{0.22em}{0ex}}I\phantom{\rule{0.44em}{0ex}}=current\phantom{\rule{0.22em}{0ex}}flowing\phantom{\rule{0.22em}{0ex}}thru\phantom{\rule{0.22em}{0ex}}conductor\\ \phantom{\rule{0.22em}{0ex}}l\phantom{\rule{0.44em}{0ex}}=\phantom{\rule{0.22em}{0ex}}vector\phantom{\rule{0.22em}{0ex}}magnitude\phantom{\rule{0.22em}{0ex}}of\phantom{\rule{0.22em}{0ex}}length\phantom{\rule{0.22em}{0ex}}of\phantom{\rule{0.22em}{0ex}}conductor\phantom{\rule{0.22em}{0ex}}\\ \phantom{\rule{0.22em}{0ex}}\stackrel{\to }{B\phantom{\rule{0.22em}{0ex}}}=magnetic\phantom{\rule{0.22em}{0ex}}field\end{array}$ Magnetic force on a moving charge (F)F=q v ⨯ B $F=q\phantom{\rule{0.22em}{0ex}}\stackrel{\to }{v\phantom{\rule{0.22em}{0ex}}}\phantom{\rule{0.22em}{0ex}}⨯\phantom{\rule{0.22em}{0ex}}\stackrel{\to }{B\phantom{\rule{0.22em}{0ex}}}$ Magnetic field due to current element (Biot-Savarat Law)𝛥B=𝜇04𝜋 I 𝛥l sin𝜃r2$𝛥B=\frac{{𝜇}_{0}}{4𝜋}\phantom{\rule{0.22em}{0ex}}\frac{I\phantom{\rule{0.22em}{0ex}}𝛥l\phantom{\rule{0.22em}{0ex}}sin𝜃}{{r}^{2}}$ Magnetic field surrounding a thin straight current carrying conductor at point PB=𝜇0 4𝜋Id (sin𝛼+sin𝛽)$B=\frac{{𝜇}_{0}\phantom{\rule{0.22em}{0ex}}}{4𝜋}\frac{I}{d}\phantom{\rule{0.22em}{0ex}}\left(sin𝛼+sin𝛽\right)$ where d=perpendicular distance d from wire𝛼 , 𝛽 = $𝛼\phantom{\rule{0.22em}{0ex}},\phantom{\rule{0.22em}{0ex}}𝛽\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}$angle subtended by end of wire ar point P against perpendicular distance Magnetic field surrounding a infinetely long thin straight current carrying conductor at point PB=𝜇0 2𝜋Id $B=\frac{{𝜇}_{0}\phantom{\rule{0.22em}{0ex}}}{2𝜋}\frac{I}{d}\phantom{\rule{0.22em}{0ex}}$ Magnetic field on the axis of a circular current loopB=𝜇0 I R22(x2+R2)3/2$B=\frac{{𝜇}_{0}\phantom{\rule{0.22em}{0ex}}I\phantom{\rule{0.22em}{0ex}}{R}^{2}}{2\left({x}^{2}+{R}^{2}{\right)}^{3/2}}$ Magnetic field at the center of the loop B = 𝜇0 I2 R$\phantom{\rule{0.22em}{0ex}}B\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}\frac{{𝜇}_{0}\phantom{\rule{0.22em}{0ex}}I}{2\phantom{\rule{0.22em}{0ex}}R}$ Magnetic field at the center due to an arc B = 𝜇04𝜋 I R 𝜃 $\phantom{\rule{0.22em}{0ex}}B\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}\frac{{𝜇}_{0}}{4𝜋\phantom{\rule{0.22em}{0ex}}}\frac{I}{\phantom{\rule{0.22em}{0ex}}R}\phantom{\rule{0.22em}{0ex}}𝜃\phantom{\rule{0.22em}{0ex}}$ where 𝜃=angle subtended by arc in radian$𝜃=angle\phantom{\rule{0.22em}{0ex}}subtended\phantom{\rule{0.22em}{0ex}}by\phantom{\rule{0.22em}{0ex}}arc\phantom{\rule{0.22em}{0ex}}in\phantom{\rule{0.22em}{0ex}}radian$R=radius of arc Magnetic field due to long solenoid (infinite length)B=𝜇onI$B={𝜇}_{o}nI$where n=number of turns per unit lengthI=current thru solenoid Magnetic field at one end of solenoid (infinite length)B=12𝜇onI$B=\frac{1}{2}{𝜇}_{o}nI$ where n=number of turns per unit lengthI=current thru solenoid Magnetic field at at point P of solenoid (finite length)B=𝜇onI2 (sin𝛼+sin𝛽)$B=\frac{{𝜇}_{o}nI}{2}\phantom{\rule{0.22em}{0ex}}\left(sin𝛼+sin𝛽\right)$ where n=number of turns per unit lengthI=current thru solenoid𝛼 , 𝛽 = $𝛼\phantom{\rule{0.22em}{0ex}},\phantom{\rule{0.22em}{0ex}}𝛽\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}$angle subtended by end of solenoid ar point P against perpendicular. Magnetic moment of current loopm =N I A $\stackrel{\to }{m}\phantom{\rule{0.22em}{0ex}}=N\phantom{\rule{0.22em}{0ex}}I\phantom{\rule{0.22em}{0ex}}\stackrel{\to }{A}\phantom{\rule{0.22em}{0ex}}$ where,I = current in loopA = area of the loop (vector)N = Number of turns Magnetic field along the axis of the current carrying loop B = 𝜇0 R22 x3= 𝜇0 4𝜋 2mx3$\phantom{\rule{0.22em}{0ex}}B\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}\frac{{𝜇}_{0}\phantom{\rule{0.22em}{0ex}}{R}^{2}}{2\phantom{\rule{0.22em}{0ex}}{x}^{3}}=\phantom{\rule{0.22em}{0ex}}\frac{{𝜇}_{0}\phantom{\rule{0.22em}{0ex}}}{4𝜋\phantom{\rule{0.22em}{0ex}}}\phantom{\rule{0.22em}{0ex}}\frac{2m}{{x}^{3}}$ where m=magnetic moment Magnetic field in the plane of the current carrying loop B = 𝜇0 R22 x3= 𝜇0 4𝜋 mx3$\phantom{\rule{0.22em}{0ex}}B\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}\frac{{𝜇}_{0}\phantom{\rule{0.22em}{0ex}}{R}^{2}}{2\phantom{\rule{0.22em}{0ex}}{x}^{3}}=\phantom{\rule{0.22em}{0ex}}\frac{{𝜇}_{0}\phantom{\rule{0.22em}{0ex}}}{4𝜋\phantom{\rule{0.22em}{0ex}}}\phantom{\rule{0.22em}{0ex}}\frac{m}{{x}^{3}}$ where m=magnetic moment Force between two long parallel conductors per unit lengthF=𝜇0IaIb2𝜋d$F=\frac{{𝜇}_{0}{I}_{a}{I}_{b}}{2𝜋d}$ Torque on Current Loop 𝜏 = I A B $\phantom{\rule{1.32em}{0ex}}𝜏\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}I\phantom{\rule{0.22em}{0ex}}A\phantom{\rule{0.22em}{0ex}}B\phantom{\rule{1.1em}{0ex}}$ 𝜏 = m ×B $\phantom{\rule{1.1em}{0ex}}\stackrel{\to }{𝜏}\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}\stackrel{\to }{m}\phantom{\rule{0.22em}{0ex}}×\stackrel{\to }{B}\phantom{\rule{1.32em}{0ex}}$ where, I=current in loopA=area of the loop ( vector)B=Magnetic fieldm $\stackrel{\to }{m}\phantom{\rule{0.22em}{0ex}}$= Magnetic moment Magnetic Pole Strength: It strength of the magnetic pole to attract magnetic materials towards itself.M=magnetic dipole momentm=pole strength2l=length of bar magnetthenM=(m)2l)$M=\left(m\right)2l\right)$Unit of pole strength=Am If magnet is cut into two equal pieces, such that length of each piece becomes half, then pole strength does not change. If magnet is cut into two equal pieces, such that width of each piece becomes half, then pole strength becomes half. Coulomb's law for magnetismF=𝜇04𝜋m1×m2r2$F=\frac{{𝜇}_{0}}{4𝜋}\frac{{m}_{1}×{m}_{2}}{{r}^{2}}$ Magnetic field strength at any point is defined as force experienced by hypothetical north pole of unit pole strengthB=Fm$B=\frac{F}{m}$ LR Circuit: growth of currenti=i0(1-e-RL t)$i={i}_{0}\left(1-{e}^{-\frac{R}{L}\phantom{\rule{0.22em}{0ex}}t}\right)$i0=ER ${i}_{0}=\frac{E}{R}\phantom{\rule{0.22em}{0ex}}$ at steady stateTime constant=𝜏L=LR$={𝜏}_{L}=\frac{L}{R}$ Time constant refers to time constant when current rises about 63% of final value. LR Circuit: decay of currenti=i0 e-RL t$i={i}_{0}\phantom{\rule{0.22em}{0ex}}{e}^{-\frac{R}{L}\phantom{\rule{0.22em}{0ex}}t}$i0=ER ${i}_{0}=\frac{E}{R}\phantom{\rule{0.22em}{0ex}}$ at steady state Cyclotron frequency of revolution of particle=𝜈c$={𝜈}_{c}$𝜈c=q B2 𝜋 m${𝜈}_{c}=\frac{q\phantom{\rule{0.22em}{0ex}}B}{2\phantom{\rule{0.22em}{0ex}}𝜋\phantom{\rule{0.22em}{0ex}}m}$ 𝜖0𝜇0=1c2${𝜖}_{0}{𝜇}_{0}=\frac{1}{{c}^{2}}$ Coefficient of coupling = K : It measures the manner in which 2 coils are coupledK=ML1L2$K=\frac{M}{\sqrt{{L}_{1}{L}_{2}}}$M=mutual inductanceK=0, if there is no couplingK=1, for maximum coupling0