UPMSP Intermediate Physics - Code 151 cheat sheet

Coulomb's law in vacuum: F = (1/4πε₀)·q₁q₂/r², with 1/4πε₀ = 9×10⁹ N·m²/C²; in a medium replace ε₀ by ε = Kε₀.

Electric field of a point charge E = kq/r², directed away from positive and toward negative charge; field lines never intersect.

Gauss's law: ∮E·dS = q_enclosed/ε₀; use it for spherical, cylindrical and infinite-sheet symmetry.

Field inside a uniformly charged spherical shell is zero; outside it behaves as a point charge at the centre.

Infinite line charge: E = λ/(2πε₀r); infinite plane sheet: E = σ/(2ε₀), independent of distance.

Electric potential V = kq/r for a point charge; E = −dV/dr, so field points from high to low potential.

Capacitance: C = Q/V; parallel plate C = ε₀A/d, and with dielectric slab C = Kε₀A/d.

Series capacitors: 1/C = Σ1/Cᵢ (same charge); parallel: C = ΣCᵢ (same potential difference).

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

Ohm's law V = IR holds for metallic conductors at constant temperature; non-ohmic devices give curved V–I graphs.

Resistance R = ρL/A; resistivity ρ depends on material and temperature, not on dimensions.

Drift velocity relation I = neAv_d; typical v_d is about 10⁻⁴ m/s while the signal travels near light speed.

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

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

Wheatstone bridge balanced when P/Q = R/S; metre bridge is its practical form, balance point l gives R/S = l/(100−l).

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

Cells in series add EMFs and internal resistances; identical cells in parallel keep ε but reduce r to r/n.

Power P = VI = I²R = V²/R; maximum power transfer to load when R = r, efficiency then 50%.

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

Straight long wire: B = μ₀I/(2πr); circular loop at centre: B = μ₀I/(2R); solenoid: B = μ₀nI.

Force on a moving charge F = qvB sinθ, always perpendicular to v and B, so magnetic force does no work.

Force on a current element F = BIL sinθ; parallel currents attract, antiparallel currents repel.

Radius of circular path r = mv/(qB); time period T = 2πm/(qB) is independent of speed.

Torque on a current loop τ = NIAB sinθ; magnetic moment m = NIA, direction by right-hand rule.

Ampere's circuital law ∮B·dl = μ₀I_enclosed; moving-coil galvanometer shows deflection ∝ current.

Conversion: shunt S = I_gG/(I − I_g) for ammeter; series R = V/I_g − G for voltmeter.

Earth's field has horizontal component B_H and angle of dip δ with tanδ = B_V/B_H; tangent galvanometer obeys I ∝ tanθ.

Faraday's law ε = −dΦ/dt; Lenz's law fixes the induced current direction so as to oppose the change.

Motional EMF for a rod moving in a field: ε = Bvl, with current I = Bvl/R in the closed circuit.

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

Mutual inductance ε₂ = −M dI₁/dt; transformer EMF ratio V_s/V_p = N_s/N_p, ideal power conserved.

AC through R: current in phase with voltage; through L: current lags by 90°; through C: current leads by 90°.

Impedance: series RLC Z = √(R² + (X_L − X_C)²), with X_L = ωL and X_C = 1/ωC.

Resonance at ω = 1/√(LC); then Z = R is minimum and current maximum; Q-factor = (1/R)√(L/C).

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

Transformer losses arise from copper (I²R) heating, eddy currents, hysteresis and flux leakage; a transformer cannot step up DC.

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

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

Relation E₀/B₀ = c; in a medium v = 1/√(με) and refractive index n = c/v.

Order of spectrum: radio, microwave, infrared, visible, ultraviolet, X-ray, gamma; visible spans about 400–700 nm.

Mirror formula 1/v + 1/u = 1/f and lens formula 1/v − 1/u = 1/f, with sign convention and f = R/2 for mirrors.

Lens maker's formula 1/f = (n − 1)(1/R₁ − 1/R₂); power P = 1/f in dioptres, combined P = ΣPᵢ.

Total internal reflection occurs when light goes from denser to rarer medium and i exceeds critical angle sinC = 1/n.

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

Single-slit diffraction gives central maximum width 2λD/a; polarisation confirms the transverse nature of light, with Brewster's tanθ_p = n.

Photoelectric equation hν = φ₀ + K_max; K_max depends on frequency, not on intensity.

Threshold frequency ν₀ = φ₀/h; stopping potential satisfies eV₀ = hν − φ₀.

Photoelectric current (saturation) is proportional to intensity; emission is instantaneous, contradicting the wave picture.

Photon energy E = hν = hc/λ; with λ in nm, E(eV) ≈ 1240/λ(nm).

de Broglie wavelength λ = h/p = h/(mv) = h/√(2mK) for a particle of kinetic energy K.

For an electron accelerated through V volts, λ ≈ 12.27/√V Å.

Davisson–Germer experiment confirmed electron diffraction, verifying matter waves.

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

Bohr postulates: quantised angular momentum mvr = nh/2π and stationary orbits without radiation.

Radius rₙ = 0.529 n²/Z Å; energy Eₙ = −13.6 Z²/n² eV; velocity vₙ ∝ Z/n.

Rydberg formula 1/λ = R(1/n₁² − 1/n₂²), R ≈ 1.097×10⁷ m⁻¹; Lyman series lies in the ultraviolet.

Ionisation energy of hydrogen is 13.6 eV; excitation energies follow from Eₙ differences.

Nuclear radius R = R₀A^(1/3), R₀ ≈ 1.2 fm; density is nearly independent of mass number.

Mass defect Δm converts to binding energy via E = Δm c²; 1 u ≈ 931.5 MeV.

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 changes neither.

Fission of heavy nuclei and fusion of light nuclei both release energy because binding energy per nucleon peaks near iron (A ≈ 56).

Intrinsic semiconductors: nₑ = n_h = nᵢ; conductivity rises with temperature because carrier number increases.

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

In a p–n junction, diffusion builds a depletion layer and barrier potential: about 0.7 V for Si, 0.3 V for Ge.

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

Half-wave rectifier uses one diode; full-wave uses two (centre-tap) or four (bridge); a filter capacitor smooths output.

Zener diode works in reverse breakdown as a voltage regulator; breakdown voltage is set by doping.

Transistor has emitter, base and collector; α = I_C/I_E and β = I_C/I_B, with β = α/(1 − α).

For a common-emitter amplifier, voltage gain A_v = β × (R_C/R_B input resistance ratio), and phase inverts by 180°.

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

Vernier calliper least count = 1 main scale division/number of vernier divisions; typical value 0.1 mm.

Screw gauge least count = pitch/number of circular scale divisions; typical value 0.01 mm, with zero error correction.

Metre bridge: unknown resistance R = S·l/(100 − l); interchange gaps and repeat to remove end errors.

Ohm's law experiment plots V against I; slope of the straight line gives resistance.

Potentiometer compares EMFs by balancing lengths: E₁/E₂ = l₁/l₂, since EMF ∝ balancing length.

Focal length of a convex mirror or lens is found by u–v method; plot 1/v against 1/u, intercepts give 1/f.

For a glass slab, refractive index n = real depth/apparent depth or from sin i/sin r.

Resonance tube: end correction e = 0.3d; speed of sound v = 2f(l₂ − l₁) using first and second resonance.

Fuse wire experiment uses the relation heat ∝ I²R and plots melting current against wire radius.

UPMSP Intermediate Physics paper (Code 151) is a 100-mark theory paper of 3 hours 15 minutes including reading time.

Objective section carries 1 mark per question; very short answer carries 2 marks; short answer 3 marks; long answer 4–5 marks.

Rough weighting: Electrostatics and Current Electricity together dominate, followed by Optics and Modern Physics.

Numerical questions are drawn mainly from electrostatics, current electricity, EMI/AC and optics; formula recall plus unit checking earns method marks.

Derivation questions typically appear as 3- or 4-mark items; draw the labelled diagram before writing the derivation.

Questions on semiconductors and communication usually appear as short-answer or objective items.

Practical-based questions ask for least count, circuit diagram, or a two-point precaution; mention instrument and observation table.

Answer length should match marks: about 30 words per 2 marks, 60–80 per 3 marks, and a full derivation for 4–5 marks.

Always write the final answer with SI unit and correct significant figures; unlabelled diagrams lose marks.

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

Product rule (uv)′ = u′v + uv′; quotient rule (u/v)′ = (u′v − uv′)/v²; chain rule for composite functions.

Integrals: ∫xⁿdx = xⁿ⁺¹/(n+1), ∫sin x dx = −cos x, ∫dx/x = ln x, ∫eˣdx = eˣ.

Small-change approximation: Δf ≈ (df/dx)Δx; used for error propagation and approximation problems.

Trigonometric identities: sin²θ + cos²θ = 1; sin2θ = 2sinθcosθ; cos2θ = 1 − 2sin²θ.

Binomial approximation (1 + x)ⁿ ≈ 1 + nx for |x| ≪ 1, used in lens and interference approximations.

Area under a graph equals the integral; slope equals the derivative, so v–t slope gives acceleration.

Logarithm rules: log(ab) = log a + log b, log(a/b) = log a − log b, log aⁿ = n log a.

Vector addition by triangle or parallelogram law; resultant R = √(A² + B² + 2AB cosθ).

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

Unit vectors: A = Aₓî + A_yĵ + A_z k̂, magnitude √(Aₓ² + A_y² + A_z²).

SI base units: metre, kilogram, second, ampere, kelvin, mole, candela; all other units are derived.

Dimensional formula examples: force [MLT⁻²], energy [ML²T⁻²], electric field [MLT⁻³A⁻¹].

Dimensional analysis checks equations and converts units, but cannot determine dimensionless constants.

Errors: absolute, relative and percentage; for sums add absolute errors, for products add relative errors.

If x = aᵖbᑫ/cʳ, relative error Δx/x = p(Δa/a) + q(Δb/b) + r(Δc/c).

Significant figures: result carries the least number of significant figures among the input data.

Modulation superimposes a low-frequency message on a high-frequency carrier; needed because antennas must be comparable to λ/4.

AM bandwidth = 2f_m; FM bandwidth by Carson's rule ≈ 2(f_m + Δf) and FM has better noise immunity.

Modulation index for AM m = A_m/A_c; sidebands at f_c ± f_m carry the information.

Skywave propagation uses ionospheric reflection (3–30 MHz); space wave is line-of-sight and used for TV and microwaves.

Optical fibre works by total internal reflection; it offers high bandwidth and immunity to electromagnetic interference.

Satellite communication uses microwaves; geostationary orbit radius is about 42,000 km with period 24 h.

LASER: stimulated emission gives coherent, monochromatic, highly directional light; population inversion is required.

Nuclear applications include radioisotopes in medicine (cobalt-60 therapy) and carbon-14 dating.

Semiconductor devices power modern communication: LEDs for optical links, photodiodes as receivers, and integrated circuits in transceivers.