BSEB Inter Physics cheat sheet

Coulomb's law: F = (1/4πε₀)·q₁q₂/r², with 1/4πε₀ = 9×10⁹ N·m²/C²; force acts along the line joining charges and is repulsive for like signs.

Electric field E = F/q₀ is a vector; field of a point charge is E = kq/r² pointing radially outward for positive q.

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

Potential V = kq/r is a scalar; E = −dV/dr, and work done in moving charge q through potential difference V is W = qV.

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

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

Dielectric slab inserted at constant charge lowers V and E by factor K; at constant battery voltage, charge and stored energy rise by K.

Conductors are equipotential surfaces; E just outside is σ/ε₀ and is normal to the surface. Field inside a hollow conductor is zero (electrostatic shielding).

Dipole moment p = q·2a; torque in uniform field τ = pE sinθ, potential energy U = −pE cosθ, axial field = 2kp/r³.

Ohm's law V = IR holds for ohmic conductors at constant temperature; drift velocity v_d = I/(nAe), with typical values ~10⁻⁴ m/s.

Current density J = I/A = nev_d = σE; resistivity ρ = m/(ne²τ) and resistance R = ρL/A.

Temperature dependence: metals ρ rises with T (ρ = ρ₀[1+αΔT]); semiconductors and electrolytes show negative α.

Series: R = ΣRᵢ, same current; parallel: 1/R = Σ1/Rᵢ, same voltage. Kirchhoff's junction rule conserves charge, loop rule conserves energy.

EMF and internal resistance: I = ε/(R+r); terminal voltage V = ε − Ir while discharging, V = ε + Ir while charging.

Wheatstone bridge balances when P/Q = R/S; metre bridge uses R/S = l/(100−l) with l the balance length in cm.

Potentiometer compares EMFs as ε₁/ε₂ = l₁/l₂ and measures internal resistance by r = R(l₁/l₂ − 1).

Cells in series give ε_eq = Σεᵢ, r_eq = Σrᵢ; in parallel with identical cells ε_eq = ε, r_eq = r/n.

Electrical power P = VI = I²R = V²/R; 1 kWh = 3.6×10⁶ J.

Biot–Savart law: dB = (μ₀/4π)·I dl sinθ/r², with μ₀/4π = 10⁻⁷ T·m/A.

Long straight wire B = μ₀I/2πr; circular loop at centre B = μ₀I/2R; solenoid B = μ₀nI; toroid B = μ₀NI/2πr.

Ampere's circuital law ∮B·dl = μ₀I_enclosed; force per unit length between parallel wires = μ₀I₁I₂/2πd, attractive for same-direction currents.

Force on a charge F = qv × B; force on a wire F = BIL sinθ. Radius of circular motion r = mv/qB, period T = 2πm/qB independent of speed.

Cyclotron frequency f = qB/2πm; maximum kinetic energy = q²B²R²/2m, limited because relativistic mass increase detunes resonance.

Torque on a current loop τ = NIAB sinθ; magnetic moment m = NIA. Moving-coil galvanometer: current sensitivity = NBA/k, voltage sensitivity = NBA/kR.

Ampere's conversion of galvanometer: shunt S = I_gG/(I−I_g) for ammeter; series R = V/I_g − G for voltmeter.

Bar magnet: axial field B = μ₀2M/4πr³, equatorial B = μ₀M/4πr³; torque τ = MB sinθ, energy U = −MB cosθ.

Earth's field has declination (angle between geographic and magnetic meridian) and dip; at magnetic poles dip is 90°.

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

Motional EMF in a rod moving with velocity v in field B is ε = Bvl, with induced current I = Bvl/R.

Self-inductance ε = −L dI/dt; solenoid L = μ₀n²Al. Mutual inductance ε₂ = −M dI₁/dt. Energy stored U = ½LI².

AC through resistor: current and voltage in phase; through inductor, voltage leads current by 90°; through capacitor, current leads voltage by 90°.

Reactances X_L = ωL, X_C = 1/ωC; impedance Z = √(R² + (X_L − X_C)²), tanφ = (X_L − X_C)/R.

Series resonance at ω₀ = 1/√(LC); at resonance Z = R is minimum and current maximum; Q-factor = ω₀L/R = (1/R)√(L/C).

RMS values: I_rms = I₀/√2, V_rms = V₀/√2; average AC power P = V_rms I_rms cosφ, where cosφ is power factor.

Transformer: V_s/V_p = N_s/N_p = I_p/I_s for an ideal device; step-up raises voltage and lowers current, power unchanged.

Eddy currents are reduced by laminating the core; a choke coil controls AC with little power loss because its phase difference is near 90°.

Maxwell's displacement current I_d = ε₀ dΦ_E/dt makes Ampere's law consistent with charge conservation and predicts EM waves.

EM wave speed c = 1/√(μ₀ε₀) ≈ 3×10⁸ m/s in vacuum; in a medium v = 1/√(με) = c/n.

In an EM wave E and B are perpendicular to each other and to the direction of propagation; E₀/B₀ = c, and both oscillate in phase.

Order of the spectrum by increasing frequency: radio, microwave, infrared, visible, ultraviolet, X-rays, gamma rays.

Intensity of EM wave: average energy density u = ½ε₀E₀², and intensity I = ½ε₀E₀²c.

Refraction: Snell's law n₁sin i = n₂sin r; critical angle sin C = 1/n for denser-to-rarer travel; total internal reflection occurs beyond C.

Lens and mirror formula: 1/v − 1/u = 1/f for a lens, 1/v + 1/u = 1/f for a mirror; magnification m = v/u (lens) or −v/u (mirror).

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

Single-slit diffraction minima at a sinθ = nλ; Brewster's law tan i_p = n; Malus's law I = I₀cos²θ for polarised light.

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

Photoelectric effect: emission occurs only above threshold frequency ν₀; maximum kinetic energy K_max = hν − φ, independent of intensity.

Stopping potential eV₀ = hν − φ; saturation photocurrent is proportional to intensity, not to frequency.

Photon momentum p = h/λ = E/c; radiation pressure follows from momentum transfer.

de Broglie wavelength λ = h/p = h/√(2mK) = h/mv; for electrons accelerated through V volts, λ ≈ 12.27/√V Å.

Davisson–Germer electron diffraction confirmed wave nature of matter; the heavier the particle, the smaller its de Broglie wavelength.

Einstein's photoelectric equation slope of V₀ versus ν graph equals h/e; intercept on ν axis gives ν₀, from which φ = hν₀.

Work function φ is a property of the metal surface; alkali metals have low φ and respond to visible light.

Bohr postulates: quantised angular momentum mvr = nh/2π and stationary orbits with no radiation; radiation only on transitions.

Bohr radius r_n = 0.53 n²/Z Å; energy E_n = −13.6 Z²/n² eV; velocity v_n = (c/137)(Z/n).

Spectral series of hydrogen: Lyman (ultraviolet, n→1), Balmer (visible, n→2), Paschen, Brackett, Pfund (infrared).

Rydberg formula 1/λ = R(1/n₁² − 1/n₂²), R = 1.097×10⁷ m⁻¹; ionisation energy of hydrogen is 13.6 eV.

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

Binding energy = Δm·c²; 1 u = 931.5 MeV. Binding energy per nucleon peaks near A = 56 (iron), so fusion of light and fission of heavy nuclei release energy.

Alpha decay reduces A by 4 and Z by 2; beta-minus raises Z by 1 with A unchanged; gamma emission changes neither A nor Z.

Half-life N = N₀(1/2)^(t/T); activity A = λN with λ = 0.693/T; mean life = 1/λ.

Mass–energy and nuclear reactions: Q-value = (mass of reactants − mass of products)c²; a positive Q means energy release.

Intrinsic semiconductors: n_e = n_h = n_i; conduction rises with temperature because more electron–hole pairs are generated.

Doping: pentavalent donors give n-type (electrons majority); trivalent acceptors give p-type (holes majority).

Mass action law n_e·n_h = n_i²; the Fermi level shifts toward the conduction band in n-type and the valence band in p-type.

p–n junction: diffusion creates a depletion layer and barrier potential (~0.7 V for Si, ~0.3 V for Ge); forward bias narrows it.

Diode I–V curve: negligible current below the knee voltage in forward bias; small reverse saturation current until breakdown.

Half-wave rectifier output frequency equals input frequency; full-wave rectifier doubles it; a filter capacitor smooths the ripple.

Zener diode works in reverse breakdown as a voltage regulator; a series resistor drops the excess voltage.

Transistor action requires emitter forward biased and collector reverse biased; current gain β = I_C/I_B, and I_E = I_B + I_C.

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

Least count of a vernier calliper = 1 main scale division − 1 vernier scale division; reading = MSR + VSD×LC, with zero error correction applied with sign.

Screw gauge least count = pitch/number of circular scale divisions; note zero error and correct the observed diameter.

Metre bridge: unknown resistance R = S·l/(100−l); keep the balance point near the middle for minimum percentage error.

Ohm's law experiment: plot V versus I; the slope gives resistance, and a straight line through the origin verifies the law.

Potentiometer needs a driver cell EMF larger than the EMF being measured and a null point on the wire; it draws no current at balance.

Focal length of a convex lens by u–v method: 1/f = 1/v − 1/u; plot 1/v against 1/u, or u+v against uv.

Refractive index by travelling microscope or prism: n = sin[(A+D_m)/2]/sin(A/2), with A the prism angle and D_m minimum deviation.

Resonance tube: end correction e = 0.6r; first and second resonances give λ = 2(l₂ − l₁), then v = νλ.

Always record instrument least count, repeat readings, and express results with proper significant figures and percentage error.

Dimensional analysis: every term in a valid equation has the same dimensions; check formulae such as v = u + at and s = ut + ½at².

Common dimensions: force MLT⁻², energy ML²T⁻², power ML²T⁻³, pressure ML⁻¹T⁻², Planck constant ML²T⁻¹.

Vector product magnitude |A×B| = AB sinθ gives a vector; dot product A·B = AB cosθ gives a scalar.

Projectile motion: range R = u²sin2θ/g, maximum height H = u²sin²θ/2g, time of flight T = 2u sinθ/g; R is maximum at 45°.

Small-angle approximations sinθ ≈ θ, tanθ ≈ θ, cosθ ≈ 1 for θ in radians help in optics and pendulum problems.

Error propagation: for z = a±b add absolute errors; for z = ab or a/b add fractional errors; for z = aⁿ multiply fractional error by n.

Graph reading: slope = Δy/Δx with units; area under a v–t graph is displacement, under a F–x graph is work.

Order-of-magnitude checks: substitute typical values (electron mass 9.1×10⁻³¹ kg, e = 1.6×10⁻¹⁹ C) to reject implausible answers.

Modulation superimposes a low-frequency message signal on a high-frequency carrier to allow efficient radiation from antennas of practical length.

Antenna length for efficient radiation is comparable to λ/4 or λ/2; higher carrier frequency means shorter antenna.

Amplitude modulation: sidebands at f_c ± f_m; bandwidth = 2f_m; modulation index m = A_m/A_c, with m ≤ 1 to avoid distortion.

Frequency modulation has constant amplitude, wider bandwidth (~2(Δf + f_m)) and better noise immunity than AM.

Bandwidths: speech ~300–3100 Hz, music ~20 kHz, TV video several MHz; optical fibre carries far higher bandwidth than coaxial cable.

Propagation: ground wave for low frequencies, sky wave by ionospheric reflection for HF, space wave (line of sight) for VHF/UHF and satellites.

Satellite communication uses geostationary orbits at about 36,000 km altitude with a round-trip delay of roughly 0.25 s.

Digital modulation and multiplexing (FDM, TDM) let many channels share one medium; optical fibre uses total internal reflection.

Modern applications include LASER (stimulated emission, coherent light), radar (echo timing), and semiconductor-based devices in mobile communication.

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

Work–energy theorem: net work equals change in kinetic energy; conservative forces allow potential energy and conservation of mechanical energy.

SI base units: metre, kilogram, second, ampere, kelvin, mole, candela; derived units combine these, e.g. newton = kg·m/s².

Thermodynamics: first law ΔQ = ΔU + ΔW; efficiency of a heat engine η = 1 − T_c/T_h for a Carnot cycle.

Waves: v = νλ; for a stretched string v = √(T/μ); sound speed in air rises with temperature and is independent of pressure at fixed T.

Simple harmonic motion: a = −ω²x, T = 2π√(m/k) for a spring and 2π√(l/g) for a pendulum, independent of amplitude for small swings.

Scientific reasoning: a hypothesis must be falsifiable; controlled experiments vary one variable; correlation alone does not establish causation.

Conservation laws (charge, energy, momentum, nucleon number) are used to test whether a proposed reaction is possible.

Significant figures and unit consistency are the first checks on any numerical answer; an answer with wrong units is wrong.