Formula Cheat Sheet

v = linear speed (m s⁻¹); ω = angular speed (rad s⁻¹); r = radius (m); T = time period of one revolution (s); f = frequency = 1/T (Hz); π = dimensionless constant ≈ 3.14159.

a_c = centripetal (radial) acceleration toward the centre (m s⁻²); v = linear speed (m s⁻¹); r = radius (m); ω = angular speed (rad s⁻¹).

F_c = centripetal net inward force in the inertial frame, and the same magnitude for the outward centrifugal pseudo-force in a rotating frame (N); m = mass (kg); v = speed (m s⁻¹); r = radius (m); ω = angular speed (rad s⁻¹).

v_max = maximum speed before slipping (m s⁻¹); μₛ = coefficient of static friction (dimensionless, 0–1); g = acceleration due to gravity (m s⁻², ≈ 9.8); r = radius of the track (m).

v_min = minimum speed to stay on a vertical wall (m s⁻¹); r = radius (m); g = gravity (m s⁻²); μₛ = coefficient of static friction (dimensionless).

v = safe speed (m s⁻¹); r = radius of the curve (m); g = gravity (m s⁻²); θ = angle of banking (rad or deg; take it in degrees with tanθ in radians convention).

v₁ = lower safe-speed limit and v₂ = upper safe-speed limit (m s⁻¹); r = radius (m); g = gravity (m s⁻²); θ = banking angle (deg/rad as above); μₛ = coefficient of static friction (dimensionless). Valid when the radicands are positive (1 − μₛ tanθ > 0 and tanθ ≥ μₛ).

T = time period (s); L = string length (m); θ = semi-vertical angle, angle between string and the vertical (deg/rad convention); g = gravity (m s⁻²); T₀ = tension in the string (N); m = bob mass (kg). π ≈ 3.14159 (dimensionless).

Subscripts A = top, B = bottom, C = horizontal midpoint. v = speed (m s⁻¹), T = tension (N) at that point, m = mass (kg), r = radius (m), g = gravity (m s⁻²). Limiting case where the particle just completes the circle; T_B − T_A = 6mg.

I = moment of inertia about the axis (kg m²); mᵢ = mass of the i-th particle (kg); rᵢ = its perpendicular distance from the axis (m); dm = mass element (kg); r = distance from axis (m); K = radius of gyration (m); M = total mass (kg).

Iₒ = moment of inertia about any axis (kg m²); I_c = moment of inertia about a parallel axis through the centre of mass (kg m²); M = total mass (kg); h = perpendicular distance between the axes (m).

For a planar lamina in the xy-plane: I_z = moment of inertia about the perpendicular z-axis, I_x and I_y about two perpendicular axes in the plane (all kg m²).

τ = torque (N m); I = moment of inertia (kg m²); α = angular acceleration (rad s⁻²); L = angular momentum (kg m² s⁻¹); ω = angular velocity (rad s⁻¹); KE = rotational kinetic energy (J); W = work (J); θ = angle rotated (rad); P = power (W).

τ_ext = net external torque (N m); L = angular momentum (kg m² s⁻¹); I = moment of inertia (kg m²); ω = angular speed (rad s⁻¹). If no external torque, L is conserved (e.g. a skater pulling arms in).

L = angular momentum about an axis (kg m² s⁻¹); m = mass (kg); v = linear speed (m s⁻¹); r = distance from the axis (m); θ = angle between position and velocity vectors.

E = total kinetic energy (J); M = mass (kg); v = centre-of-mass speed (m s⁻¹); K = radius of gyration (m); R = rolling (outer) radius (m); a = acceleration down the incline (m s⁻²); θ = incline angle (deg/rad); g = gravity (m s⁻²); h = vertical height (m). K² = I/M.

P = pressure (Pa = N m⁻²); F = normal force (N); A = area (m²); P₀ = pressure at the surface (Pa); ρ = fluid density (kg m⁻³); g = gravity (m s⁻²); h = depth (m).

F_b = buoyant force (N); ρ_f = density of the displaced fluid (kg m⁻³); V = volume of fluid displaced (m³); g = gravity (m s⁻²).

A₁, A₂ = cross-section areas (m²); v₁, v₂ = flow speeds (m s⁻¹); Q = volume flow rate (m³ s⁻¹) measured through area A at speed v.

P = pressure (Pa); ρ = density (kg m⁻³); v = flow speed (m s⁻¹); g = gravity (m s⁻²); h = height (m). Holds along a streamline for steady, incompressible, non-viscous flow.

v = speed of liquid emerging from an opening (m s⁻¹); g = gravity (m s⁻²); h = height of the free surface above the opening (m).

F = viscous force (N); η = coefficient of viscosity (Pa s = kg m⁻¹ s⁻¹; 1 Pa s = 10 poise); A = area of the layer (m²); dv/dx = velocity gradient (s⁻¹).

F = viscous drag (N); π ≈ 3.14159 (dimensionless); r = sphere radius (m); η = viscosity (Pa s); v = sphere speed (m s⁻¹). Valid for slow motion at low Reynolds number.

v_t = terminal (constant) fall speed (m s⁻¹); r = sphere radius (m); ρₛ = density of the sphere (kg m⁻³); ρ_f = density of the fluid (kg m⁻³); g = gravity (m s⁻²); η = viscosity (Pa s); 9 dimensionless coefficient.

T = surface tension (N m⁻¹); F = force along the surface (N); L = length of the surface line on which the force acts (m).

dW = work done to increase surface area (J); T = surface tension (N m⁻¹); dA = increase in surface area (m²). For a soap film with two surfaces the relevant area doubles.

ΔP = excess pressure just inside the curved surface over outside (Pa); T = surface tension (N m⁻¹); r = radius of curvature (m). A bubble has two surfaces, hence the extra factor of 2.

h = capillary rise (m); T = surface tension (N m⁻¹); θ = angle of contact (deg/rad; wetting when < 90°, non-wetting when > 90°); r = tube radius (m); ρ = liquid density (kg m⁻³); g = gravity (m s⁻²).

P = pressure (Pa); V = volume (m³); n = number of moles (mol); R = universal gas constant (8.314 J mol⁻¹ K⁻¹); T = absolute temperature (K); N = number of molecules (dimensionless count); k_B = Boltzmann constant (1.38 × 10⁻²³ J K⁻¹).

P = pressure (Pa); ρ = gas density (kg m⁻³); v_rms = root-mean-square speed (m s⁻¹); n = molecule number density (m⁻³); m = mass of one molecule (kg).

v_rms = root-mean-square speed (m s⁻¹); R = gas constant (8.314 J mol⁻¹ K⁻¹); T = absolute temperature (K); M = molar mass (kg mol⁻¹); k_B = Boltzmann constant (J K⁻¹); m = molecule mass (kg).

v_avg = mean speed (m s⁻¹); v_mp = most probable speed (m s⁻¹); v_rms = rms speed (m s⁻¹); R = gas constant (J mol⁻¹ K⁻¹); T = temperature (K); M = molar mass (kg mol⁻¹); π ≈ 3.14159 (dimensionless).

m = molecule mass (kg); v_rms = rms speed (m s⁻¹); k_B = Boltzmann constant (1.38 × 10⁻²³ J K⁻¹); T = absolute temperature (K); overline KE = mean translational kinetic energy per mole (J); R = gas constant (8.314 J mol⁻¹ K⁻¹).

U = internal energy (J); f = degrees of freedom (dimensionless integer); n = number of moles (mol); R = gas constant (J mol⁻¹ K⁻¹); T = temperature (K); γ = ratio of specific heats (dimensionless); C_P, C_V = molar specific heats at constant pressure/volume (J mol⁻¹ K⁻¹).

E = power radiated per unit surface area (W m⁻²); σ_SB = Stefan-Boltzmann constant (5.67 × 10⁻⁸ W m⁻² K⁻⁴); T = absolute temperature (K); P_net = net radiated power (W); A = surface area (m²); T₀ = surrounding temperature (K).

λ_max = wavelength of peak emission (m); T = absolute temperature (K); b = Wien's constant (2.898 × 10⁻³ m K).

e_λ = spectral emissive power (dimensionless fraction of ideal blackbody at wavelength λ); a_λ = spectral absorptivity (dimensionless, fraction absorbed at the same wavelength and direction). A good absorber is a good emitter.

Féry's blackbody: a hollow box with a small opening traps incoming rays by repeated reflection, so the hole absorbs (almost) all incident radiation and radiates as a blackbody.

dQ = heat added to the system (J); dU = change in internal energy (J); dW = work done by the system (J). Convention: dQ > 0 heat in; dW > 0 work done by the system.

U = internal energy (J); f = degrees of freedom (dimensionless); n = moles (mol); R = gas constant (J mol⁻¹ K⁻¹); T = temperature (K); dU = change in internal energy (J); C_V = molar specific heat at constant volume (J mol⁻¹ K⁻¹); dT = change in temperature (K).

C_P, C_V = molar specific heats at constant pressure and volume (J mol⁻¹ K⁻¹); R = gas constant (8.314 J mol⁻¹ K⁻¹); γ = ratio of specific heats (dimensionless).

W = work done by the gas in an isothermal (constant temperature) process (J); n = moles (mol); R = gas constant (J mol⁻¹ K⁻¹); T = absolute temperature (K); V₁, V₂ = initial and final volumes (m³); ln = natural logarithm (dimensionless).

P = pressure (Pa); V = volume (m³); γ = Cp/Cv (dimensionless); T = absolute temperature (K); W = work done by the gas (J); P₁V₁, P₂V₂ = initial and final (pressure, volume) states. Adiabatic means no heat exchange.

W = work (J); P = constant pressure (Pa); ΔV = change in volume (m³); Q = heat added (J); ΔU = change in internal energy (J). Isochoric means constant volume (no work).

η = efficiency (dimensionless, 0–1); Q₁ = heat absorbed from the hot reservoir (J); Q₂ = heat rejected to the cold reservoir (J); W = useful work = Q₁ − Q₂ (J).

η = maximum (Carnot) efficiency (dimensionless); T₁ = absolute temperature of the hot reservoir (K); T₂ = absolute temperature of the cold reservoir (K).

COP = coefficient of performance (dimensionless); Q₂ = heat extracted from the cold space (J); W = work input (J); T₂ = cold-reservoir temperature (K); T₁ = hot-reservoir temperature (K).

dS = entropy change (J K⁻¹); dQ_rev = heat added reversibly (J); T = absolute temperature (K). Total entropy of an isolated system never decreases.

F = force between two point charges (N); π ≈ 3.14159 (dimensionless); ε₀ = permittivity of free space (8.85 × 10⁻¹² C² N⁻¹ m⁻²); q₁, q₂ = charges (C); r = separation (m).

E = electric field (N C⁻¹ = V m⁻¹); F = force on the charge (N); q₀ = positive test charge (C); π, ε₀ as above; q = source charge (C); r = distance (m).

E = electric field (V m⁻¹); dV/dr = rate of change of electric potential with distance along the field (V m⁻¹); the minus sign means E points toward decreasing potential.

∮E·dA = electric flux through a closed surface (N m² C⁻¹); E = electric field (N C⁻¹); dA = area element vector (m²); q_enclosed = net charge inside the surface (C); ε₀ = permittivity of free space (C² N⁻¹ m⁻²).

E_inside = field within a conductor/spherical shell (N C⁻¹), zero in electrostatic equilibrium; E_outside = field at distance r (N C⁻¹); Q = charge on the shell (C); r = distance from the centre, r ≥ radius (m); ε₀ = permittivity of free space.

E_in = field inside at radius r (N C⁻¹, r ≤ R); E_out = field outside (N C⁻¹, r ≥ R); Q = total charge (C); r = distance from the centre (m); R = sphere radius (m); ε₀ = permittivity of free space.

E_line = field of an infinite charged line (N C⁻¹); λ = linear charge density (C m⁻¹); r = perpendicular distance (m); E_sheet = field of an infinite plane sheet (N C⁻¹); σ = surface charge density (C m⁻²); ε₀ = permittivity of free space. Between two oppositely charged parallel plates, E = σ/ε₀.

V = electric potential (V = J C⁻¹); q = charge (C); r = distance (m); ε₀ = permittivity of free space.

V_d = potential of a dipole far away (V); p = dipole moment = qd (C m); θ = angle from the dipole axis (rad/deg); r = distance (m); k = 1/(4πε₀) ≈ 9 × 10⁹ (N m² C⁻²); E_axis = field on the dipole axis and E_eq = equatorial field (V m⁻¹). Valid approximation for r ≫ separation d.

τ = torque (N m); p = dipole moment (C m); E = field (V m⁻¹); θ = angle between p and E (rad/deg); U = potential energy (J). Stable at θ = 0, unstable at θ = 180°.

C = capacitance (F); Q = charge stored (C); V = potential difference (V); ε₀ = permittivity of free space (F m⁻¹); A = plate area (m²); d = plate separation (m).

C = capacitance with dielectric (F); κ = relative permittivity (dimensionless, > 1); C₀ = capacitance without dielectric (F); ε₀ = permittivity of free space; A = plate area (m²); d = separation (m). Full filling of the gap is assumed.

C_eq = equivalent capacitance (F); C_i = individual capacitances (F). Series: reciprocal sum; parallel: direct sum.

U = stored electrostatic energy (J); Q = charge (C); V = potential difference (V); C = capacitance (F); u = energy density (J m⁻³); ε₀ = permittivity of free space; E = field between plates (V m⁻¹).

V = potential difference (V); I = current (A); R = resistance (Ω); ρ = resistivity (Ω m); l = length (m); A = cross-section area (m²). Conductance G = 1/R (S).

R_eq = equivalent resistance (Ω); R_i = individual resistors (Ω). Series: direct sum (same current); parallel: reciprocal sum (same voltage).

I = current (A); n = number density of free electrons (m⁻³); A = cross-section area (m²); e = magnitude of electron charge (1.6 × 10⁻¹⁹ C); v_d = drift velocity (m s⁻¹); J = current density (A m⁻²); σ = conductivity (S m⁻¹); E = electric field (V m⁻¹).

V = terminal potential difference (V); ε = emf of the cell (V); I = current (A); r = internal resistance of the cell (Ω); R = external load resistance (Ω).

Junction rule (charge conservation): the algebraic sum of currents at a junction is zero, I in ampere (A). Loop rule (energy conservation): around any closed loop, the sum of emfs ε (V) equals the sum of IR drops, I = current (A), R = resistance (Ω).

P, Q, S, R = the four arm resistances (Ω) in the diamond. Balanced (galvanometer reads zero) when P/Q = R/S.

X = unknown resistance (Ω); R = known resistance-box value (Ω); l_x = balancing length on the X side (cm/m); l_r = balancing length on the R side (cm/m), with l_x + l_r = 100 cm.

K = potential gradient (V m⁻¹); V = potential difference across the wire (V); L = wire length (m); E₁, E₂ = emfs (V); l₁, l₂ = balancing lengths (m). Also E = K·l.

E₁, E₂ = emfs (V); l₁, l₂ = balancing lengths (m); r = internal resistance of the cell (Ω); R = external resistance box value (Ω).

P = power (W); V = voltage (V); I = current (A); R = resistance (Ω); W = electrical energy (J); t = time (s). Energy in kWh = P(kW) × t(h).

S = shunt resistance, connected in parallel (Ω); I_g = full-scale current of the galvanometer (A); G = galvanometer resistance (Ω); I = new ammeter range (A).

R = series resistance added (Ω); V = new voltmeter range (V); I_g = full-scale galvanometer current (A); G = galvanometer resistance (Ω).

dB = magnetic field contribution of a current element (T); μ₀ = 4π × 10⁻⁷ (N A⁻²); I = current (A); dl = length of element (m); θ = angle between dl and the radius vector (rad/deg); r = distance (m); π ≈ 3.14159.

B = magnetic field at distance r (T); μ₀ = 4π × 10⁻⁷ (N A⁻²); I = current (A); π ≈ 3.14159; r = perpendicular distance from a very long wire (m).

B = magnetic field (T); μ₀ = 4π × 10⁻⁷ (N A⁻²); N = number of turns (dimensionless); I = current (A); R = loop radius (m); x = distance along the axis from the centre (m). At the centre (x = 0) the axis formula reduces to the centre value.

∮B·dl = line integral of the field around a closed loop (T m); B = magnetic field (T); dl = path element (m); μ₀ = 4π × 10⁻⁷ (N A⁻²); I_enclosed = net current threading the loop (A).

B = field inside an ideal, very long solenoid (T); μ₀ = 4π × 10⁻⁷ (N A⁻²); n = N/L = turns per unit length (m⁻¹); I = current (A). Nearly zero field outside.

B = field inside the toroid at radius r (T); μ₀ = 4π × 10⁻⁷ (N A⁻²); N = total turns (dimensionless); I = current (A); π ≈ 3.14159; r = distance from the axis inside the toroid (m).

F = magnetic force (N); q = moving charge (C); v = speed (m s⁻¹); B = magnetic field (T); θ = angle between v and B (rad/deg). Force is perpendicular to both (does no work).

F = magnetic force (N); B = field (T); I = current (A); l = wire length in the field (m); θ = angle between the wire and B (rad/deg).

F/l = force per unit length (N m⁻¹); μ₀ = 4π × 10⁻⁷ (N A⁻²); I₁, I₂ = currents (A); π ≈ 3.14159; d = separation between the wires (m). Like (parallel) currents attract; opposite currents repel.

θ = deflection (rad or division); N = number of turns (dimensionless); B = radial field (T); A = coil area (m²); C = restoring (torsional) constant (N m rad⁻¹); I = current (A).

S = shunt resistance in parallel to make an ammeter (Ω); I_g = full-scale galvanometer current (A); G = galvanometer resistance (Ω); I = ammeter range (A); R = series resistance to make a voltmeter (Ω); V = voltmeter range (V).

m = magnetic dipole moment (A m² = J T⁻¹); q_m = pole strength (A m); d = vector from the south to the north pole, length = pole separation (m).

F = force between two magnetic poles (N); μ₀ = 4π × 10⁻⁷ (N A⁻²); q_m1, q_m2 = pole strengths (A m); r = separation (m); π ≈ 3.14159.

M = magnetization = net dipole moment per unit volume (A m⁻¹); m_net = net dipole moment (A m²); V = volume (m³); H = magnetic intensity, the applied field (A m⁻¹).

B = total magnetic flux density inside the material (T); μ₀ = 4π × 10⁻⁷ (N A⁻²); H = magnetic intensity (A m⁻¹); M = magnetization (A m⁻¹); χ = magnetic susceptibility (dimensionless).

χ = magnetic susceptibility (dimensionless), ratio of magnetization M (A m⁻¹) to magnetic intensity H (A m⁻¹); μ_r = relative permeability (dimensionless); μ = absolute permeability (N A⁻²); μ₀ = 4π × 10⁻⁷ (N A⁻²).

τ = torque (N m); m = dipole moment (A m²); B = field (T); θ = angle between m and B (rad/deg); U = potential energy (J). τ_max = mB at θ = 90°; zero at θ = 0°, 180°; stable at 0°, unstable at 180°.

T = time period of small angular oscillations (s); π ≈ 3.14159; I = moment of inertia of the magnet (kg m²); m = magnetic dipole moment (A m²); B = magnetic field (T). Valid for small oscillations about equilibrium.

I = orbital current (A); e = magnitude of electron charge (1.6 × 10⁻¹⁹ C); T = period of revolution (s); v = orbital speed (m s⁻¹); r = orbital radius (m); m_orb = orbital magnetic dipole moment (A m²).

m_orb = orbital magnetic moment (A m²); L = orbital angular momentum (kg m² s⁻¹); e = electron charge magnitude (1.6 × 10⁻¹⁹ C); m_e = electron mass (9.11 × 10⁻³¹ kg). Value ≈ 8.8 × 10¹⁰ C kg⁻¹.

μ_B = Bohr magneton (A m²); e = electron charge (1.6 × 10⁻¹⁹ C); h = Planck constant (6.626 × 10⁻³⁴ J s); π ≈ 3.14159; m_e = electron mass (9.11 × 10⁻³¹ kg).

χ = magnetic susceptibility (dimensionless); C = Curie constant (K, a material property); T = absolute temperature (K).

χ = susceptibility (dimensionless); C = Curie constant (K); T = absolute temperature (K); T_c = Curie temperature (K). Valid only for T > T_c, above which ferromagnetic order is lost and the material is paramagnetic.

χ_dia = diamagnetic susceptibility, small and negative (dimensionless); χ_para = paramagnetic, small positive; χ_ferro = ferromagnetic, large positive and field-dependent. μ_r is slightly below 1 (dia), slightly above 1 (para), and much greater than 1 (ferro).