Physics Formulas Sheet

phy-1109
PHY-1109 Physics

I. Oscillations

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

Simple Harmonic Motion (SHM)

SHM Differential Equation

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 $k/m$). $\frac{d^{2}y}{d{t}^2}+{\omega }^{2}y=0$

• $y$: displacement.
• $\omega$: angular frequency.

Angular Frequency ($\omega$)

This formula relates the angular frequency of the oscillator to the physical properties of the system (stiffness $k$ and inertia $m$). $\omega =\sqrt{\frac{k}{m}}$

• $k$: spring constant or restoring constant.
• $m$: mass of the particle.

Displacement in SHM

This gives the instantaneous displacement $y$ of a particle undergoing SHM at time $t$. $y=a\sin (\omega t+\Phi )$

• $a$: Amplitude (maximum displacement).
• $\omega t+\Phi$: total phase.
• $\Phi$: initial phase/epoch.
• $t$: time.

Velocity in SHM

This determines the instantaneous speed based on displacement. Maximum velocity occurs at the equilibrium point ($y=0$), where $V_{max}=\omega a$. $v=\omega \sqrt{a^2-y^2}$

• $v$: instantaneous velocity.
• $a$: amplitude.
• $y$: displacement.
• $\omega$: angular frequency.

Total Energy (T.E.) in SHM

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. $\text{T.E.}=\frac{1}{2}k{a}^2$

• $k$: restoring constant.
• $a$: amplitude.

Potential Energy (P.E.) in SHM

This represents the energy stored in the system due to its displacement from the equilibrium position. $\text{P.E.}=\frac{1}{2}k{y}^2$

• $k$: restoring constant.
• $y$: instantaneous displacement.

Average Total Energy (SHM)

This shows that the average total energy and the instantaneous total energy are the same. (It is related to the average KE/PE: ${\text{KE}}_{\text{Avg}}=\frac{1}{4}k{a}^2$ and ${\text{PE}}_{\text{Avg}}=\frac{1}{4}k{a}^2$). $E_{\text{Avg}}=\frac{1}{2}k{a}^2$

• $k$: restoring constant.
• $a$: amplitude.

Resultant Amplitude (Superposition)

This gives the amplitude resulting from the superposition of two simple harmonic vibrations of the same frequency. $A=\sqrt{a_1^2+a_2^2+2a_1{a}_2\cos ({\alpha }_1-{\alpha }_2)}$

• $A$: resultant amplitude.
• $a_1,a_2$: individual amplitudes.
• ${\alpha }_1-{\alpha }_2$: phase difference.

Damped Harmonic Oscillator Differential Eq.

This is the differential equation governing damped motion, showing forces proportional to acceleration, velocity (damping, $-b\frac{dy}{dt}$), and displacement (restoring, $-ay$). $\frac{d^{2}y}{d{t}^2}+2\lambda \frac{dy}{dt}+{\omega }^{2}y=0$

• $\lambda$: damping constant (where $2\lambda =b/m$).
• $\omega$: natural angular frequency.
• $y$: displacement.

Wave and Group Velocities

Group Velocity ($u$) - Wave Number form

The speed at which the envelope of a wave packet travels. $u=\frac{d\omega }{dk}$

• $u$: group velocity.
• $\omega$: angular frequency.
• $k$: wave number.

Phase Velocity ($w$) - Wave form

The speed at which a point of constant phase travels through the medium. $w=\frac{\omega }{k}$

• $w$: phase velocity.
• $\omega$: angular frequency.
• $k$: wave number.

Relation between Group ($u$) and Phase Velocity ($v$)

This 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 ($\frac{dv}{d\lambda }$). (This is an External Formula used for PYQ calculations). $u=v-\lambda \frac{dv}{d\lambda }$

• $u$: group velocity.
• $v$: phase velocity.
• $\lambda$: wavelength.

Acoustics and Sound

Intensity of Plane Progressive Wave

Calculates the rate of energy flow per unit area, proportional to density, velocity, and the square of the frequency and amplitude. $I=2{\pi }^2\rho a^2{f}^{2}v$

• $I$: 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.

Acoustic Intensity Level (Decibels)

Measures the loudness of sound relative to a reference intensity, $I_0$. $L=10{\log }_{10}\frac{I}{I_0}\text{ dB}$

• $L$: intensity level (loudness).
• $I$: intensity of the sound.
• $I_0$: reference intensity.

Beats Frequency

The frequency of the amplitude modulation observed when two waves of slightly different frequencies interfere. $\text{Number of beats per second}=n_1-n_2$

• $n_1,n_2$: frequencies of the two notes.

Sabine’s Reverberation Time

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. $T=\frac{0.16V}{A}$

• $T$: reverberation time (seconds).
• $V$: volume of the room ($m^3$).
• $A$: Total absorption (Sabines).

Doppler Effect (Source Moving Towards Observer)

Calculates the perceived frequency when the source moves towards a stationary observer, resulting in an increase in apparent frequency ($n^{\prime }>n$). (This is an External Formula used for PYQ calculations). $n^{\prime }=n\left(\frac{V}{V-V_s}\right)$

• $n^{\prime }$: apparent frequency.
• $n$: actual frequency.
• $V$: velocity of sound.
• $V_s$: velocity of the source.

Doppler Effect (Observer Moving Towards Source)

Calculates the perceived frequency when the observer moves towards a stationary source, resulting in an increase in apparent frequency ($n^{\prime }>n$). (This is an External Formula used for PYQ calculations). $n^{\prime }=n\left(\frac{V+V_o}{V}\right)$

• $n^{\prime }$: apparent frequency.
• $n$: actual frequency.
• $V$: velocity of sound.
• $V_o$: velocity of the observer (labeled $b$ in source derivation).

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II. Optics

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

Interference

Phase Difference ($\delta$) / Path Difference ($x$)

This relates the path difference between two waves traveling from coherent sources to their phase difference. $\delta =\frac{2\pi }{\lambda }\cdot x$

• $\delta$: phase difference.
• $\lambda$: wavelength.
• $x$: path difference.

Intensity (Two Equal Amplitude Waves)

This shows that intensity, proportional to the square of the resultant amplitude ($R^2$), depends sinusoidally on the phase difference $\delta$. $I=4a^2{\cos }^2(\delta /2)$

• $I$: resultant intensity.
• $a$: amplitude of individual wave.
• $\delta$: phase difference.

Fringe Width (Young’s Experiment)

This determines the spacing of the interference fringes based on the experimental geometry and the wavelength used. $\beta =\frac{\lambda D}{d}$

• $\beta$: fringe width (distance between successive maxima/minima).
• $\lambda$: wavelength.
• $D$: distance from slits to screen.
• $d$: slit separation.

Wavelength (Fresnel’s Biprism Method)

Used experimentally to determine the wavelength of light once the fringe width $\beta$ and the geometry parameters ($d,D$) are measured. $\lambda =\frac{\beta d}{D}$

• $\lambda$: wavelength.
• $\beta$: measured fringe width.
• $d$: effective distance between the two virtual sources.
• $D$: distance from source plane to screen/eyepiece.

Virtual Source Separation (Fresnel’s Biprism)

Calculates the effective separation $d$ required in the fringe width formula, based on the material properties and geometry of the biprism. (This is an External Formula used for PYQ calculations). $d=2(\mu -1)\alpha y_1$

• $d$: 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.

Condition for Bright Film (Reflected Light)

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. $2\mu t\cos r=(2n+1)\frac{\lambda }{2}$

• $\mu$: refractive index of the film.
• $t$: thickness of the film.
• $r$: angle of refraction inside the film.
• $n$: integer (0, 1, 2…).

Condition for Dark Film (Reflected Light)

Condition for destructive interference (dark appearance) in a thin film due to reflected light. $2\mu t\cos r=n\lambda$

• $\mu ,t,r$: film properties.
• $n$: integer (0, 1, 2…).

Wavelength Determination (Newton’s Dark Rings)

Used to calculate the wavelength of light by measuring the diameters of two non-consecutive dark rings in Newton’s ring setup. $\lambda =\frac{D_{n+m}^2-D_n^2}{4mR}$

• $\lambda$: 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.

Refractive Index ($\mu$) of Liquid (Newton’s Rings)

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. (This is an External Formula used for PYQ calculations). $\mu =\frac{D_{n+m}^2\text{(air)}-D_n^2\text{(air)}}{D_{n+m}^2\text{(liquid)}-D_n^2\text{(liquid)}}$

• $D^2$: square of the ring diameter (dark rings).
• Subscripts $n,n+m$: ring orders.
• (air) and (liquid): denote the medium filling the film.

Polarization and Refraction

Brewster’s Law (Polarizing Angle)

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. $\mu =\tan i_p$

• $\mu$: refractive index of the medium.
• $i_p$: angle of polarization.

Snell’s Law (Standard Refraction)

Relates the angle of incidence and refraction for light passing between two media. $\mu =\frac{\sin i}{\sin r}$

• $\mu$: refractive index.
• $i$: angle of incidence.
• $r$: angle of refraction.

Malus’s Law (Intensity of Transmitted Polarized Light)

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. $E_1=E{\cos }^2\theta$

• $E_1$: intensity of transmitted light.
• $E$: intensity of incident polarized light.
• $\theta$: angle between the plane of the polarizer and the plane of the analyzer.

Specific Rotation ($S$)

Defines the characteristic rotating property of an optically active substance normalized by length and concentration. $S_{\lambda }^t=\frac{10\Theta }{lc}$

• $S_{\lambda }^t$: 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).

Diffraction Grating

Grating Maxima Condition

Gives the angles $\theta$ at which principal maxima occur in the Fraunhofer diffraction pattern formed by a diffraction grating. $d\sin \theta =m\lambda$

• $d$: grating element (spacing between adjacent slits).
• $\theta$: angle of diffraction.
• $m$: order of the maximum (integer).
• $\lambda$: wavelength.

Grating Dispersion

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. (This is an External Formula used for PYQ calculations). $D=\frac{d\theta }{d\lambda }=\frac{m}{d\cos \theta }$

• $D$: angular dispersion.
• $m$: order of spectrum.
• $d$: grating element.
• $\theta$: angle of diffraction.

Resolving Power of Grating

Defines the ability of the grating to separate two closely spaced wavelengths ($\lambda /\Delta \lambda$). $R=mN$

• $R$: resolving power.
• $m$: order of the spectrum.
• $N$: total number of rulings (slits) on the grating surface.

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III. Atomic Physics

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

Reduced Mass ($m^{\prime }$)

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. $m^{\prime }=\frac{mM}{M+m}$

• $m^{\prime }$: reduced mass.
• $m$: electron mass.
• $M$: nuclear mass.

Electrostatic Force (Hydrogenic Atom)

The attractive force responsible for holding the electron in orbit around a nucleus with charge $+Ze$. $F_e=\frac{1}{4\pi {ϵ}_o}\frac{Z{e}^2}{r^2}$

• $F_e$: electrostatic force.
• ${ϵ}_o$: permittivity of free space.
• $Z$: atomic number.
• $e$: elementary charge.
• $r$: distance between nucleus and electron.

Electron Energy Level (Hydrogenic Atom, with $m^{\prime }$)

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 ($m^{\prime }$). $E_n=-\frac{m^{\prime }{Z}^2{e}^4}{8{ϵ}_o^2{h}^2}\left(\frac{1}{n^2}\right)$

• $E_n$: energy of the $n$-th orbit.
• $m^{\prime }$: reduced mass.
• $Z,e,{ϵ}_o,h$: constants.
• $n$: principal quantum number.

Wavelength of Spectral Lines (Hydrogenic Atom)

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. $\frac{1}{\lambda }=\frac{m^{\prime }{Z}^2{e}^4}{8{ϵ}_o^{2}c{h}^3}\left(\frac{1}{n_f^2}-\frac{1}{n_i^2}\right)$

• $\lambda$: 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.

Bohr Quantization Condition (Angular Momentum)

Bohr’s condition stating that the angular momentum of the electron is quantized, being an integer multiple of $\hslash$. $mvr=\frac{nh}{2\pi }=n\hslash$

• $m$: electron mass.
• $v$: velocity.
• $r$: orbital radius.
• $n$: principal quantum number.
• $\hslash$: reduced Planck constant.

Bohr Correspondence Principle (Classical Frequency)

This formula represents the frequency of electromagnetic radiation expected from classical physics when an electron revolves in the $n$-th orbit. $f=\frac{m{e}^4}{8{ϵ}_o^2{h}^3}\left(\frac{2}{n^3}\right)$

• $f$: classical frequency of revolution of the electron.
• $n$: quantum number of the orbit.

Bohr Correspondence Principle (Quantum Frequency)

The frequency derived from quantum mechanics for a transition between two energy levels. $\nu =\frac{m{e}^4}{8{ϵ}_o^2{h}^3}\left(\frac{1}{n_f^2}-\frac{1}{n_i^2}\right)$

• $\nu$: frequency of emitted photon.
• $n_i,n_f$: initial and final quantum numbers.

Spectral Series (General Rydberg Formula, Hydrogen $Z=1$)

This general formula covers all spectral series (Lyman, Balmer, Paschen, etc.) based on the electron transitioning from $n_i$ to a final level $n_f$. (This is an External Formula used for PYQ calculations). $\frac{1}{\lambda }=R\left(\frac{1}{n_f^2}-\frac{1}{n_i^2}\right)$

• $\frac{1}{\lambda }$: wavenumber.
• $R$: Rydberg constant ($1.097\times 10^7{\text{ m}}^{-1}$).
• $n_f$: final orbit level.
• $n_i$: initial orbit level ($n_i>n_f$).

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IV. Modern Physics

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

Quantum Mechanics and Uncertainty

Heisenberg Uncertainty Principle (Position-Momentum)

States that the product of the experimental uncertainties in simultaneously measuring a particle’s position and momentum must be greater than or equal to $\hslash /2$. $\Delta P\cdot \Delta x\ge \frac{1}{2}\hslash$

• $\Delta P$: uncertainty in momentum.
• $\Delta x$: uncertainty in position.
• $\hslash$: reduced Planck constant ($h/2\pi$).

Heisenberg Uncertainty Principle (Momentum Lower Bound)

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. $\Delta P\ge \frac{\hslash }{2\Delta x}$

• $\Delta P$: minimum momentum uncertainty.
• $\Delta x$: confinement uncertainty (e.g., radius of the nucleus/atom).

Kinetic Energy (Classical approximation)

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). $T=\frac{p^2}{2m}$

• $T$: kinetic energy.
• $p$: momentum.
• $m$: mass.

Photoelectric Effect

Energy of a Photon (Quantum)

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. $E=h\nu$

• $E$: energy of the photon.
• $h$: Planck’s constant.
• $\nu$: frequency of the light.

Einstein’s Photoelectric Equation

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. $h\nu =T_{max}+\varphi$

• $h\nu$: energy of incident photon.
• $T_{max}$: maximum kinetic energy of the emitted photoelectron.
• $\varphi$: work function ($\varphi =h{\nu }_0$).

Maximum Kinetic Energy (Stopping Potential)

Relates the maximum kinetic energy of the emitted electrons to the voltage required to stop them completely. (This is an External Formula used for PYQ calculations). $T_{max}=e{V}_0$

• $T_{max}$: maximum kinetic energy.
• $e$: electron charge.
• $V_0$: stopping potential.

Work Function ($\varphi$)

Represents the minimum energy required for an electron to escape from a particular metal surface. $\varphi =h{\nu }_0$

• $\varphi$: work function.
• ${\nu }_0$: threshold frequency.
• $h$: Planck’s constant.

De Broglie Waves

De Broglie Wavelength ($\lambda$)

Establishes the wave nature of matter, stating that any moving particle has an associated wavelength inversely proportional to its momentum. $\lambda =\frac{h}{p}=\frac{h}{mv}$

• $\lambda$: de Broglie wavelength.
• $h$: Planck’s constant.
• $p$: momentum ($mv$).
• $m$: relativistic mass.
• $v$: velocity.

Phase Velocity ($w$) for De Broglie Waves

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. $w=\frac{c^2}{v}$

• $w$: phase velocity.
• $c$: speed of light.
• $v$: particle velocity.

De Broglie Wavelength (Kinetic Energy form)

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). (This is an External Formula used for PYQ calculations). $\lambda =\frac{h}{\sqrt{2mE}}$

• $\lambda$: de Broglie wavelength.
• $h$: Planck’s constant.
• $m$: mass of neutron/particle.
• $E$: kinetic energy (KE).

Compton Effect

Compton Shift ($\Delta \lambda$)

Calculates the increase in wavelength (red shift) observed when a photon scatters off a free electron, confirming the particle nature of light. ${\lambda }^{\prime }-\lambda =\frac{h}{m_{0}c}(1-\cos \Phi )$

• ${\lambda }^{\prime }$: 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.

Compton Wavelength (${\lambda }_c$)

A characteristic constant wavelength derived from the fundamental constants $h$, $m_0$, and $c$. The Compton shift can be expressed simply as $\Delta \lambda ={\lambda }_c(1-\cos \Phi )$. ${\lambda }_c=\frac{h}{m_{0}c}$

• ${\lambda }_c$: Compton wavelength of the scattering particle.
• $h,m_0,c$: physical constants.

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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.

• 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). The group velocity is equal to the actual speed of the particle ($u=v$).

~ Nakib 2409007