Std. 12
Physics

Electric charge is a fundamental property responsible for electrical interaction. Charge is conserved and quantised: an isolated body's net charge changes in integral multiples of the elementary charge.
q = ne

The force between two point charges is directly proportional to the product of their charges and inversely proportional to the square of their separation. It acts along the line joining the charges.
F = (1/4πε₀) |q₁q₂|/r²

Electric field intensity at a point is the force experienced per unit positive test charge placed at that point.
E = F/q₀ = (1/4πε₀) q/r²

An electric dipole consists of two equal and opposite charges separated by a small distance. Dipole moment points from negative to positive charge.
p = q(2a)

The two charges experience equal and opposite forces. Their resultant force is zero, but they form a couple that tends to rotate the dipole.
τ = pE sinθ
U = −pE cosθ

Electric flux measures the electric field passing through a surface. Gauss's theorem relates total flux through a closed surface to the net charge enclosed.
Φ_E = ∮E·dA = Q_enclosed/ε₀

ISC prescribes applications to an infinitely long straight charged wire, a uniformly charged infinite plane sheet and a uniformly charged thin spherical shell. Symmetry is the key reason Gauss's law becomes useful.

Potential at a point is the work done per unit positive test charge in bringing it from infinity to that point without acceleration.
V = W/q; V_point = (1/4πε₀)q/r

Potential energy of a system of charges is the work associated with assembling the charges from infinity.
U_pair = (1/4πε₀)q₁q₂/r

Capacitance tells how much charge a conductor system stores for a given potential difference.
C = Q/V; C_parallel plate = ε₀A/d

Energy is stored in the electric field between the plates.
U = 1/2 CV² = 1/2 QV = Q²/(2C)

In parallel, capacitances add because the potential difference is common. In series, reciprocal capacitances add because the charge magnitude is common.
Parallel: C_eq = C₁ + C₂ + …
Series: 1/C_eq = 1/C₁ + 1/C₂ + …

Current is the rate at which charge crosses a chosen cross-section. Conventional current direction is the direction of motion of positive charge.
I = dQ/dt

In a conductor, free electrons have random thermal motion. An applied electric field produces a small average drift velocity superposed on this random motion.
I = neAv_d

Current density is current per unit cross-sectional area and has direction along conventional current.
J = I/A = nev_d

Mobility measures drift velocity produced per unit electric field. Conductivity measures how readily a material conducts current.
v_d = μE; J = σE

For an ohmic conductor under specified physical conditions, current is proportional to potential difference.
V = IR; R = ρL/A

Electrical power is the rate of electrical energy transfer. Use the form most convenient for the known quantities.
P = VI = I²R = V²/R
W = Pt

The emf of a cell is energy supplied by the source per unit charge. Terminal voltage is lower than emf while a real cell supplies current because of internal resistance.
I = ε/(R+r); V = ε−Ir

Junction law follows conservation of charge. Loop law follows conservation of energy.
ΣI = 0; ΣΔV = 0

A Wheatstone bridge is balanced when the galvanometer carries no current. At balance, the ratio of resistances in one arm equals the ratio in the other.
R₁/R₂ = R₃/R₄

The metre bridge is a practical form of the Wheatstone bridge using a uniform resistance wire. At balance, resistance ratios equal the corresponding wire-length ratios.
R₃/R₄ = l₁/l₂

A potentiometer compares potential differences by balancing an unknown potential against a known potential drop along a uniform wire. At balance, the test source supplies no current through the galvanometer branch.
V = Kl; ε₁/ε₂ = l₁/l₂

The magnetic field contribution from a small current element depends on current, element length, distance and orientation.
dB = (μ₀/4π) I(dℓ×r̂)/r²

The magnetic field of a current-carrying circular coil is obtained by integrating the Biot–Savart contribution. ISC includes the centre and axial point cases.

For a closed path, the circulation of magnetic field is related to the current enclosed. It is especially useful for high-symmetry arrangements.
∮B·dl = μ₀I_enclosed

A moving charge in a magnetic field experiences a force perpendicular to its velocity and the magnetic field. Therefore a magnetic field alone does no work on the charge.
F = q(v×B); |F| = qvB sinθ

For v perpendicular to B, magnetic force acts as centripetal force and the particle moves in a circular path.
r = mv/(qB); T = 2πm/(qB)

A conductor carrying current in a magnetic field experiences a force whose direction follows the cross-product rule.
F = I(L×B); |F| = BIL sinθ

Two long parallel current-carrying conductors exert forces on each other. Same-direction currents attract; opposite-direction currents repel.
F/L = μ₀I₁I₂/(2πd)

A current loop behaves like a magnetic dipole. Its magnetic moment is perpendicular to the plane of the loop.
m = NIA; τ = mB sinθ

A current-carrying coil in a magnetic field experiences torque. The balance between magnetic torque and restoring torque gives the instrument's deflection.
τ = NIAB

Diamagnetic, paramagnetic and ferromagnetic materials respond differently to an applied magnetic field. Magnetisation is magnetic moment per unit volume.
B = μ₀(H+M); χ_m = M/H; μ_r = 1+χ_m

Magnetic flux through a surface depends on magnetic field, area and the angle between the field and the area normal.
Φ_B = BA cosθ

An emf is induced whenever the magnetic flux linked with a circuit changes. The magnitude depends on the rate of change of flux.
ε = −dΦ_B/dt

The negative sign in Faraday's law represents Lenz's law: the induced current produces an effect that opposes the change in flux responsible for it.

When a conductor moves through a magnetic field so that magnetic force separates charges, a potential difference is produced across the conductor.
ε = Blv

A changing current in a coil changes its own magnetic flux and induces an emf that opposes the current change.
ε = −L dI/dt; Φ = LI

For a long solenoid, inductance depends on permeability, number of turns, cross-sectional area and length.
L = μ₀N²A/l

A changing current in one coil can change the flux linked with a nearby coil and induce an emf in it.
ε₂ = −M dI₁/dt

A sinusoidal alternating voltage changes direction periodically. Peak value, instantaneous value, rms value and mean value describe different aspects.
v = V₀ sinωt; V_rms = V₀/√2; I_rms = I₀/√2

Inductors and capacitors oppose alternating current through frequency-dependent reactance.
X_L = ωL; X_C = 1/(ωC)

The impedance combines resistance with the difference between inductive and capacitive reactances.
Z = √[R²+(X_L−X_C)²]

The phase angle describes the lead or lag between voltage and current. Only the in-phase component contributes to average power.
tanφ = (X_L−X_C)/R; P = V_rms I_rms cosφ

In a series LCR circuit, resonance occurs when inductive and capacitive reactances are equal. Impedance is then minimum and current is maximum.
ω₀ = 1/√(LC); f₀ = 1/(2π√LC)

A transformer transfers AC electrical energy between circuits through mutual induction. An ideal transformer changes voltage and current in inverse proportion.
V_s/V_p = N_s/N_p; V_pI_p = V_sI_s

A changing electric field contributes to the electromagnetic interaction. This concept completes the symmetry between changing electric and magnetic fields in Maxwell's framework.

Electromagnetic waves are transverse. The electric field and magnetic field are mutually perpendicular and both are perpendicular to the direction of propagation.
c = 1/√(μ₀ε₀)

Frequency, wavelength and speed are related for electromagnetic waves. In vacuum the speed is c.
c = νλ

The spectrum is arranged by frequency or wavelength: radio waves, microwaves, infrared, visible, ultraviolet, X-rays and gamma rays.

ISC requires qualitative understanding of the production, detection, properties and common uses of the different regions of the electromagnetic spectrum.

A spherical mirror forms images through reflection from a curved surface. Use the prescribed Cartesian sign convention consistently.
1/f = 1/v + 1/u; R = 2f; m = −v/u

Refraction is the change in direction of light when it enters another medium because its speed changes.
n₁ sin i = n₂ sin r; n = c/v

Total internal reflection occurs when light travels from a denser to a rarer medium and the angle of incidence exceeds the critical angle.
sin C = n₂/n₁

A prism changes the direction of light through two refractions. At minimum deviation the path through the prism is symmetric.
δ = i₁+i₂−A; n = sin[(A+δ_m)/2]/sin(A/2)

The spherical-surface relation connects object distance, image distance, refractive indices and radius of curvature.
n₂/v − n₁/u = (n₂−n₁)/R

For a thin lens, focal length depends on refractive index of lens material and the radii of curvature of its surfaces.
1/f = (n−1)(1/R₁−1/R₂)

The lens formula connects object distance, image distance and focal length. Magnification compares image size with object size.
1/f = 1/v − 1/u; m = v/u

Power measures the converging or diverging ability of a lens. For thin lenses in contact, powers add algebraically.
P = 1/f (f in metre); P_eq = P₁+P₂

ISC includes the simple and compound microscope, refracting telescope, reflecting telescope, ray diagrams, magnifying power and resolving power where prescribed.

Every point on a wavefront can be treated as a source of secondary wavelets. The envelope of these wavelets gives the new wavefront. ISC requires proof of reflection and refraction laws using this principle.

Interference occurs when coherent waves overlap. Bright fringes arise from constructive interference and dark fringes from destructive interference.
Bright: Δ=nλ; Dark: Δ=(n+1/2)λ; β=λD/d

Diffraction is the spreading of waves when they pass through a narrow aperture. ISC treats the Fraunhofer single-slit pattern qualitatively, including the central maximum and intensity distribution.
a sinθ = nλ (minima)

Polarisation demonstrates the transverse nature of light. Malus' law relates transmitted intensity to the angle between the transmission axes of the polarisers.
I = I₀ cos²θ

When suitable-frequency light falls on a metal surface, electrons may be emitted. Emission depends on frequency and intensity in the characteristic ways observed experimentally.
K_max = hν − W₀

The stopping potential is the reverse potential just sufficient to stop the most energetic photoelectrons.
K_max = eV_s

Threshold frequency is the minimum frequency required for photoemission. Work function is the minimum energy needed to remove an electron from the metal surface.
W₀ = hν₀

A photon has zero rest mass but carries energy and momentum, showing the particle aspect of electromagnetic radiation.
p = E/c = h/λ

Matter particles also exhibit wave nature. The wavelength associated with a particle is inversely proportional to its momentum.
λ = h/p = h/mv

Electron diffraction provides experimental evidence for the wave nature of matter.

Alpha-particle scattering showed that most of the atom is empty space while positive charge and most mass are concentrated in a tiny nucleus. ISC treats the scattering experiment qualitatively; mathematical scattering theory is not required.

Electrons occupy allowed stationary orbits and emit or absorb radiation when they transition between allowed energy states.
mvr = nh/(2π)

For hydrogen-like atoms, allowed orbit radius varies as n² while the magnitude of energy decreases with increasing principal quantum number.
r_n ∝ n²; E_n = −13.6/n² eV

Spectral lines arise when an electron changes between quantised energy levels. The Rydberg equation gives the wavelength.
1/λ = R_H(1/n₁² − 1/n₂²), n₂>n₁

The atomic mass number A is the total number of protons and neutrons. Atomic number Z is the number of protons.
A = Z + N

A bound nucleus has slightly less mass than the total mass of its separated constituent nucleons. The missing mass corresponds to binding energy.
BE = Δmc²

Binding energy per nucleon is a useful measure of nuclear stability. The characteristic curve explains why both fusion of light nuclei and fission of heavy nuclei can release energy.

Fission is splitting a heavy nucleus into lighter nuclei with energy release. Fusion combines light nuclei into a more tightly bound nucleus and can release large energy.

In solids, closely spaced atomic energy levels form bands. Conductors have overlapping/partially filled bands, while semiconductors have a small forbidden energy gap and insulators have a larger gap.

A pure semiconductor has thermally generated electrons and holes. At a given temperature their numbers are equal in the intrinsic material.

Doping introduces controlled impurity atoms. Donor impurities produce n-type material with electrons as majority carriers; acceptor impurities produce p-type material with holes as majority carriers.

Joining p-type and n-type semiconductor creates a depletion region and an internal barrier potential. Biasing changes the depletion region and current flow.

Forward bias reduces the barrier and permits substantial current after the knee region. Reverse bias widens the depletion region and gives a small reverse current until breakdown.

The I–V curve shows the different current behaviour in forward and reverse bias. Numerical and graph-based interpretation must follow the prescribed ISC scope.

A rectifier converts AC into unidirectional pulsating current. ISC includes half-wave and full-wave rectification and the working of the prescribed circuits.

An LED emits light when forward biased. A photodiode detects light and is normally operated in reverse bias. A Zener diode is designed to operate in reverse breakdown.

A Zener diode connected in reverse breakdown can maintain an approximately constant voltage across a load while the input or load changes within its regulation range.

A solar cell converts incident light energy into electrical energy using the photovoltaic effect.