Physics Equations and Concepts

Physical Constants

• Speed of light: $c = 3 × 10^8 m/s$
• Planck constant: $h = 6.63 × 10^{-34} J s$
• Gravitation constant: $G = 6.67×10^{-11} m^3 kg^{-1} s ^{-2}$
• Boltzmann constant: $k = 1.38 × 10^{-23} J/K$
• Molar gas constant: $R = 8.314 J/(mol K)$
• Avogadro’s number: $NA = 6.023 × 10^{23} mol^{-1}$
• Charge of electron: $e = 1.602 × 10^{-19} C$

Mechanics

Vectors

• Notation: $\vec{a} = a<em>x \hat{i} + a</em>y \hat{j} + a_z \hat{k}$
• Magnitude: $a = |\vec{a}| = \sqrt{a<em>x^2 + a</em>y^2 + a_z^2}$
• Dot product: $\vec{a} \cdot \vec{b} = a<em>xb</em>x + a<em>yb</em>y + a<em>zb</em>z = ab \cos \theta$
• Cross product: $|\vec{a} × \vec{b}| = ab \sin \theta$

Kinematics

• Average and Instantaneous Velocity and Acceleration:
  • $\vec{v}<em>{av} = \frac{\Delta \vec{r}}{\Delta t}$, $\vec{v}</em>{inst} = \frac{d\vec{r}}{dt}$
  • $\vec{a}<em>{av} = \frac{\Delta \vec{v}}{\Delta t}$, $\vec{a}</em>{inst} = \frac{d\vec{v}}{dt}$
• Motion in a straight line with constant $a$:
  • $v = u + at$, $s = ut + \frac{1}{2}at^2$, $v^2 - u^2 = 2as$
• Relative Velocity: $\vec{v}<em>{A/B} = \vec{v}</em>A - \vec{v}_B$
• Projectile Motion:
  • $x = ut \cos \theta$, $y = ut \sin \theta - \frac{1}{2}gt^2$
  • $T = \frac{2u \sin \theta}{g}$, $R = \frac{u^2 \sin 2\theta}{g}$, $H = \frac{u^2 \sin^2 \theta}{2g}$

Newton’s Laws and Friction

• Linear momentum: $\vec{p} = m\vec{v}$
• Newton’s second law: $\vec{F} = \frac{d\vec{p}}{dt}$, $\vec{F} = m\vec{a}$
• Newton’s third law: $\vec{F}<em>{AB} = -\vec{F}</em>{BA}$
• Frictional force: $f<em>{static, max} = \mu</em>sN$, $f<em>{kinetic} = \mu</em>kN$
• Centripetal force: $F<em>c = \frac{mv^2}{r}$, $a</em>c = \frac{v^2}{r}$

Work, Power and Energy

• Work: $W = \vec{F} \cdot \vec{S} = FS \cos \theta$, $W = \int \vec{F} \cdot d\vec{S}$
• Kinetic energy: $K = \frac{1}{2}mv^2 = \frac{p^2}{2m}$
• Potential energy: $F = -\frac{\partial U}{\partial x}$
  • $U<em>{gravitational} = mgh$, $U</em>{spring} = \frac{1}{2}kx^2$
• Work-energy theorem: $W = \Delta K$
• Power: $P<em>{av} = \frac{\Delta W}{\Delta t}$, $P</em>{inst} = \vec{F} \cdot \vec{v}$

Centre of Mass and Collision

• Centre of mass: $x<em>{cm} = \frac{\sum x</em>im<em>i}{\sum m</em>i}$, $x_{cm} = \frac{\int x dm}{\int dm}$
• Impulse: $\vec{J} = \int \vec{F} dt = \Delta \vec{p}$
• Momentum conservation: $m<em>1v</em>1 + m<em>2v</em>2 = m<em>1v'</em>1 + m<em>2v'</em>2$
• Coefficient of restitution: $e = -\frac{(v'<em>1 - v'</em>2)}{v<em>1 - v</em>2}$
  • $e = 1$ (completely elastic), $e = 0$ (completely in-elastic)

Rigid Body Dynamics

• Angular velocity: $\omega_{av} = \frac{\Delta \theta}{\Delta t}$, $\omega = \frac{d\theta}{dt}$, $\vec{v} = \vec{\omega} × \vec{r}$
• Angular Acceleration: $\alpha_{av} = \frac{\Delta \omega}{\Delta t}$, $\alpha = \frac{d\omega}{dt}$, $\vec{a} = \vec{\alpha} × \vec{r}$
• Moment of Inertia: $I = \sum<em>i m</em>ir_i^2$, $I = \int r^2dm$
• Theorem of Parallel Axes: $I = I_{cm} + md^2$
• Theorem of Perpendicular Axes: $I<em>z = I</em>x + I_y$
• Angular Momentum: $\vec{L} = \vec{r} × \vec{p}$, $\vec{L} = I\vec{\omega}$
• Torque: $\vec{\tau} = \vec{r} × \vec{F}$, $\vec{\tau} = \frac{d\vec{L}}{dt}$, $\tau = I\alpha$
• Kinetic Energy: $K_{rot} = \frac{1}{2} I \omega^2$

Gravitation

• Gravitational force: $F = G \frac{m<em>1m</em>2}{r^2}$
• Potential energy: $U = - G \frac{Mm}{r}$
• Gravitational acceleration: $g = \frac{GM}{R^2}$
• Escape velocity: $v_e = \sqrt{\frac{2GM}{R}}$
• Kepler’s laws: $T^2 \propto a^3$

Simple Harmonic Motion

• Hooke’s law: $F = -kx$
• Acceleration: $a = \frac{d^2x}{dt^2} = - \frac{k}{m} x = -\omega^2x$
• Time period: $T = \frac{2\pi}{\omega} = 2\pi \sqrt{\frac{m}{k}}$
• Displacement: $x = A \sin(\omega t + \phi)$
• Velocity: $v = A\omega \cos(\omega t + \phi) = \pm \omega \sqrt{A^2 - x^2}$
• Potential energy: $U = \frac{1}{2} kx^2$
• Kinetic energy: $K = \frac{1}{2}mv^2$
• Total energy: $E = U + K = \frac{1}{2}m\omega^2A^2$
• Simple pendulum: $T = 2\pi\sqrt{\frac{l}{g}}$
• Springs in series: $\frac{1}{k<em>{eq}} = \frac{1}{k</em>1} + \frac{1}{k_2}$
• Springs in parallel: $k<em>{eq} = k</em>1 + k_2$

Properties of Matter

• Young’s modulus: $Y = \frac{F/A}{\Delta l/l}$
• Bulk modulus: $B = -V \frac{\Delta P}{\Delta V}$
• Shear modulus: $\eta = \frac{F}{A\theta}$
• Compressibility: $K = \frac{1}{B} = - \frac{1}{V} \frac{dV}{dP}$
• Poisson’s ratio: $\sigma = \frac{\text{lateral strain}}{\text{longitudinal strain}} = \frac{\Delta D/D}{\Delta l/l}$
• Surface tension: $S = \frac{F}{l}$
• Excess pressure in bubble: $\Delta p<em>{air} = \frac{2S}{R}$, $\Delta p</em>{soap} = \frac{4S}{R}$
• Capillary rise: $h = \frac{2S \cos \theta}{r\rho g}$
• Hydrostatic pressure: $p = \rho gh$
• Buoyant force: $F_B = \rho V g$
• Equation of continuity: $A<em>1v</em>1 = A<em>2v</em>2$
• Bernoulli’s equation: $p + \frac{1}{2} \rho v^2 + \rho gh = \text{constant}$
• Viscous force: $F = -\eta A \frac{dv}{dx}$
• Stoke’s law: $F = 6\pi \eta rv$
• Terminal velocity: $v_t = \frac{2r^2(\rho - \sigma)g}{9\eta}$

Waves

Wave Motion

• General equation of a wave: $\frac{\partial^2y}{\partial x^2} = \frac{1}{v^2} \frac{\partial^2y}{\partial t^2}$
• Progressive wave travelling with speed $v$: $y=f(t - x/v)$, $y=f(t + x/v)$
• Progressive sine wave: $y = A \sin(kx - \omega t) = A \sin(2\pi (x/\lambda - t/T))$

Waves on a String

• Speed of waves on a string: $v = \sqrt{\frac{T}{\mu}}$
• Transmitted power: $P_{av} = 2\pi^2\mu vA^2\nu^2$
• Interference: $y = y<em>1 + y</em>2$
• Standing Waves: $y = (2A \cos kx) \sin \omega t$
• String fixed at both ends: $L = n \frac{\lambda}{2}$, $\nu = n \frac{1}{2L} \sqrt{\frac{T}{\mu}}$, $n = 1, 2, 3, …$
• String fixed at one end: $L = (2n + 1) \frac{\lambda}{4}$, $\nu = \frac{2n+1}{4L} \sqrt{\frac{T}{\mu}}$, $n = 0, 1, 2, …$

Sound Waves

• Displacement wave: $s = s_0 \sin \omega(t - x/v)$
• Pressure wave: $p = p<em>0 \cos \omega(t - x/v)$, $p</em>0 = (B\omega/v)s_0$
• Speed of sound waves:
  • $v_{liquid} = \sqrt{\frac{B}{\rho}}$,
  • $v_{solid} = \sqrt{\frac{Y}{\rho}}$,
  • $v_{gas} = \sqrt{\frac{\gamma P}{\rho}}$
• Intensity: $I = \frac{p_0^2}{2\rho v}$
• Closed organ pipe: $L = (2n + 1) \frac{\lambda}{4}$, $\nu = (2n + 1) \frac{v}{4L}$
• Open organ pipe: $L = n \frac{\lambda}{2}$, $\nu = n \frac{v}{2L}$
• Beats: $\omega = (\omega<em>1 + \omega</em>2)/2$, $\Delta \omega = \omega<em>1 - \omega</em>2$
• Doppler Effect: $\nu = \frac{v + u<em>o}{v - u</em>s} \nu_0$

Light Waves

• Plane Wave: $E = E<em>0 \sin \omega(t - x v )$, $I = I</em>0$
• Spherical Wave: $E = \frac{aE<em>0}{r} \sin \omega(t - \frac{r}{v})$, $I = \frac{I</em>0}{r^2}$
• Path difference: $\Delta x = \frac{dy}{D}$
• Phase difference: $\delta = \frac{2\pi}{\lambda} \Delta x$
• Interference Conditions:
  • $\delta = 2n\pi$, constructive
  • $\delta = (2n + 1)\pi$, destructive

Optics

Reflection of Light

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

Refraction of Light

• Refractive index: $\mu = \frac{\text{speed of light in vacuum}}{\text{speed of light in medium}} = \frac{c}{v}$
• Snell’s Law: $\frac{\sin i}{\sin r} = \frac{\mu<em>2}{\mu</em>1}$
• Critical angle: $\theta_c = \sin^{-1} \frac{1}{\mu}$
• Deviation by a prism: $\delta = i + i' - A$
• Refraction at spherical surface: $\frac{\mu<em>2}{v} - \frac{\mu</em>1}{u} = \frac{\mu<em>2 - \mu</em>1}{R}$
• Lens maker’s formula: $\frac{1}{f} = (\mu - 1) (\frac{1}{R<em>1} - \frac{1}{R</em>2})$
• Lens formula: $\frac{1}{v} - \frac{1}{u} = \frac{1}{f}$
• Power of the lens: $P = \frac{1}{f}$

Optical Instruments

• Simple microscope: $m = D/f$
• Compound microscope: $m = \frac{v}{u} \frac{D}{f_e}$
• Astronomical telescope: $m = - \frac{f<em>o}{f</em>e}$, $L = f<em>o + f</em>e$

Dispersion

• Cauchy’s equation: $\mu = \mu_0 + \frac{A}{\lambda^2}$
• Mean deviation: $\delta<em>y = (\mu</em>y - 1)A$
• Angular dispersion: $\theta = (\mu<em>v - \mu</em>r)A$
• Dispersive power: $\omega = \frac{\mu<em>v - \mu</em>r}{\mu<em>y - 1} \approx \frac{\theta}{\delta</em>y}$

Heat and Thermodynamics

Heat and Temperature

• Temperature scales: $F = 32 + \frac{9}{5}C$, $K = C + 273.16$
• Ideal gas equation: $pV = nRT$
• Thermal expansion: $L = L<em>0(1 + \alpha\Delta T)$, $A = A</em>0(1 + \beta\Delta T)$, $V = V_0(1 + \gamma\Delta T)$

Kinetic Theory of Gases

• RMS speed: $v_{rms} = \sqrt{\frac{3kT}{m}} = \sqrt{\frac{3RT}{M}}$
• Average speed: $v = \sqrt{\frac{8kT}{\pi m}} = \sqrt{\frac{8RT}{\pi M}}$
• Most probable speed: $v_p = \sqrt{\frac{2kT}{m}}$
• Pressure: $p = \frac{1}{3} \rho v^2_{rms}$

Specific Heat

• Specific heat: $s = \frac{Q}{m\Delta T}$
• Latent heat: $L = Q/m$
• Specific heat at constant volume: $C<em>v = \frac{\Delta Q}{n\Delta T}|</em>V$
• Specific heat at constant pressure: $C<em>p = \frac{\Delta Q}{n\Delta T}|</em>p$
• Relation between $C<em>p$ and $C</em>v$: $C<em>p - C</em>v = R$
• Ratio of specific heats: $\gamma = C<em>p/C</em>v$
• Relation between $U$ and $C<em>v$: $\Delta U = nC</em>v\Delta T$

Thermodynamic Processes

• First law of thermodynamics: $\Delta Q = \Delta U + \Delta W$
• Work done by the gas: $\Delta W = p\Delta V$, $W = \int<em>{V</em>1}^{V_2} pdV$
• Efficiency of the heat engine: $\eta = \frac{\text{work done by the engine}}{\text{heat supplied to it}} = \frac{Q<em>1 - Q</em>2}{Q_1}$
• Coefficient of performance of refrigerator: $COP = \frac{Q<em>2}{W} = \frac{Q</em>2}{Q<em>1-Q</em>2}$
• Entropy: $\Delta S = \frac{\Delta Q}{T}$

Heat Transfer

• Conduction: $\frac{\Delta Q}{\Delta t} = -KA \frac{\Delta T}{x}$
• Stefan-Boltzmann law: $\frac{\Delta Q}{\Delta t} = \sigma eAT^4$
• Newton’s law of cooling: $\frac{dT}{dt} = -bA(T - T_0)$

Electricity and Magnetism

Electrostatics

• Coulomb’s law: $\vec{F} = \frac{1}{4\pi \epsilon<em>0} \frac{q</em>1q_2}{r^2} \hat{r}$
• Electric field: $\vec{E}(\vec{r}) = \frac{1}{4\pi \epsilon_0} \frac{q}{r^2} \hat{r}$
• Electrostatic potential: $V = \frac{1}{4\pi \epsilon_0} \frac{q}{r}$
• Electric dipole moment: $\vec{p} = q \vec{d}$
• Potential of a dipole: $V = \frac{1}{4\pi \epsilon_0} \frac{p \cos \theta}{r^2}$
• Torque on a dipole: $\vec{\tau} = \vec{p} × \vec{E}$
• Potential energy of a dipole: $U = -\vec{p} \cdot \vec{E}$

Gauss’s Law and its Applications

• Electric flux: $\phi = \oint \vec{E} \cdot d\vec{S}$
• Gauss’s law: $\oint \vec{E} \cdot d\vec{S} = \frac{q<em>{in}}{\epsilon</em>0}$
• Field of a uniformly charged sphere:
  • $E = \frac{1}{4\pi \epsilon_0} \frac{Qr}{R^3}$, for r < R
  • $E = \frac{1}{4\pi \epsilon_0} \frac{Q}{r^2}$, for $r \geq R$
• Field of a line charge: $E = \frac{\lambda}{2\pi \epsilon_0r}$
• Field of an infinite sheet: $E = \frac{\sigma}{2\epsilon_0}$

Capacitors

• Capacitance: $C = q/V$
• Parallel plate capacitor: $C = \frac{\epsilon_0A}{d}$
• Energy stored in capacitor: $U = \frac{1}{2}CV^2 = \frac{Q^2}{2C}$
• Capacitor with dielectric: $C = \frac{\epsilon_0KA}{d}$

Current electricity

• Current density: $j = i/A = \sigma E$
• Drift speed: $v_d = \frac{1}{2} \frac{eE}{m} \tau$
• Resistance of a wire: $R = \rho l/A$
• Ohm’s law: $V = iR$
• Electric Power: $P = V^2/R = I^2R = IV$

Magnetism

• Lorentz force: $\vec{F} = q(\vec{v} × \vec{B}) + q\vec{E}$
• Force on a current carrying wire: $\vec{F} = i (\vec{l} × \vec{B})$
• Magnetic moment of a current loop: $\mu = iA$
• Torque on a dipole: $\vec{\tau} = \vec{\mu}×\vec{B}$
• Energy of a magnetic dipole: $U = -\vec{\mu} \cdot \vec{B}$

Magnetic Field due to Current

• Biot-Savart law: $d\vec{B} = \frac{\mu_0}{4\pi} \frac{i d\vec{l}×\vec{r}}{r^3}$
• Field due to a straight conductor: $B = \frac{\mu<em>0i}{4\pi d} (\cos \theta</em>1 - \cos \theta_2)$
• Field due to an infinite straight wire: $B = \frac{\mu_0i}{2\pi d}$
• Ampere’s law: $\oint \vec{B} \cdot d\vec{l} = \mu<em>0I</em>{in}$
• Field inside a solenoid: $B = \mu_0ni$
• Field inside a toroid: $B = \frac{\mu_0N i}{2\pi r}$

Electromagnetic Induction

• Magnetic flux: $\phi = \oint \vec{B} \cdot d\vec{S}$
• Faraday’s law: $e = - \frac{d\phi}{dt}$
• Self inductance: $\phi = Li$, $e = -L\frac{di}{dt}$
• Growth of current in LR circuit: $i = \frac{e}{R} (1 - e^{-t \frac{R}{L}})$
• Decay of current in LR circuit: $i = i_0e^{-t \frac{R}{L}}$
• Energy stored in an inductor: $U = \frac{1}{2}Li^2$
• Alternating current: $i = i_0 \sin(\omega t + \phi)$

Modern Physics

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.: $\nu_0 = \frac{\phi}{h}$
• de Broglie wavelength: $\lambda = h/p$

The Atom

• Energy in nth Bohr’s orbit: $E_n = - \frac{13.6Z^2}{n^2} eV$
• Radius of the nth Bohr’s orbit: $r<em>n = \frac{n^2a</em>0}{Z}$
• Photon energy in state transition: $E<em>2 - E</em>1 = h\nu$

The Nucleus

• Nuclear radius: $R = R_0A^{1/3}$
• Decay rate: $\frac{dN}{dt} = -\lambda N$
• Half life: $t_{1/2} = \frac{0.693}{\lambda}$
• Mass defect: $\Delta m = [Zmp + (A - Z)mn] - M$
• Binding energy: $B = [Zmp + (A - Z)mn - M] c^2$