Physics Formulas Table

phy-1109
PHY-1109 Physics

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I. Oscillations

This section covers Simple Harmonic Motion (SHM), superposition, damped/forced systems, acoustics, and mechanical waves.

Concept	Formula	Symbols Defined	Concept Explanation
Simple Harmonic Motion (SHM) Differential Equation	$\frac{{\mathbf{d}}^2\mathbf{y}}{\mathbf{d}{\mathbf{t}}^2}+{\omega }^2\mathbf{y}=0$	$y$: displacement. $\omega$: angular frequency.	This is the second-order homogeneous differential equation representing SHM, where the acceleration is directly proportional and opposite in direction to the displacement (since ${\omega }^2$ is a positive constant $k/m$).
Angular Frequency ($\omega$)	$\omega =\sqrt{\frac{\mathbf{k}}{\mathbf{m}}}$	$k$: spring constant or restoring constant. $m$: mass of the particle.	Relates the angular frequency of the oscillator to the physical properties of the system (stiffness $k$ and inertia $m$).
Displacement in SHM	$\mathbf{y}=\mathbf{a}\sin (\omega \mathbf{t}+\mathbf{\Phi })$	$a$: Amplitude (maximum displacement). $\omega t+\Phi$: total phase. $\Phi$: initial phase/epoch. $t$: time.	Gives the instantaneous displacement $y$ of a particle undergoing SHM at time $t$.
Velocity in SHM	$\mathbf{v}=\omega \sqrt{{\mathbf{a}}^2-{\mathbf{y}}^2}$	$v$: instantaneous velocity. $a$: amplitude. $y$: displacement. $\omega$: angular frequency.	Determines the instantaneous speed based on displacement. Maximum velocity occurs at the equilibrium point ($y=0$), where $V_{max}=\omega a$.
Total Energy in SHM	$T.E.=\frac{1}{2}\mathbf{k}{\mathbf{a}}^2$	$k$: restoring constant. $a$: amplitude.	The total mechanical energy (Kinetic Energy + Potential Energy) of a simple harmonic oscillator is proportional to the square of the amplitude and is always constant.
Potential Energy (PE) in SHM	$\text{P.E.}=\frac{1}{2}\mathbf{k}{\mathbf{y}}^2$	$k$: restoring constant. $y$: instantaneous displacement.	The energy stored in the system due to its displacement from the equilibrium position.
Average Total Energy (SHM)	${\mathbf{E}}_{\text{Avg}}=\frac{1}{2}\mathbf{k}{\mathbf{a}}^2$	$k$: restoring constant. $a$: amplitude.	Shows that the average total energy and the instantaneous total energy are the same. (Related also to average KE/PE: ${\text{KE}}_{\text{Avg}}=\frac{1}{4}k{a}^2$ and ${\text{PE}}_{\text{Avg}}=\frac{1}{4}k{a}^2$).
Resultant Amplitude (Superposition)	$\mathbf{A}=\sqrt{{\mathbf{a}}_1^2+{\mathbf{a}}_2^2+2{\mathbf{a}}_1{\mathbf{a}}_2\cos ({\alpha }_1-{\alpha }_2)}$	$A$: resultant amplitude. $a_1,a_2$: individual amplitudes. ${\alpha }_1-{\alpha }_2$: phase difference.	Gives the amplitude resulting from the superposition of two simple harmonic vibrations of the same frequency.
Damped Harmonic Oscillator Differential Eq.	$\frac{{\mathbf{d}}^2\mathbf{y}}{\mathbf{d}{\mathbf{t}}^2}+2\lambda \frac{\mathbf{dy}}{\mathbf{dt}}+{\omega }^2\mathbf{y}=0$	$\lambda$: damping constant (where $2\lambda =b/m$). $\omega$: natural angular frequency. $y$: displacement.	This is the differential equation governing damped motion, showing forces proportional to acceleration, velocity (damping, $-b\frac{dy}{dt}$), and displacement (restoring, $-ay$).
Group Velocity ($u$) - Wave Number form	$\mathbf{u}=\frac{\mathbf{d}\omega }{\mathbf{dk}}$	$u$: group velocity. $\omega$: angular frequency. $k$: wave number.	The speed at which the envelope of a wave packet travels.
Phase Velocity ($w$) - Wave form	$\mathbf{w}=\frac{\omega }{\mathbf{k}}$	$w$: phase velocity. $\omega$: angular frequency. $k$: wave number.	The speed at which a point of constant phase travels through the medium.
Relation between Group ($u$) and Phase Velocity ($v$)	$\mathbf{u}=\mathbf{v}-\lambda \frac{\mathbf{dv}}{\mathbf{d}\lambda }$	$u$: group velocity. $v$: phase velocity. $\lambda$: wavelength.	Relates the speed of the wave packet (group velocity) to the speed of individual component waves (phase velocity) and how phase velocity changes with wavelength ($\frac{dv}{d\lambda }$). [External Formula (to solve PYQs)] (Used in PYQ calculations).
Intensity of Plane Progressive Wave	$\mathbf{I}=2{\pi }^2\rho {\mathbf{a}}^2{\mathbf{f}}^2\mathbf{v}$	$I$: intensity (energy current per unit area). $\rho$: density of the medium. $a$: amplitude of the sound wave. $f$: frequency. $v$: velocity of the sound wave.	Calculates the rate of energy flow per unit area, proportional to density, velocity, and the square of the frequency and amplitude.
Acoustic Intensity Level (Decibels)	$\mathbf{L}=10{\log }_{10}\frac{\mathbf{I}}{{\mathbf{I}}_0}\text{ dB}$	$L$: intensity level (loudness). $I$: intensity of the sound. $I_0$: reference intensity.	Measures the loudness of sound relative to a reference intensity, $I_0$.
Beats Frequency	$\text{Number of beats per second}$ $=n_1-n_2$	$n_1,n_2$: frequencies of the two notes.	The frequency of the amplitude modulation observed when two waves of slightly different frequencies interfere.
Sabine’s Reverberation Time	$\mathbf{T}=\frac{0.16V}{\mathbf{A}}$	$T$: reverberation time (seconds). $V$: volume of the room ($m^3$). $A$: Total absorption (Sabines).	Gives the time taken for sound energy density to decay by a factor of $10^6$ (60 dB). The total absorption $A$ is calculated as $A=\sum aS$, where $a$ is the absorption coefficient and $S$ is the surface area.
Doppler Effect (Source Moving Towards Observer)	${\mathbf{n}}^{\prime }=\mathbf{n}\left(\frac{\mathbf{V}}{\mathbf{V}-{\mathbf{V}}_{\mathbf{s}}}\right)$	$n^{\prime }$: apparent frequency. $n$: actual frequency. $V$: velocity of sound. $V_s$: velocity of the source.	Calculates the perceived frequency when the source moves towards a stationary observer, resulting in an increase in apparent frequency ($n^{\prime }>n$). [External Formula (to solve PYQs)]
Doppler Effect (Observer Moving Towards Source)	${\mathbf{n}}^{\prime }=\mathbf{n}\left(\frac{\mathbf{V}+{\mathbf{V}}_{\mathbf{o}}}{\mathbf{V}}\right)$	$n^{\prime }$: apparent frequency. $n$: actual frequency. $V$: velocity of sound. $V_o$: velocity of the observer (labeled $b$ in source derivation).	Calculates the perceived frequency when the observer moves towards a stationary source, resulting in an increase in apparent frequency ($n^{\prime }>n$). [External Formula (to solve PYQs)]

II. Optics

This section covers Interference, Diffraction, Polarization, and Specific Rotation.

Concept	Formula	Symbols Defined	Concept Explanation
Phase Difference ($\delta$) / Path Difference ($x$)	$\delta =\frac{2\pi }{\lambda }\cdot \mathbf{x}$	$\delta$: phase difference. $\lambda$: wavelength. $x$: path difference.	Relates the path difference between two waves traveling from coherent sources to their phase difference.
Intensity (Two Equal Amplitude Waves)	$\mathbf{I}=4{\mathbf{a}}^2{\cos }^2(\delta /2)$	$I$: resultant intensity. $a$: amplitude of individual wave. $\delta$: phase difference.	Shows that intensity, proportional to the square of the resultant amplitude ($R^2$), depends sinusoidally on the phase difference $\delta$.
Fringe Width (Young’s Experiment)	$\beta =\frac{\lambda \mathbf{D}}{\mathbf{d}}$	$\beta$: fringe width (distance between successive maxima/minima). $\lambda$: wavelength. $D$: distance from slits to screen. $d$: slit separation.	Determines the spacing of the interference fringes based on the experimental geometry and the wavelength used.
Wavelength (Fresnel’s Biprism Method)	$\lambda =\frac{\beta \mathbf{d}}{\mathbf{D}}$	$\lambda$: wavelength. $\beta$: measured fringe width. $d$: effective distance between the two virtual sources. $D$: distance from source plane to screen/eyepiece.	Used experimentally to determine the wavelength of light once the fringe width $\beta$ and the geometry parameters ($d,D$) are measured.
Virtual Source Separation (Fresnel’s Biprism)	$\mathbf{d}=2(\mu -1)\alpha {\mathbf{y}}_1$	$d$: separation of virtual sources. $\mu$: refractive index of the prism material. $\alpha$: angle of the prism at the vertex. $y_1$: distance from the actual slit to the biprism.	Calculates the effective separation $d$ required in the fringe width formula, based on the material properties and geometry of the biprism. [External Formula (to solve PYQs)]
Condition for Bright Film (Reflected Light)	$2\mu \mathbf{t}\cos \mathbf{r}=(2n+1)\frac{\lambda }{2}$	$\mu$: refractive index of the film. $t$: thickness of the film. $r$: angle of refraction inside the film. $n$: integer (0, 1, 2…).	Condition for constructive interference (bright appearance) in a thin film due to reflected light, accounting for the $\lambda /2$ phase shift upon reflection at the denser medium.
Condition for Dark Film (Reflected Light)	$2\mu \mathbf{t}\cos \mathbf{r}=\mathbf{n}\lambda$	$\mu ,t,r$: film properties. $n$: integer (0, 1, 2…).	Condition for destructive interference (dark appearance) in a thin film due to reflected light.
Wavelength Determination (Newton’s Dark Rings)	$\lambda =\frac{{\mathbf{D}}_{\mathbf{n}+\mathbf{m}}^2-{\mathbf{D}}_{\mathbf{n}}^2}{4mR}$	$\lambda$: wavelength. $D_{n+m},D_n$: diameters of the $(n+m)$-th and $n$-th dark rings. $m$: difference in ring number. $R$: radius of curvature of the plano-convex lens.	Used to calculate the wavelength of light by measuring the diameters of two non-consecutive dark rings in Newton’s ring setup.
Refractive Index ($\mu$) of Liquid (Newton’s Rings)	$\mu =\frac{{\mathbf{D}}_{\mathbf{n}+\mathbf{m}}^2\text{(air)}-{\mathbf{D}}_{\mathbf{n}}^2\text{(air)}}{{\mathbf{D}}_{\mathbf{n}+\mathbf{m}}^2\text{(liquid)}-{\mathbf{D}}_{\mathbf{n}}^2\text{(liquid)}}$	$D^2$: square of the ring diameter (dark rings). Subscripts: $n,n+m$ are ring orders; (air) and (liquid) denote the medium filling the film.	Calculates the refractive index of a liquid by comparing the squares of the diameters of corresponding dark rings when the film is filled with air versus the liquid. [External Formula (to solve PYQs)]
Brewster’s Law (Polarizing Angle)	$\mu =\tan {\mathbf{i}}_{\mathbf{p}}$	$\mu$: refractive index of the medium. $i_p$: angle of polarization.	States that when unpolarized light is incident at the polarizing angle $i_p$, the tangent of $i_p$ is numerically equal to the refractive index of the medium, resulting in completely polarized reflected light.
Snell’s Law (Standard Refraction)	$\mu =\frac{\sin \mathbf{i}}{\sin \mathbf{r}}$	$\mu$: refractive index. $i$: angle of incidence. $r$: angle of refraction.	Relates the angle of incidence and refraction for light passing between two media.
Malus’s Law (Intensity of Transmitted Polarized Light)	${\mathbf{E}}_1=\mathbf{E}{\cos }^2\theta$	$E_1$: intensity of transmitted light. $E$: intensity of incident polarized light. $\theta$: angle between the plane of the polarizer and the plane of the analyzer.	The intensity of polarized light transmitted through an analyzer varies as the square of the cosine of the angle between the planes of transmission of the analyzer and the polarizer.
Specific Rotation ($S$)	${\mathbf{S}}_{\lambda }^{\mathbf{t}}=\frac{10\Theta }{\mathbf{lc}}$	$S_{\lambda }^t$: specific rotation at temperature $t$ for wavelength $\lambda$. $\Theta$: observed rotation of the plane of vibration (degrees). $l$: length of the column/tube (decimeters, $10$ cm). $c$: concentration (grams of active substance per cubic centimeter of solution).	Defines the characteristic rotating property of an optically active substance normalized by length and concentration.
Grating Maxima Condition	$\mathbf{d}\sin \theta =\mathbf{m}\lambda$	$d$: grating element (spacing between adjacent slits). $\theta$: angle of diffraction. $m$: order of the maximum (integer). $\lambda$: wavelength.	Gives the angles $\theta$ at which principal maxima occur in the Fraunhofer diffraction pattern formed by a diffraction grating.
Grating Dispersion	$\mathbf{D}=\frac{\mathbf{d}\theta }{\mathbf{d}\lambda }=\frac{\mathbf{m}}{\mathbf{d}\cos \theta }$	$D$: angular dispersion. $m$: order of spectrum. $d$: grating element. $\theta$: angle of diffraction.	Measures the angular separation ($d\theta$) of two nearby wavelengths ($d\lambda$) for a given order, indicating the ability of the grating to spread out the spectrum. [External Formula (to solve PYQs)]
Resolving Power of Grating	$\mathbf{R}=\mathbf{mN}$	$R$: resolving power. $m$: order of the spectrum. $N$: total number of rulings (slits) on the grating surface.	Defines the ability of the grating to separate two closely spaced wavelengths ($\lambda /\Delta \lambda$).

III. Atomic Physics

This section covers the Bohr model, spectral series, nuclear motion corrections, and the correspondence principle.

Concept	Formula	Symbols Defined	Concept Explanation
Reduced Mass ($m^{\prime }$)	${\mathbf{m}}^{\prime }=\frac{\mathbf{mM}}{\mathbf{M}+\mathbf{m}}$	$m^{\prime }$: reduced mass. $m$: electron mass. $M$: nuclear mass.	Accounts for the motion of the nucleus around the center of mass. This mass replaces the simple electron mass in accurate quantum calculations for atoms.
Electrostatic Force (Hydrogenic Atom)	${\mathbf{F}}_{\mathbf{e}}=\frac{1}{4\pi {ϵ}_{\mathbf{o}}}\frac{\mathbf{Z}{\mathbf{e}}^2}{{\mathbf{r}}^2}$	$F_e$: electrostatic force. ${ϵ}_o$: permittivity of free space. $Z$: atomic number. $e$: elementary charge. $r$: distance between nucleus and electron.	The attractive force responsible for holding the electron in orbit around a nucleus with charge $+Ze$.
Electron Energy Level (Hydrogenic Atom, with $m^{\prime }$ )	${\mathbf{E}}_{\mathbf{n}}=-\frac{{\mathbf{m}}^{\prime }{\mathbf{Z}}^2{\mathbf{e}}^4}{8{ϵ}_{\mathbf{o}}^2{\mathbf{h}}^2}\left(\frac{1}{{\mathbf{n}}^2}\right)$	$E_n$: energy of the $n$-th orbit. $m^{\prime }$: reduced mass. $Z,e,{ϵ}_o,h$: constants. $n$: principal quantum number.	Calculates the quantized total energy (kinetic + potential) of an electron in a hydrogenic atom (any single-electron system with nuclear charge $Z$), incorporating the correction for nuclear motion ($m^{\prime }$).
Wavelength of Spectral Lines (Hydrogenic Atom)	$\frac{1}{\lambda }=\frac{{\mathbf{m}}^{\prime }{\mathbf{Z}}^2{\mathbf{e}}^4}{8{ϵ}_{\mathbf{o}}^2\mathbf{c}{\mathbf{h}}^3}\left(\frac{1}{{\mathbf{n}}_{\mathbf{f}}^2}-\frac{1}{{\mathbf{n}}_{\mathbf{i}}^2}\right)$	$\lambda$: wavelength of emitted light. $c$: speed of light. $n_i,n_f$: initial and final quantum numbers ($n_i>n_f$). $m^{\prime },Z,e,{ϵ}_o,h$: constants.	The general formula, derived from energy level transitions, used to predict the wavelengths of all spectral lines emitted by hydrogenic atoms, corrected for nuclear motion.
Bohr Quantization Condition (Angular Momentum)	$\mathbf{mvr}=\frac{\mathbf{nh}}{2\pi }=\mathbf{n\hslash }$	$m$: electron mass. $v$: velocity. $r$: orbital radius. $n$: principal quantum number. $\hslash$: reduced Planck constant.	Bohr’s condition stating that the angular momentum of the electron is quantized, being an integer multiple of $\hslash$.
Bohr Correspondence Principle (Classical Frequency)	$\mathbf{f}=\frac{\mathbf{m}{\mathbf{e}}^4}{8{ϵ}_{\mathbf{o}}^2{\mathbf{h}}^3}\left(\frac{2}{{\mathbf{n}}^3}\right)$	$f$: classical frequency of revolution of the electron. $n$: quantum number of the orbit.	This formula represents the frequency of electromagnetic radiation expected from classical physics when an electron revolves in the $n$-th orbit.
Bohr Correspondence Principle (Quantum Frequency)	$\nu =\frac{\mathbf{m}{\mathbf{e}}^4}{8{ϵ}_{\mathbf{o}}^2{\mathbf{h}}^3}\left(\frac{1}{{\mathbf{n}}_{\mathbf{f}}^2}-\frac{1}{{\mathbf{n}}_{\mathbf{i}}^2}\right)$	$\nu$: frequency of emitted photon. $n_i,n_f$: initial and final quantum numbers.	The frequency derived from quantum mechanics for a transition between two energy levels.
Spectral Series (General Rydberg Formula, Hydrogen $Z=1$)	$\frac{1}{\lambda }=\mathbf{R}\left(\frac{1}{{\mathbf{n}}_{\mathbf{f}}^2}-\frac{1}{{\mathbf{n}}_{\mathbf{i}}^2}\right)$	$\frac{1}{\lambda }$: wavenumber. $R$: Rydberg constant ($1.097\times 10^7{\text{ m}}^{-1}$). $n_f$: final orbit level. $n_i$: initial orbit level ($n_i>n_f$).	This general formula covers all spectral series (Lyman, Balmer, Paschen, etc.) based on the electron transitioning from $n_i$ to a final level $n_f$. [External Formula (to solve PYQs)]

IV. Modern Physics

This section covers Wave Mechanics, Quantum Effects (Photoelectric, Compton), and Uncertainty Principles.

Concept	Formula	Symbols Defined	Concept Explanation
Heisenberg Uncertainty Principle (Position-Momentum)	$\mathbf{\Delta P}\cdot \mathbf{\Delta x}\ge \frac{1}{2}\mathbf{\hslash }$	$\Delta P$: uncertainty in momentum. $\Delta x$: uncertainty in position. $\hslash$: reduced Planck constant ($h/2\pi$).	States that the product of the experimental uncertainties in simultaneously measuring a particle’s position and momentum must be greater than or equal to $\hslash /2$.
Heisenberg Uncertainty Principle (Momentum Lower Bound)	$\mathbf{\Delta P}\ge \frac{\mathbf{\hslash }}{2\Delta x}$	$\Delta P$: minimum momentum uncertainty. $\Delta x$: confinement uncertainty (e.g., radius of the nucleus/atom).	Used to estimate the minimum momentum required for a particle confined within a specific spatial dimension $\Delta x$. This is crucial in determining if an electron can exist inside a nucleus.
Kinetic Energy (Classical approximation)	$\mathbf{T}=\frac{{\mathbf{p}}^2}{2m}$	$T$: kinetic energy. $p$: momentum. $m$: mass.	Used for calculating the kinetic energy of a particle when its velocity $v$ is much less than the speed of light $c$ (non-relativistic limit).
Energy of a Photon (Quantum)	$\mathbf{E}=\mathbf{h}\nu$	$E$: energy of the photon. $h$: Planck’s constant. $\nu$: frequency of the light.	Defines the quantized energy carried by a single photon, confirming light energy is concentrated in small packets rather than spread continuously over a wave front.
Einstein’s Photoelectric Equation	$\mathbf{h}\nu ={\mathbf{T}}_{\mathbf{max}}+\varphi$	$h\nu$: energy of incident photon. $T_{max}$: maximum kinetic energy of the emitted photoelectron. $\varphi$: work function ($\varphi =h{\nu }_0$).	States that the energy of the incident photon is used partly to liberate the electron (work function $\varphi$) and the remaining energy becomes the maximum kinetic energy of the photoelectron.
Maximum Kinetic Energy (Stopping Potential)	${\mathbf{T}}_{\mathbf{max}}=\mathbf{e}{\mathbf{V}}_0$	$T_{max}$: maximum kinetic energy. $e$: electron charge. $V_0$: stopping potential.	Relates the maximum kinetic energy of the emitted electrons to the voltage required to stop them completely. [External Formula (to solve PYQs)]
Work Function ($\varphi$)	$\varphi =\mathbf{h}{\nu }_0$	$\varphi$: work function. ${\nu }_0$: threshold frequency. $h$: Planck’s constant.	Represents the minimum energy required for an electron to escape from a particular metal surface.
De Broglie Wavelength ($\lambda$)	$\lambda =\frac{\mathbf{h}}{\mathbf{p}}=\frac{\mathbf{h}}{\mathbf{mv}}$	$\lambda$: de Broglie wavelength. $h$: Planck’s constant. $p$: momentum ($mv$). $m$: relativistic mass. $v$: velocity.	Establishes the wave nature of matter, stating that any moving particle has an associated wavelength inversely proportional to its momentum.
Phase Velocity ($w$) for De Broglie Waves	$\mathbf{w}=\frac{{\mathbf{c}}^2}{\mathbf{v}}$	$w$: phase velocity. $c$: speed of light. $v$: particle velocity.	The phase velocity of the matter wave is always greater than the speed of light ($w>c$), which explains why it does not physically represent the particle speed.
Kinetic Energy (Thermal Neutron)	$\lambda =\frac{\mathbf{h}}{\sqrt{2mE}}$	$\lambda$: de Broglie wavelength. $h$: Planck’s constant. $m$: mass of neutron/particle. $E$: kinetic energy (KE).	An external form derived by substituting $p=\sqrt{2mE}$ into $\lambda =h/p$. This formula is essential for calculating the wavelength of a particle given its kinetic energy (especially for non-relativistic particles like thermal neutrons). [External Formula (to solve PYQs)]
Compton Shift ($\Delta \lambda$)	${\lambda }^{\prime }-\lambda =\frac{\mathbf{h}}{{\mathbf{m}}_0\mathbf{c}}(1-\cos \mathbf{\Phi })$	${\lambda }^{\prime }$: scattered photon wavelength. $\lambda$: incident photon wavelength. $h$: Planck’s constant. $m_0$: rest mass of the scattered particle (electron). $c$: speed of light. $\Phi$: scattering angle of the photon.	Calculates the increase in wavelength (red shift) observed when a photon scatters off a free electron, confirming the particle nature of light.
Compton Wavelength (${\lambda }_c$)	${\lambda }_{\mathbf{c}}=\frac{\mathbf{h}}{{\mathbf{m}}_0\mathbf{c}}$	${\lambda }_c$: Compton wavelength of the scattering particle. $h,m_0,c$: physical constants.	A characteristic constant wavelength derived from the fundamental constants $h$, $m_0$, and $c$. The Compton shift can be expressed simply as $\Delta \lambda ={\lambda }_c(1-\cos \Phi )$.

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Clarification Analogy: Group vs. Phase Velocity

The difference between phase velocity ($w$) and group velocity ($u$) is crucial in Modern Physics. Imagine a parade marching down the street:

• Phase Velocity ($w$): This is the speed of an individual soldier’s waving arm. In the quantum world, this speed ($c^2/v$) can exceed the speed limit ($c$) because it carries no information or mass, much like an individual wave crest.
• Group Velocity ($u$): This is the speed of the marching band as a whole. This group (the wave packet) is what carries the mass, energy, and information (the actual particle). In both classical and quantum mechanics, the group velocity is equal to the actual speed of the particle ($u=v$).