RBSE SS 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 attractive for unlike signs.

Electric field E = F/q₀ is a vector pointing away from positive charge; field of a point charge is E = kq/r².

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

Electric potential V = kq/r is a scalar; potential energy of two charges is U = kq₁q₂/r, and E = −dV/dr.

Conductor in electrostatic equilibrium: E inside is zero, charge resides on the outer surface, and field just outside is σ/ε₀ normal to the surface.

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 u = ½ε₀E².

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

Ohm's law V = IR holds for ohmic conductors at constant temperature; drift velocity v_d = I/(nAe) and current density J = nev_d = σE.

Resistivity ρ = m/(ne²τ); temperature dependence ρ = ρ₀(1 + αΔT), with α negative for semiconductors.

Series: R = ΣRᵢ, same current; parallel: 1/R = Σ1/Rᵢ, same voltage.

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

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

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

Power P = I²R = V²/R; maximum power transfer to a load occurs when R = r, giving P = ε²/4r.

Cells in series add EMFs (ε₁+ε₂) with r₁+r₂; in parallel with equal EMF, ε_eq = ε and r_eq = r/n.

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

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

Force on a moving charge F = qvB sinθ; on a current element F = BIL sinθ; direction by Fleming's left-hand rule.

Ampere's circuital law ∮B·dl = μ₀I_enclosed; parallel wires carrying currents in the same direction attract with F/L = μ₀I₁I₂/2πd.

Torque on a current loop τ = NIAB sinθ; magnetic moment m = NIA, and U = −mB cosθ.

Moving-coil galvanometer: I = (k/NAB)θ; conversion to ammeter uses a low shunt S = I_gG/(I−I_g); to voltmeter uses high series R = V/I_g − G.

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

Magnetic materials: χ small positive for paramagnets, large positive for ferromagnets, small negative for diamagnets; Curie law χ ∝ 1/T.

Faraday's law ε = −dΦ/dt for a single turn; for N turns ε = −N dΦ/dt; Lenz's law fixes the direction opposing the flux change.

Motional EMF: ε = Bvl for a rod, and ε = BAω sinωt for a coil rotating in a field.

Self-inductance ε = −L dI/dt; solenoid L = μ₀n²Al; energy stored U = ½LI².

Mutual inductance ε₂ = −M dI₁/dt; for two coaxial solenoids M = μ₀n₁n₂Al.

AC through R: I and V in phase; through L: V leads I by 90° (X_L = ωL); through C: I leads V by 90° (X_C = 1/ωC).

Series LCR: Z = √(R² + (X_L − X_C)²), tanφ = (X_L − X_C)/R; resonance at ω₀ = 1/√(LC), Q-factor = (1/R)√(L/C).

Average power P = V_rms I_rms cosφ; wattless current flows when φ = 90°.

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

Maxwell's displacement current I_d = ε₀ dΦ_E/dt makes the Ampere–Maxwell law consistent with charge conservation.

EM waves are transverse: E, B and the propagation direction k form a right-handed set; E and B oscillate in phase.

Relation c = 1/√(μ₀ε₀); in a medium v = 1/√(με) = c/n.

Ratio of magnitudes E₀/B₀ = c; intensity I = ½ε₀E₀²c = P/4πr² for a point source.

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

Radiation pressure: p = I/c for full absorption and p = 2I/c for perfect reflection.

Energy of a photon E = hν = hc/λ; momentum p = h/λ; h = 6.63×10⁻³⁴ J·s.

Mirror formula 1/v + 1/u = 1/f with f = R/2; lens formula 1/v − 1/u = 1/f; magnification m = v/u (mirror) or v/u (lens).

Refraction: n₁ sin i = n₂ sin r; critical angle sin C = 1/n for denser-to-rarer travel; total internal reflection occurs above C.

Lens maker's formula 1/f = (n − 1)(1/R₁ − 1/R₂); power P = 1/f in dioptres with f in metres; thin lenses in contact P = P₁ + P₂.

Prism: deviation δ = i₁ + i₂ − A; at minimum deviation n = sin[(A + δ_m)/2]/sin(A/2).

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

Single slit diffraction: first minimum at a sinθ = λ; central maximum width is twice the secondary fringe width.

Malus's law I = I₀cos²θ; Brewster angle tanθ_p = n; polarisation proves light is transverse.

Resolving power of a telescope = D/1.22λ; Rayleigh criterion for two just-resolved points is angular separation 1.22λ/D.

Photoelectric equation hν = φ₀ + K_max, with K_max = eV₀ and threshold ν₀ = φ₀/h.

Stopping potential depends on frequency, not intensity; saturation photocurrent depends on intensity, not frequency.

Photoelectric emission is instantaneous (no time lag) and requires ν > ν₀ regardless of intensity.

de Broglie wavelength λ = h/p = h/√(2mK) = h/√(2mqV) for an accelerated charge.

Photon energy in eV: E(eV) = 1240/λ(nm).

Electron wavelength at 100 V is about 0.123 nm, comparable to atomic spacing, hence electron diffraction.

Work function values are a few eV for metals; alkali metals have the lowest thresholds.

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

Rydberg formula 1/λ = R(1/n₁² − 1/n₂²) with R = 1.097×10⁷ m⁻¹; Lyman series is in the UV, Balmer in the visible.

Ionisation energy of hydrogen is 13.6 eV; energy needed from n = 1 to n = 2 is 10.2 eV.

Nuclear radius R = R₀A^(1/3) with R₀ ≈ 1.2 fm; density is nearly constant across nuclei.

Mass defect Δm = Zm_p + (A−Z)m_n − M; binding energy BE = Δm c², with 1 u = 931.5 MeV.

Radioactive decay N = N₀e^(−λt); half-life T₁/₂ = 0.693/λ; activity A = λN.

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

Fission of U-235 releases about 200 MeV per nucleus; fusion of light nuclei releases energy because binding energy per nucleon peaks near Fe-56.

Intrinsic semiconductor: n_e = n_h = n_i; conductivity rises with temperature because carrier density increases.

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

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

Forward bias narrows the depletion layer and lowers resistance; reverse bias widens it and gives a small leakage current.

Half-wave rectifier output frequency equals the input frequency; full-wave rectifier doubles it.

Zener diode works in reverse breakdown as a voltage regulator; the breakdown voltage is nearly constant.

Transistor in CE configuration: current gain β = I_C/I_B, α = I_C/I_E, and β = α/(1 − α).

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

Vernier calliper least count = 1 main scale division − 1 vernier division, typically 0.1 mm; reading = MSR + VSD × LC.

Screw gauge least count = pitch/number of circular scale divisions, typically 0.01 mm; account for zero error with sign.

Metre bridge: unknown resistance R = S·l/(100 − l), where l is the balancing length in cm on the unknown side.

Ohm's law experiment: plot V versus I; the slope of the best-fit line gives resistance.

Potentiometer compares EMFs by balancing lengths: E₁/E₂ = l₁/l₂, since the potential drop per unit length is constant.

Focal length of a convex lens by u–v method uses 1/f = 1/v − 1/u; plot 1/v against 1/u to get a straight line.

Focal length of a concave mirror by the u–v method uses 1/f = 1/v + 1/u with sign conventions applied.

Galvanometer figure of merit k = I/θ, and resistance by half-deflection method G = R·S/(R − S).

Vector addition by components; magnitude |A| = √(A_x² + A_y² + A_z²); unit vector  = A/|A|.

Dot product A·B = AB cosθ is a scalar; cross product |A×B| = AB sinθ is a vector perpendicular to both.

Derivative rules: d/dx(xⁿ) = nxⁿ⁻¹, d/dx(sin x) = cos x, d/dx(eˣ) = eˣ, d/dx(ln x) = 1/x.

Integration reverses differentiation; ∫xⁿdx = xⁿ⁺¹/(n+1) + C for n ≠ −1, and ∫dx/x = ln x + C.

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

Scalar and vector triple products: A·(B×C) is a scalar volume; A×(B×C) = B(A·C) − C(A·B).

Graphical methods: slope of a straight line gives the rate constant; area under a curve gives the integral quantity.

SI base units: metre, kilogram, second, ampere, kelvin, mole, candela; derived units combine these.

Dimensional analysis checks equations and converts units; dimensionless quantities include refractive index and strain.

Absolute error Δa = |a_mean − a_i|; mean absolute error is the average of these; relative error = Δa/a_mean; percentage error = relative × 100.

Combination rules: sums and differences add absolute errors; products and quotients add relative errors; powers multiply relative error by the exponent.

Significant figures: the result of multiplication or division carries the least number of significant figures among inputs.

Random errors reduce by averaging repeated readings; systematic errors shift all readings and need instrument calibration.

Precision describes reproducibility; accuracy describes closeness to the true value.

A communication system has three blocks: transmitter, channel, and receiver; the channel adds noise.

Modulation superimposes a message on a carrier; it enables antenna radiation and frequency multiplexing.

Antenna length for efficient radiation is about λ/4 or λ/2; audio frequencies (20 Hz–20 kHz) need impractically long antennas, hence modulation.

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

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

Space communication uses line-of-sight propagation; the maximum range d = √(2Rh) for an antenna of height h.

Optical fibre works by total internal reflection; it carries high bandwidth with low loss.

Propagation modes: ground wave (low frequencies), sky wave (ionospheric reflection, 3–30 MHz), space wave (VHF/UHF, line of sight).