CBSE Class 12 Physics
Chapters: 9
1. Electrostatics
Electric Charges and Fields · Electrostatic Potential and Capacitance
- Electric Charges and Fields – Charge comes in two kinds, is conserved and comes in whole-number packets of e = 1.6 × 10⁻¹⁹ C. Two point charges push or pull with F = kq₁q₂/r² (k = 9 × 10⁹ N m² C⁻²). Forces and fields from many charges add as vectors (superposition). The field E = F/q₀ of a point charge is kq/r²; field lines show it. A dipole (±q a distance 2a apart) has moment p = q·2a; in a uniform field it feels zero net force but a torque τ = pE sinθ. Electric flux Φ = E·A, and Gauss's law says the flux out of any closed surface is q_enclosed/ε₀, which quickly gives E for a long wire (λ/2πε₀r), a plane sheet (σ/2ε₀) and a thin spherical shell (kq/r² outside, 0 inside).
- Electrostatic Potential and Capacitance – Electrostatic potential V at a point is the work done per unit positive charge to bring it from infinity: V = kq/r for a point charge, and potentials of many charges add as plain numbers. Potential difference V_B − V_A = W_AB/q. Equipotential surfaces join equal-V points; E is perpendicular to them and E = −dV/dr. Potential energy of two charges is U = kq₁q₂/r, and of a dipole in a field U = −pE cosθ. Conductors have free charges (E inside = 0); dielectrics have bound charges that polarise and cut the field to E₀/K. A capacitor stores charge Q = CV. A parallel plate capacitor has C = ε₀A/d, and KC with a dielectric. In series 1/C = 1/C₁ + 1/C₂ + …; in parallel C = C₁ + C₂ + …. Energy stored U = ½CV² = Q²/2C = ½QV.
2. Current Electricity
Current Electricity
- Current in a Metal: Drift Velocity and Mobility – In a metal, free electrons move very fast in random directions, so on average they go nowhere. An electric field adds a small, steady shift opposite to the field: the drift velocity vd = eEτ/m. The current is I = n e A vd, and the current density is j = n e vd. Mobility μ = vd/E tells how easily a charge drifts. Drift speed is only about a millimetre per second, yet a bulb lights at once because the field is set up in the whole wire almost instantly.
- EMF, Internal Resistance, Power and Combination of Cells – A cell does work on charges; the work per coulomb is its emf ε. Inside the cell there is a small internal resistance r. With current I, the terminal voltage is V = ε − Ir and I = ε/(R + r). Electrical power P = VI = I²R = V²/R, and energy W = Pt. Power delivered to R is largest when R = r. Cells in series: εeq = ε1 + ε2, req = r1 + r2. Cells in parallel: εeq = (ε1r2 + ε2r1)/(r1 + r2), 1/req = 1/r1 + 1/r2.
- Kirchhoff's Rules and the Wheatstone Bridge – Junction rule: at any junction, the sum of currents entering equals the sum leaving (ΣI = 0), because charge is conserved. Loop rule: around any closed loop, the algebraic sum of potential changes is zero (ΣΔV = 0), because energy is conserved. Sign rules: a resistor crossed along the current gives −IR; a cell crossed from − to + gives +ε. A Wheatstone bridge of four resistors P, Q, R, S is balanced (no galvanometer current) when P/Q = R/S; this lets us find an unknown resistance, as in the metre bridge.
- Ohm's Law, Resistivity and Effect of Temperature – Using drift velocity, V = IR with R = ml/(ne²τA) = ρl/A. Resistivity ρ = m/(ne²τ) depends only on the material and temperature; conductivity σ = 1/ρ. Ohm's law in vector form is j = σE. Metals, wires and resistors give a straight V–I line (ohmic); bulbs, diodes and devices like GaAs give curved or one-way graphs (non-ohmic). For metals ρ rises with temperature: ρT = ρ0[1 + α(T − T0)]; alloys like nichrome change very little; semiconductors get lower ρ when hot.
3. Magnetic Effects of Current and Magnetism
Moving Charges and Magnetism · Magnetism and Matter
- Lorentz Force, Torque on a Loop and the Moving Coil Galvanometer – A charge q moving with velocity v feels F = q(E + v × B), the Lorentz force. The magnetic part qvB sinθ is at right angles to v, so it bends the path into a circle (r = mv/qB) without changing speed. A wire feels F = IL × B. Parallel wires attract if the currents are in the same direction: F/L = μ₀I₁I₂/2πd. A loop in a field feels a torque τ = m × B with magnetic moment m = NIA, so a loop acts as a magnetic dipole. A moving coil galvanometer uses this torque: its deflection φ = (NAB/k)I. A small shunt turns it into an ammeter; a large series resistance turns it into a voltmeter.
- Magnetic Field due to a Current: Biot-Savart Law and Ampere's Law – A current makes a magnetic field around it (Oersted, 1820). The Biot-Savart law gives the field of a tiny piece of wire: dB = (μ₀/4π)·I dl sinθ / r². Adding all pieces of a circular loop gives B = μ₀NI / 2R at the centre and μ₀NIR² / 2(R² + x²)^{3/2} on the axis. Ampere's circuital law ∮B·dl = μ₀I gives B = μ₀I / 2πr for a long straight wire. A long solenoid has a strong, uniform field B = μ₀nI inside and almost none outside.
- Magnetism and Matter – A bar magnet behaves like a solenoid: tiny current loops of electrons inside it line up. Its field on the axis (2m/r³ form) is twice the field on the equator at the same distance, and points the other way. In a uniform field a magnet feels a torque τ = m × B that turns it to line up with B. Field lines are closed loops that never cross. Materials react differently: diamagnetic ones are pushed out of a field (χ small and negative), paramagnetic ones are pulled in weakly (χ small and positive), ferromagnetic ones are pulled in strongly (χ very large). Magnetisation M is the magnetic moment per unit volume; B = μ₀(H + M), χ = M/H, μᵣ = 1 + χ. Heating reduces magnetism: paramagnets follow Curie law χ = C/T, and a ferromagnet turns paramagnetic above its Curie temperature.
4. Electromagnetic Induction and Alternating Currents
Electromagnetic Induction · Alternating Current
- Electromagnetic Induction – When the magnetic flux through a coil changes, an emf is produced in it: ε = −N dΦ/dt (Faraday). The minus sign is Lenz's law: the induced current always opposes the change that made it. Moving a rod in a field gives motional emf ε = Blv. A changing current makes an emf in its own coil (self-induction, L) and in a nearby coil (mutual induction, M).
- AC Generator and Transformer – An AC generator turns a coil in a magnetic field. The flux through it keeps changing, so it makes an emf e = NBAω sin ωt with peak NBAω. A transformer uses mutual induction between two coils on one iron core: Vs/Vp = Ns/Np. A step-up transformer raises voltage and lowers current; a step-down does the opposite. Power stays nearly the same (Vp Ip ≈ Vs Is).
- Alternating Current – An alternating current (AC) changes size and direction again and again: I = I₀ sin ωt. Its rms value is I₀/√2, the steady DC that gives the same heating. A resistor keeps V and I in step; an inductor makes I lag by 90° (Xʟ = ωL); a capacitor makes I lead by 90° (Xᴄ = 1/ωC). In a series LCR circuit Z = √(R² + (Xʟ − Xᴄ)²), resonance happens when Xʟ = Xᴄ, and average power is P = Vrms Irms cos φ.
5. Electromagnetic Waves
Electromagnetic Waves
- Electromagnetic Waves – A changing electric field acts like a current, called the displacement current, and it makes a magnetic field. A changing magnetic field makes an electric field. Together they travel as an electromagnetic (EM) wave at c = 3 × 10⁸ m/s, even through empty space. E and B are at right angles to each other and to the direction of travel, so EM waves are transverse. Sorted by wavelength, they form the spectrum: radio, microwave, infrared, visible, ultraviolet, X-rays and gamma rays.
6. Optics
Ray Optics and Optical Instruments · Wave Optics
- Spherical Mirrors and the Mirror Formula – A spherical mirror is a piece cut from a shiny ball. Its focal length is half its radius of curvature (f = R/2). With the Cartesian sign convention, the object distance u, image distance v and focal length f are linked by 1/v + 1/u = 1/f, and the magnification is m = h′/h = −v/u. Concave mirrors have negative f; convex mirrors have positive f.
- Optical Instruments: Microscopes and Telescopes – How big a thing looks depends on the angle it makes at the eye. A simple microscope (one convex lens) gives m = 1 + D/f (image at D) or D/f (image at infinity). A compound microscope uses a short-focus objective and an eyepiece: m = mₒ × mₑ ≈ (L/fₒ)(D/fₑ). An astronomical telescope uses a long-focus objective and short-focus eyepiece: m = fₒ/fₑ in normal adjustment, with tube length fₒ + fₑ. Reflecting telescopes use a concave mirror as objective.
- Thin Lenses: Lens Maker's Formula, Lens Formula and Power – At one curved surface, n₂/v − n₁/u = (n₂ − n₁)/R. Two such surfaces make a thin lens with 1/f = (n − 1)(1/R₁ − 1/R₂) (lens maker's formula). Object and image distances follow 1/v − 1/u = 1/f, magnification m = v/u, power P = 1/f (in dioptres when f is in metres), and thin lenses in contact add their powers: P = P₁ + P₂.
- Refraction, Total Internal Reflection and the Prism – Light bends when it crosses from one medium to another: n₁ sin i = n₂ sin r. Going from a denser to a rarer medium it bends away from the normal; beyond the critical angle C (sin C = 1/n) no light escapes and all of it reflects back: total internal reflection. Optical fibres use this to carry light. In a prism, r₁ + r₂ = A and deviation δ = i + e − A; at minimum deviation n = sin((A + δm)/2) / sin(A/2).
- Wavefronts and Huygens' Principle – A wavefront is a surface on which every point of a wave is in the same phase. A point source gives spherical wavefronts; a far source gives plane wavefronts; rays are at right angles to wavefronts. Huygens' principle: every point on a wavefront acts as a source of secondary wavelets, and the forward envelope of these wavelets is the new wavefront. Using it, reflection gives i = r and refraction gives sin i / sin r = v₁/v₂ = n₂/n₁.
- Interference (Young's Double Slit) and Single Slit Diffraction – Two coherent sources (same frequency, fixed phase difference) make a steady pattern of bright and dark fringes. At a point on the screen the path difference is Δ = yd/D: bright where Δ = nλ, dark where Δ = (n + ½)λ. All fringes have equal width β = λD/d. A single slit of width a gives diffraction: a bright central maximum of width 2λD/a, with first minima where a sin θ = λ, and weaker side maxima.
7. Dual Nature of Radiation and Matter
Dual Nature of Radiation and Matter
- Matter Waves and the de Broglie Relation – Light behaves like a wave and a particle. In 1924 Louis de Broglie said matter does the same: every moving particle has a wave with wavelength λ = h/p = h/mv. For a body of kinetic energy K, λ = h/√(2mK); for an electron accelerated through V volts, λ = 1.227/√V nm. Everyday objects have far too tiny a λ to notice, but electrons have λ about the size of atoms.
- Photoelectric Effect and Einstein's Equation – When light of high enough frequency falls on a metal, electrons jump out at once. Light comes in packets called photons, each with energy E = hν. One photon gives all its energy to one electron: hν = φ + KEmax, where φ is the work function. Below the threshold frequency ν₀ = φ/h no electron comes out, however bright the light.
8. Atoms and Nuclei
Atoms · Nuclei
- Atoms: From Alpha Scattering to the Bohr Model – Rutherford shot alpha particles at thin gold foil and found that an atom is mostly empty, with a tiny, heavy, positive nucleus in the middle. Bohr then said the electron in hydrogen can move only on fixed orbits where its angular momentum is nh/2π. In orbit n the radius is 0.529 n² Å, the speed is (2.19 × 10⁶)/n m/s and the energy is −13.6/n² eV. When the electron jumps down, the energy difference comes out as light of one exact colour, which gives the line spectrum of hydrogen.
- Nuclei: Size, Nuclear Force, Binding Energy, Fission and Fusion – A nucleus is made of Z protons and N neutrons (A = Z + N nucleons). Its radius is R = R₀A^(1/3) with R₀ ≈ 1.2 fm, so every nucleus has almost the same huge density. A very strong, short-range nuclear force holds the nucleons together. The nucleus weighs a little less than its loose parts; this mass defect Δm is the binding energy, E = Δm c² (1 u = 931.5 MeV). Binding energy per nucleon is highest near iron (A ≈ 56), so heavy nuclei give energy when they split (fission) and light nuclei give energy when they join (fusion).
9. Electronic Devices
Semiconductor Electronics: Materials, Devices and Simple Circuits
- Semiconductors and the p-n Junction Diode – A semiconductor has a small energy gap (about 1 eV), so a little heat frees some electrons. Pure silicon is intrinsic (electrons = holes). Adding a 5-valence atom makes n-type; a 3-valence atom makes p-type. Joining p and n makes a junction with a depletion layer and a barrier (about 0.7 V for Si). The diode conducts in forward bias, almost not in reverse bias, so it can change AC into one-way DC (rectifier).