BSEH SS Physics cheat sheet

Coulomb's law: F = kq₁q₂/r² with k = 1/(4πε₀) ≈ 9×10⁹ N·m²/C²; force is along the line joining charges and is repulsive for like signs.

Electric field of a point charge E = kq/r²; field lines leave positive charge and enter negative charge, never crossing each other.

Gauss's law: ∮E·dA = q_enclosed/ε₀; use it for spheres, infinite lines (E = λ/2πε₀r) and infinite sheets (E = σ/2ε₀).

Potential from a point charge V = kq/r; E = −dV/dr, so field points from high to low potential and is zero inside a conductor in equilibrium.

Capacitance: C = Q/V; parallel plate C = ε₀A/d, with dielectric C = Kε₀A/d; series 1/C = Σ1/Cᵢ, parallel C = ΣCᵢ.

Energy stored U = ½CV² = ½QV = Q²/2C; energy density in field u = ½ε₀E².

Conductor facts: charge resides on the outer surface, field just outside is σ/ε₀, and sharp points have higher surface charge density.

Dipole moment p = q·2a; torque τ = pE sinθ, potential energy U = −pE cosθ; dipole field falls as 1/r³.

Ohm's law V = IR holds for ohmic conductors at constant temperature; drift velocity v_d = I/(nAe) is typically ~10⁻⁴ m/s while signal speed is near c.

Resistivity ρ = RA/L; for metals ρ rises with temperature, for semiconductors it falls; R = ρL/A for a uniform wire.

Series: same current, R_eq = ΣR; parallel: same voltage, 1/R_eq = Σ1/R.

Kirchhoff's rules: junction rule ΣI = 0 (charge conservation) and loop rule ΣΔV = 0 (energy conservation).

EMF and internal resistance: I = ε/(R + r); terminal voltage V = ε − Ir; maximum power transfer to load when R = r.

Wheatstone bridge balanced when P/Q = R/S (no galvanometer current); metre bridge uses R/S = l/(100 − l).

Power P = VI = I²R = V²/R; heat H = I²Rt (Joule's law).

Potentiometer compares EMFs by balance lengths: ε₁/ε₂ = l₁/l₂, because at balance no current is drawn from the cell.

Biot–Savart: dB = (μ₀/4π)(I dl × r̂)/r²; long straight wire B = μ₀I/2πr; centre of a circular loop B = μ₀I/2R.

Solenoid interior B = μ₀nI; toroid B = μ₀NI/2πr; Ampere's law ∮B·dl = μ₀I_enclosed.

Force on moving charge F = qv × B, giving circular motion of radius r = mv/qB and cyclotron frequency f = qB/2πm.

Force on a current element F = IL × B; parallel currents attract, antiparallel repel, with force per length μ₀I₁I₂/2πd.

Torque on a current loop τ = NIAB sinθ; magnetic moment m = NIA; potential energy U = −m·B.

Magnetic dipole field falls as 1/r³; bar magnet equivalents: field lines run from N to S outside the magnet.

Earth's magnetism: declination, dip (inclination), and horizontal component B_H = B cosδ; tangent galvanometer and moving-coil galvanometer deflection obey I ∝ tanθ and I ∝ sinθ respectively.

Conversion of galvanometer to ammeter uses a low shunt S = I_gG/(I − I_g); to voltmeter uses a high series R = V/I_g − G.

Faraday's law: ε = −dΦ/dt; Lenz's law fixes the direction so induced current opposes the change in flux.

Motional EMF: ε = Bvl for a rod moving perpendicular to B; ε = Blv sinθ in general.

Self-inductance ε = −L dI/dt, energy U = ½LI²; solenoid L = μ₀n²Al; mutual inductance ε₂ = −M dI₁/dt.

AC through R: current in phase; through L: current lags by 90°; through C: current leads by 90°; reactances X_L = ωL, X_C = 1/ωC.

Series LCR impedance Z = √(R² + (X_L − X_C)²); resonance at ω = 1/√(LC), where Z = R and current is maximum.

Power in AC: P = V_rms I_rms cosφ; power factor cosφ = R/Z; wattless current flows when φ = 90°.

Transformer: V_s/V_p = N_s/N_p = I_p/I_s (ideal); step-up raises voltage, lowers current, and power is conserved.

RMS values: V_rms = V₀/√2, I_rms = I₀/√2; average of AC over a full cycle is zero.

Maxwell's displacement current I_d = ε₀ dΦ_E/dt completes Ampere's law and predicts electromagnetic waves.

EM waves are transverse: E ⊥ B ⊥ direction of propagation, with E and B in phase; speed c = 1/√(μ₀ε₀) ≈ 3×10⁸ m/s in vacuum.

In a medium v = 1/√(με) = c/n; E₀/B₀ = c, and intensity I = ½ε₀cE₀².

Energy is shared equally on average between electric and magnetic fields; momentum p = U/c and radiation pressure = I/c (absorbed) or 2I/c (reflected).

Spectrum order by increasing wavelength: gamma, X-ray, UV, visible (400–700 nm), IR, microwave, radio.

Displacement current arises where E changes with time, e.g. between capacitor plates, and produces the same magnetic effect as conduction current.

Mirror formula 1/v + 1/u = 1/f with f = R/2; lens formula 1/v − 1/u = 1/f; magnification m = v/u (sign conventions per Cartesian rule).

Refraction: Snell's law n₁sinθ₁ = n₂sinθ₂; critical angle sinθ_c = 1/n; apparent depth = real depth/n.

Lens maker's formula 1/f = (n − 1)(1/R₁ − 1/R₂); power P = 1/f (dioptre, f in metres); thin lenses in contact add powers.

Prism: deviation A(n − 1) for small angles; at minimum deviation n = sin((A + D_m)/2)/sin(A/2); dispersion gives angular separation of colours.

Young's double slit: fringe width β = λD/d; bright when path difference = nλ, dark when (2n − 1)λ/2.

Single-slit diffraction: first minimum at a sinθ = λ; resolving power of a telescope = D/1.22λ.

Polarisation by reflection obeys Brewster's law tanθ_p = n; Malus's law I = I₀cos²θ through an analyser.

Huygens' principle: each wavefront point is a secondary source; it explains reflection, refraction, and diffraction but not polarisation.

Photoelectric effect: maximum kinetic energy K_max = hν − φ, with threshold frequency ν₀ = φ/h; current depends on intensity, K_max on frequency.

Photon energy E = hν = hc/λ; h = 6.63×10⁻³⁴ J·s; 1 eV = 1.6×10⁻¹⁹ J; λ(nm) ≈ 1240/E(eV).

Stopping potential eV₀ = hν − φ; the graph of V₀ versus ν has slope h/e.

de Broglie wavelength λ = h/p = h/mv; for an electron accelerated by V volts, λ ≈ 1.227/√V nm.

Davisson–Germer experiment confirmed electron diffraction, verifying wave nature of matter.

Photoelectric emission is instantaneous (no time lag) and one photon ejects one electron, contradicting classical wave predictions.

Bohr model: angular momentum mvr = nh/2π; radius r_n = 0.53n²/Z Å; energy E_n = −13.6Z²/n² eV.

Hydrogen spectral series: Lyman (UV), Balmer (visible), Paschen, Brackett, Pfund (IR); 1/λ = R(1/n₁² − 1/n₂²), R = 1.097×10⁷ m⁻¹.

Nuclear radius R = R₀A^(1/3) with R₀ ≈ 1.2 fm; density is roughly constant for all nuclei (~2.3×10¹⁷ kg/m³).

Binding energy per nucleon peaks near A ≈ 56 (iron), so fusion of light nuclei and fission of heavy nuclei both release energy.

Radioactive decay: N = N₀e^(−λt); half-life T½ = 0.693/λ; activity A = λN.

Alpha decay reduces A by 4 and Z by 2; beta-minus raises Z by 1 with A unchanged; gamma emission changes neither.

Mass–energy: E = Δmc²; 1 u = 931.5 MeV; Q value of a reaction equals the mass defect times c².

Intrinsic semiconductors: n_e = n_h; doping with pentavalent donors gives n-type (majority electrons), trivalent acceptors give p-type (majority holes).

At the p–n junction a depletion layer and barrier potential form (≈0.7 V for Si, ≈0.3 V for Ge); forward bias narrows it, reverse bias widens it.

Diode I–V: negligible current below the knee, sharp rise in forward bias, small saturation current in reverse bias until breakdown.

Half-wave rectifier output frequency equals input frequency; full-wave rectifier output frequency is twice the input frequency.

Zener diode works in reverse breakdown as a voltage regulator; a photodiode is reverse biased; an LED is forward biased.

Transistor: I_E = I_B + I_C; current gains α = I_C/I_E and β = I_C/I_B, with β = α/(1 − α); active mode needs emitter–base forward, collector–base reverse bias.

Logic gates: AND (Y = A·B), OR (Y = A + B), NOT (Y = Ā); NAND and NOR are universal gates.

Energy gap: conductors have overlapping bands, insulators a large gap (>3 eV), semiconductors a small gap (~1 eV, Si ≈ 1.1 eV).

Least count: vernier = 1 main scale division/number of vernier divisions; screw gauge = pitch/number of circular scale divisions.

Zero error corrections: vernier zero error is added with opposite sign; screw gauge zero error is subtracted from the observed reading.

Metre bridge: unknown resistance R = S·l/(100 − l); interchange gaps and average to cancel end corrections.

Ohm's law experiment plots V versus I; the slope gives resistance, and a straight line through the origin confirms ohmic behaviour.

Galvanometer figure of merit k = E/((R + G)θ) from a half-deflection or full-deflection setup; use it to find G.

Focal length of a convex mirror or concave lens uses a convex lens auxiliary; plot u–v graphs and use 1/f = 1/v − 1/u.

Error propagation: for Z = A·B/C, fractional errors add; for Z = A ± B, absolute errors add.

Significant figures and instrument precision determine reporting; a result should not claim more digits than the least precise measurement.

Vector dot product A·B = AB cosθ (scalar); cross product |A × B| = AB sinθ (vector, right-hand rule).

Kinematics and dynamics with calculus: v = dx/dt, a = dv/dt; use v dv = a dx when time is not given.

Differentiation rules: d/dx(xⁿ) = nxⁿ⁻¹, d/dx(sin x) = cos x, product and chain rules for composite expressions.

Integrals: ∫xⁿ dx = xⁿ⁺¹/(n+1), ∫dx/x = ln x, ∫sin x dx = −cos x; definite integrals give areas and average values.

Small-angle approximations sinθ ≈ θ, tanθ ≈ θ, cosθ ≈ 1 − θ²/2 for θ in radians.

Graph interpretation: slope of x–t is velocity, area under v–t is displacement, slope of V–I is resistance.

Dimensional analysis checks equations and converts units; dimensional consistency is necessary but not sufficient.

Newton's laws: F = ma in inertial frames; action–reaction pairs act on different bodies and never cancel on one body.

Work–energy theorem W = ΔK; conservative forces have potential energy, and mechanical energy is conserved when only they act.

Momentum p = mv is conserved in all collisions; kinetic energy is conserved only in elastic collisions.

Circular motion: centripetal acceleration a = v²/r; banking angle tanθ = v²/rg.

SHM: x = A sin(ωt), ω = 2π/T; for a spring ω = √(k/m), for a pendulum ω = √(g/L) for small swings.

Thermodynamics: ΔQ = ΔU + ΔW; efficiency of a heat engine η = 1 − T_c/T_h (Carnot).

Waves: v = fλ; beats frequency = |f₁ − f₂|; Doppler shift Δf/f = v_source/v for a receding source.

Gravitation: F = Gm₁m₂/r²; orbital speed v = √(GM/r); escape speed v_e = √(2GM/R) = √2 v_orbit.