MDCAT Physics Formula Sheets

Unit XIII · Electronics

Rectifiers & diode current

Formula	Use
$I_f = \frac{V_f}{r_f}$	Forward current
$I_r = \frac{V_r}{r_r}$	Reverse current
$\text{Form factor} = \frac{V_{\mathrm{rms}}}{V_{\mathrm{av}}}$	Form factor
$\text{Half-wave:}\quad T_{\mathrm{out}}=T_{\mathrm{in}}$ $f_{\mathrm{out}}=f_{\mathrm{in}}$	Time period and frequency
$\text{Half-wave:}\quad V_{\mathrm{rms}}=\frac{V_0}{2}$ $V_{\mathrm{av}}=\frac{V_0}{\pi}$	RMS and average output voltage
$\text{Full-wave:}\quad T_{\mathrm{out}}=\frac{T_{\mathrm{in}}}{2}$ $f_{\mathrm{out}}=2f_{\mathrm{in}}$	Time period and frequency
$\text{Full-wave:}\quad V_{\mathrm{rms}}=\frac{V_0}{\sqrt{2}}$ $V_{\mathrm{av}}=\frac{2V_0}{\pi}$	RMS and average output voltage

Units X–XI · Magnetism & electromagnetic induction

Magnetic field, force & charged particles

Formula	Use
$\Phi=\vec{B}\cdot\vec{A}=BA\cos\theta$	Magnetic flux through a vector area
$\Phi=BA\sin\theta$	Magnetic flux through a plane area
$B=\frac{\Phi}{A}$	Magnetic flux density for perpendicular field
$B=\frac{\mu_0 I}{2\pi r}$	Field outside a long straight conductor
$B=\frac{\mu_0 Ir}{2\pi R^2}$	Field inside a solid long straight conductor
$B=\frac{\mu_0 I}{2R}$	Field at centre of one circular loop
$B=\frac{\mu_0 NI}{2R}$	Field at centre of N-turn coil
$B=\mu_0 nI$	Field inside a long solenoid
$B_{\mathrm{end}}=\frac{\mu_0 nI}{2}$	Field at end of a solenoid
$F=BIL\sin\theta$	Force on a current-carrying conductor
$\tau=NIBA\cos\alpha$	Torque on a current coil
$\frac{F}{L}=\frac{\mu_0 I_1 I_2}{2\pi d}$	Force per unit length between parallel currents
$F=qvB\sin\theta$	Force on a moving charge in magnetic field
$r=\frac{mv}{qB}=\frac{p}{qB}$	Cyclotron radius
$r=\frac{\sqrt{2m\,\mathrm{KE}}}{qB}$	Cyclotron radius in terms of kinetic energy
$\frac{q}{m}=\frac{2V}{B^2r^2}$	Charge-to-mass ratio from cyclotron potential
$\vec{F}=q\vec{E}+q(\vec{v}\times\vec{B})$	Lorentz force
$v=\frac{E}{B}$	Velocity selected by crossed E and B fields

Electrostatics

Charge, field, potential & capacitors

Formula	Use
$Q=ne$	Quantisation of charge
$F=\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{r^2}$	Coulomb force in vacuum
$F=\frac{1}{4\pi\varepsilon_0\varepsilon_r}\frac{q_1q_2}{r^2}$	Coulomb force in a medium
$k=\frac{1}{4\pi\varepsilon_0}$	Coulomb constant
$E=\frac{F}{q_0}=\frac{kq}{r^2}$	Electric field strength
$\Phi_E=\vec{E}\cdot\vec{A}=EA\cos\theta$	Electric flux through vector area
$\Phi_E=EA\sin\theta$	Electric flux through plane area
$\Phi_E=\frac{Q}{\varepsilon_0}$	Gauss's law in vacuum
$E=\frac{\sigma}{2\varepsilon_0}$	Field of an infinite charged sheet
$E=\frac{\sigma}{\varepsilon_0}$	Field between oppositely charged sheets
$V=\frac{W}{q_0}=\frac{kq}{r}$	Potential at a point charge
$E=-\frac{\Delta V}{\Delta r}$	Electric field as potential gradient
$\Delta V=kq\left(\frac{1}{r_1}-\frac{1}{r_2}\right)$	Potential difference between two points
$C=\frac{Q}{V}$	Capacitance
$C=\frac{\varepsilon_0 A}{d}$ $C=\frac{\varepsilon_0\varepsilon_r A}{d}$	Parallel-plate capacitance without / with dielectric
$W=\frac{1}{2}QV=\frac{1}{2}CV^2=\frac{Q^2}{2C}$	Energy stored in a capacitor
$\frac{1}{C_{\mathrm{eq}}}=\frac{1}{C_1}+\frac{1}{C_2}+\cdots$	Capacitors in series
$C_{\mathrm{eq}}=C_1+C_2+\cdots$	Capacitors in parallel
$\tau=RC$	RC time constant
$\text{Charging:}\quad q=q_0\left(1-e^{-t/(RC)}\right)$ $V=V_0\left(1-e^{-t/(RC)}\right)$ $I=I_0e^{-t/(RC)}$	Charging capacitor
$\text{Discharging:}\quad q=q_0e^{-t/(RC)}$ $V=V_0e^{-t/(RC)}$ $I=I_0e^{-t/(RC)}$	Discharging capacitor (magnitude)

Rotational & circular motion

Circular motion & gravity

Formula	Use
$F_c=\frac{mv^2}{r}$	Centripetal force
$\frac{mv^2}{r}=\frac{GMm}{r^2}$	Satellite in circular orbit
$\frac{mv^2}{r}=\mu N$	Car on an unbanked circular road
$\frac{mv^2}{r}=\frac{kqQ}{r^2}$	Electron around nucleus (classical model)
$\frac{mv^2}{r}=qvB$	Charged particle in magnetic field
$\frac{mv^2}{r}=T$	Object tied to a string in horizontal circle
$v=\sqrt{rg\tan\theta}$	Critical speed for a banked road
$g_{\mathrm{inside}}=\frac{GM(R-d)}{R^3}$	Gravitational acceleration at depth d
$g_{\mathrm{surface}}=\frac{GM}{R^2}$	Gravitational acceleration at Earth surface
$g_{\mathrm{outside}}=\frac{GM}{(R+h)^2}$	Gravitational acceleration at height h

Unit XVI · Nuclear physics

Nucleus, decay & radiation dose

Formula	Use
$R=R_0 A^{1/3}$	Nuclear radius
$V=\frac{4}{3}\pi R_0^3 A$	Nuclear volume
$\text{Activity } A=-\frac{\Delta N}{\Delta t}=\lambda N$	Activity and decay rate
$N=N_0 e^{-\lambda t}=N_0\left(\frac{1}{2}\right)^n$	Undecayed nuclei
$N_{\mathrm{decayed}}=N_0\left[1-\left(\frac{1}{2}\right)^n\right]$	Nuclei decayed after n half-lives
$n=\frac{t}{T_{1/2}}$	Number of half-lives
$\lambda=\frac{\ln 2}{T_{1/2}}=\frac{0.693}{T_{1/2}}$	Decay constant
$\tau=\frac{1}{\lambda}=\frac{T_{1/2}}{\ln 2}$	Mean life
$D=\frac{E}{m}$	Absorbed dose
$D_{\mathrm{equivalent}}=D_{\mathrm{absorbed}}\times\mathrm{RBE}$	Equivalent dose
$\alpha:\quad {}^{A}_{Z}\mathrm{X}\rightarrow{}^{A-4}_{Z-2}\mathrm{Y}+{}^{4}_{2}\mathrm{He}$	Alpha decay
$\beta^-:\quad {}^{A}_{Z}\mathrm{X}\rightarrow{}^{A}_{Z+1}\mathrm{Y}+{}^{0}_{-1}\mathrm{e}+\bar{\nu}$	Beta-minus decay
$\beta^+:\quad {}^{A}_{Z}\mathrm{X}\rightarrow{}^{A}_{Z-1}\mathrm{Y}+{}^{0}_{+1}\mathrm{e}+\nu$	Beta-plus decay
$\gamma:\quad {}^{A}_{Z}\mathrm{X}^{*}\rightarrow{}^{A}_{Z}\mathrm{X}+{}^{0}_{0}\gamma$	Gamma decay

Unit XII · Alternating current

Alternating current

Formula	Use
$V=V_0\sin\omega t$ $I=I_0\sin\omega t$	Instantaneous AC voltage and current
$V_{\mathrm{rms}}=\frac{V_0}{\sqrt{2}}$ $I_{\mathrm{rms}}=\frac{I_0}{\sqrt{2}}$	RMS values
$\theta=\omega t=2\pi ft=\frac{2\pi}{T}t$	Phase of AC
$c=\frac{1}{\sqrt{\varepsilon_0\mu_0}}$ $c_{\mathrm{medium}}=\frac{c}{n}$	Speed of light in vacuum / medium
$Z=\sqrt{R^2+(X_L-X_C)^2}$	Impedance of a series RLC circuit
$X_C=\frac{1}{\omega C}=\frac{1}{2\pi fC}$	Capacitive reactance
$X_L=\omega L=2\pi fL$	Inductive reactance
$P_{\mathrm{av}}=V_{\mathrm{rms}}I_{\mathrm{rms}}\cos\phi=\frac{V_0I_0}{2}\cos\phi$	Average power in AC circuit
$\text{Power factor}=\cos\phi$	Power factor
$P_{\mathrm{av}}=0$	Average power in a pure capacitor or inductor

Unit VII · Waves & oscillations

SHM, waves, sound & optics

Formula	Use
$\vec{F}\propto-\vec{x}$ $\vec{a}\propto-\vec{x}$ $a=-\omega^2x$	Restoring force and acceleration in SHM
$\text{S in one vibration}=4A$ $T=\frac{1}{f}$ $\omega=2\pi f$	Basic SHM relations
$x=x_0\sin\omega t$ $v=v_0\cos\omega t$	SHM starting from mean position
$x=x_0\cos\omega t$ $v=-v_0\sin\omega t$	SHM starting from extreme position
$\mathrm{PE}=\frac{1}{2}kx^2$ $\mathrm{KE}=\frac{1}{2}kx_0^2\left(1-\frac{x^2}{x_0^2}\right)$	Energy in SHM
$v=f\lambda$ $\text{wave number}=\frac{1}{\lambda}$	Wave equation and wave number
$I\propto A^2$ $A\propto\sqrt{I}$	Intensity–amplitude relation
$v=\sqrt{\frac{F}{\mu}}$	Speed of transverse wave on a string
$v=\sqrt{\frac{E}{\rho}}$	Speed of longitudinal wave
$\lambda_{\text{de Broglie}}=\frac{h}{p}=\frac{h}{mv}=\frac{h}{\sqrt{2m\,\mathrm{KE}}}$	de Broglie wavelength
$n=\frac{c}{v}$ $n_1\sin\theta_1=n_2\sin\theta_2$	Refractive index and Snell's law
$v_{\mathrm{sound}}=\sqrt{\frac{\gamma P}{\rho}}$ $\gamma=\frac{C_p}{C_v}$	Laplace formula for speed of sound
$\frac{v_t}{v_0}=\sqrt{\frac{T}{T_0}}$ $v_t=v_0+0.61t$	Speed of sound versus temperature

Unit XI · Electromagnetism

Induction, transformers & generator

Formula	Use
$\varepsilon=-\frac{\Delta\Phi}{\Delta t}$ $\varepsilon=-\frac{N\Delta\Phi}{\Delta t}$	Induced EMF (Faraday's law)
$I=-\frac{N\Delta\Phi}{R\Delta t}$ $q=-\frac{N\Delta\Phi}{R}$	Induced current and charge
$\varepsilon=-vBL\sin\theta$	Motional EMF
$\text{Flux linkage}=N\Phi=NBA\cos\theta$	Flux linkage
$M=\frac{N_s\Phi_s}{I_p}=-\frac{\varepsilon}{\Delta I_p/\Delta t}$	Mutual inductance
$L=\frac{N\Phi}{I}=-\frac{\varepsilon}{\Delta I/\Delta t}$	Self inductance
$\frac{N_s}{N_p}=\frac{V_s}{V_p}=\frac{I_p}{I_s}$	Ideal transformer ratios
$V_s I_s=V_p I_p$	Ideal transformer power relation
$\eta=\frac{P_{\mathrm{out}}}{P_{\mathrm{in}}}\times100$	Transformer efficiency
$\varepsilon=N\omega AB\sin\theta$ $\varepsilon_0=N\omega AB$	Instantaneous and maximum generator EMF
$I=\frac{\varepsilon}{R}=\frac{\varepsilon_0\sin\theta}{R}$ $I_0=\frac{\varepsilon_0}{R}$	Generator current

Unit II · Force & motion

Projectile motion

Formula	Use
$h=\frac{1}{2}gt^2$	Vertical drop / height from rest
$v_x=u\cos\theta$ $v_y=u\sin\theta-gt$	Projectile velocity components
$t_{\text{to summit}}=\sqrt{\frac{2H}{g}}=\frac{u\sin\theta}{g}$	Time to maximum height
$H=\frac{u^2\sin^2\theta}{2g}$	Maximum height of projectile
$R=\frac{u^2\sin2\theta}{g}$	Range on level ground
$T=\frac{2u\sin\theta}{g}$	Time of flight
$\mathrm{KE}_{\mathrm{summit}}=\mathrm{KE}_{\mathrm{initial}}\cos^2\theta$	Kinetic energy at summit
$\mathrm{PE}_{\mathrm{summit}}=\mathrm{KE}_{\mathrm{initial}}\sin^2\theta$	Potential energy at summit
$R\tan\theta=4H$ $R\tan\theta=\frac{1}{2}gT^2$ $H=\frac{gT^2}{8}$	Useful range–height–time relations

Unit XIV · Dawn of modern physics

Relativity, photons & photoelectric effect

Formula	Use
$t=\frac{t_0}{\sqrt{1-\frac{v^2}{c^2}}}$	Time dilation
$l=l_0\sqrt{1-\frac{v^2}{c^2}}$	Length contraction
$\lambda=\frac{h}{p}=\frac{h}{mv}=\frac{h}{\sqrt{2m\,\mathrm{KE}}}=\frac{h}{\sqrt{2mqV}}$	de Broglie wavelength
$E=hf=\frac{hc}{\lambda}$	Photon energy
$P=n\left(\frac{hf}{t}\right)=\frac{nhc}{\lambda t}$	Power of n photons
$E(\mathrm{eV})=\frac{12400}{\lambda(\mathring{\mathrm{A}})}=\frac{1240}{\lambda(\mathrm{nm})}$	Photon energy from wavelength
$\Delta\lambda=\frac{h}{m_0c}(1-\cos\theta)$	Compton shift
$\phi=hf_0=\frac{hc}{\lambda_0}$	Work function
$hf=\phi+\frac{1}{2}mv_{\mathrm{max}}^2$ $eV_0=hf-\phi$	Einstein photoelectric equation
$v_{\mathrm{max}}=6\times10^5\sqrt{V_0}$	Maximum photoelectron speed from stopping potential

Unit VI · Fluid dynamics

Fluid dynamics

Formula	Use
$Q=\frac{\Delta V}{\Delta t}=Av$	Volume flow rate
$\dot{m}=\frac{\Delta m}{\Delta t}=\rho Av$	Mass flow rate
$A_1v_1=A_2v_2$ $Q_1=Q_2$	Continuity equation
$\mathrm{Re}=\frac{\rho vD}{\mu}$	Reynolds number
$F=6\pi\mu rv$	Stokes’ law
$P=\rho gh$	Fluid pressure at depth h
$v_t=\frac{2gr^2\rho}{9\mu}$	Terminal velocity of a small sphere
$P+\frac{1}{2}\rho v^2+\rho gh=\text{constant}$	Bernoulli equation
$v_{\mathrm{efflux}}=\sqrt{2gh}$	Efflux speed (Torricelli)
$Q=\frac{\pi r^4\Delta P}{8\eta L}$	Poiseuille’s law

Unit VII · Waves & oscillations

Stationary waves & Doppler effect

Formula	Use
$\lambda_n=\frac{2L}{n}$ $f_n=\frac{nv}{2L}=nf_1$	String fixed at both ends
$f_1=\frac{1}{2L}\sqrt{\frac{F}{\mu}}$	Fundamental frequency of a string
$\text{Open pipe:}\quad\lambda_n=\frac{2L}{n}$ $f_n=\frac{nv}{2L}$	Open air column
$\text{Closed pipe:}\quad\lambda_n=\frac{4L}{n}$ $f_n=\frac{nv}{4L},\quad n\text{ odd}$	Closed air column
$v'=v\pm v_w$	Speed of sound with/against wind
$f_a=f_0\left(\frac{v\pm v_o}{v\mp v_s}\right)$	General Doppler formula (choose signs for approach)
$\frac{\Delta\lambda}{\lambda_0}=\frac{v_s}{c}$	Doppler shift for light at low speeds

Unit I · Vectors & equilibrium

Vectors

Formula	Use
$\|\vec{r}\|=\sqrt{a^2+b^2}$ $\|\vec{r}\|=\sqrt{a^2+b^2+c^2}$	Magnitude in 2D / 3D
$\vec{r}=a\hat{\imath}+b\hat{\jmath}\;(+c\hat{k})$	Position vector
$R_x=R\cos\theta$ $R_y=R\sin\theta$	Rectangular components
$R=\sqrt{R_x^2+R_y^2}$ $\theta=\tan^{-1}\left(\frac{R_y}{R_x}\right)$	Resultant magnitude and direction
$R=\sqrt{A^2+B^2+2AB\cos\theta}$	Law of cosines for vector resultant
$\vec{A}\cdot\vec{B}=AB\cos\theta=A_xB_x+A_yB_y+A_zB_z$	Dot product
$\vec{A}\times\vec{B}=AB\sin\theta\,\hat{n}$	Cross product

Unit II · Force & motion

Kinematics, momentum & horizontal projectiles

Formula	Use
$\vec{v}_{\mathrm{av}}=\frac{\Delta\vec{d}}{\Delta t}$ $v=\frac{\Delta S}{\Delta t}$ $a_{\mathrm{av}}=\frac{\Delta v}{\Delta t}$	Velocity, speed and acceleration
$v_f=v_i+at$ $S=v_it+\frac{1}{2}at^2$ $v_f^2-v_i^2=2aS$	Equations of motion
$\vec{F}=m\vec{a}=\frac{\Delta\vec{p}}{\Delta t}$ $\vec{J}=\vec{F}\Delta t=\Delta\vec{p}$	Newton’s second law and impulse
$\vec{p}=m\vec{v}$ $m_1u_1+m_2u_2=m_1v_1+m_2v_2$	Momentum and its conservation
$F_{\mathrm{friction}}=\mu_s N$	Limiting static friction
$U=mgh$ $\text{gravitational potential}=gh$	Gravitational potential energy / potential
$v_{fx}=v_i$ $v_{fy}=gt$ $v_f=\sqrt{v_{fx}^2+v_{fy}^2}$	Horizontal projectile motion
$\theta=\tan^{-1}\left(\frac{v_{fy}}{v_{fx}}\right)$ $R=vt$	Direction and range of horizontal projectile

Unit III · Work & energy

Work, energy & power

Formula	Use
$W=Fd\cos\theta$ $W=\tau\omega$	Work in linear / rotational motion
$W=P\Delta V$	Pressure–volume work
$W_{\mathrm{spring}}=\frac{1}{2}kx^2$ $E_{\mathrm{inductor}}=\frac{1}{2}LI^2$	Stored energy in spring / inductor
$\mathrm{KE}=\frac{1}{2}mv^2=\frac{1}{2}pv=\frac{p^2}{2m}$	Kinetic energy forms
$Fd=\frac{1}{2}m(v_f^2-v_i^2)$ $Fd=mg(h_2-h_1)$	Work–energy principle
$v=\sqrt{2gh}$	Speed after falling through height h in vacuum
$U=-\frac{GMm}{r}$ $V_{\mathrm{grav}}=-\frac{GM}{r}$	Gravitational potential energy / potential
$P=\frac{W}{t}=Fv\cos\theta=\tau\omega$	Power
$\eta=\frac{\text{output}}{\text{input}}\times100$	Efficiency

Unit XV · Atomic spectra

Bohr model & atomic spectra

Formula	Use
$r_n=\frac{n^2a_0}{Z}$	Radius of nth Bohr orbit for hydrogen-like atom
$E_n=-\frac{13.6Z^2}{n^2}\,\mathrm{eV}$	Energy of nth Bohr orbit
$v_n=2.18\times10^6\frac{Z}{n}\,\mathrm{m\,s^{-1}}$	Electron speed in nth orbit
$L_n=\frac{nh}{2\pi}$	Angular momentum quantisation
$\mathrm{KE}=\frac{ke^2}{2r_n}$ $\mathrm{PE}=-\frac{ke^2}{r_n}$ $\mathrm{TE}=-\frac{ke^2}{2r_n}$	Energy relations in Bohr orbit
$f\propto\frac{Z^2}{n^3}$ $T\propto\frac{n^3}{Z^2}$	Orbital frequency and period trends
$\bar{\nu}=R_H\left(\frac{1}{p^2}-\frac{1}{n^2}\right)$	Rydberg wavenumber relation
$\lambda_{\mathrm{limit}}=\frac{p^2}{R_H}$	Series-limit wavelength
$N_{\mathrm{max}}=\frac{(n-p+1)(n-p)}{2}$	Maximum transitions between two orbits

Unit IX · Current electricity

Current electricity

Formula	Use
$I=\frac{\Delta Q}{\Delta t}=\frac{ne}{\Delta t}$ $I=fe$	Electric current
$v_d=\frac{I}{nAe}$	Drift velocity
$V=IR$ $R=\frac{\rho L}{A}$ $G=\frac{\sigma A}{L}$	Ohm’s law, resistance and conductance
$R=\frac{\rho L^2}{V}=\frac{\rho V}{A^2}$	Resistance forms using volume
$R_t=R_0(1+\alpha\Delta T)$ $\rho_t=\rho_0(1+\alpha\Delta T)$	Temperature dependence
$P=IV=\frac{V^2}{R}=I^2R$	Electric power
$E_{\mathrm{heat}}=\frac{V^2t}{R}=I^2Rt$	Electrical energy / heat
$R_{\mathrm{series}}=R_1+R_2+\cdots$ $\frac{1}{R_{\mathrm{parallel}}}=\frac{1}{R_1}+\frac{1}{R_2}+\cdots$	Equivalent resistance
$V_n=V\left(\frac{R_n}{R_{\mathrm{total}}}\right)$ $I_n=I\left(\frac{R_{\mathrm{eq}}}{R_n}\right)$	Voltage divider and current divider
$E_{\mathrm{series}}=E_1+E_2+\cdots$ $r_{\mathrm{series}}=r_1+r_2+\cdots$	Cells in series

Formula sheets.