Std. 12
Physics

Rotational motion describes motion about a fixed axis. Angular displacement, angular velocity and angular acceleration describe how the body turns.
ω=dθ/dt α=dω/dt

Torque is the turning effect of a force about an axis. Its magnitude is force times perpendicular distance.
τ=r×F |τ|=rF sinθ

Moment of inertia measures resistance to angular acceleration.
τ=Iα I=Mk²

A rotating body has kinetic energy because its particles are moving.
K=½Iω² L=Iω

Pressure is normal force per unit area. In a liquid, pressure increases with depth.
P=F/A P=ρgh

Pressure applied to an enclosed fluid is transmitted equally in all directions. Hydraulic lifts and brakes use this principle.
F₁/A₁=F₂/A₂

Surface tension makes a liquid surface behave like a stretched membrane. Capillarity describes rise or fall in a narrow tube.
h=2T cosθ/(ρgr)

Viscosity is resistance between neighbouring fluid layers. Bernoulli's equation connects pressure, speed and height.
F=ηA(dv/dx) P+½ρv²+ρgh=constant

An ideal gas is a model in which molecules occupy negligible volume and interact negligibly except during collisions.
PV=nRT=NkT

Gas pressure results from collisions of molecules with the container walls.
P=⅓ρvᵣₘₛ² ½mvᵣₘₛ²=3kT/2

Mean free path is the average distance a molecule travels between two successive collisions.
It depends on molecular size and number density.

Hot bodies emit electromagnetic radiation. A black body is an ideal absorber and emitter.
P=σAT⁴ λₘT=b

Temperature describes thermal state; heat is energy transferred because of temperature difference.
The zeroth law provides the basis for defining temperature.

Energy supplied as heat becomes internal-energy change and/or work done by the system.
ΔQ=ΔU+ΔW

In isothermal, adiabatic, isobaric and isochoric processes, different quantities remain constant.
PV^γ=constant for an ideal-gas adiabatic process.

A heat engine takes heat from a high-temperature reservoir, does work and rejects some heat.
η=1−Q꜀/Qₕ ηCarnot=1−T꜀/Tₕ

A periodic motion repeats after a fixed time T.
f=1/T ω=2πf

In SHM, acceleration is proportional to displacement and directed towards the mean position.
a=−ω²x

The amplitude A is maximum displacement from the mean position.
x=A sin(ωt+φ) v²=ω²(A²−x²)

For a spring and simple pendulum, the period depends on mass/stiffness and length/gravity respectively.
T=2π√(m/k) T=2π√(l/g)

A travelling wave transfers energy while the disturbance travels through the medium.
y=A sin(ωt−kx+φ) v=fλ

When two or more waves overlap, the resultant displacement is the algebraic sum of their individual displacements.
This principle explains interference and beats.

Beats are periodic variations in intensity caused by two waves of slightly different frequencies.
fᵦ=|f₁−f₂|

A stationary wave has nodes of zero amplitude and antinodes of maximum amplitude.
fₙ=nv/(2L) for a string fixed at both ends.

Every point on a wavefront acts as a source of secondary wavelets. Their envelope forms the new wavefront.
This gives a wave-based explanation of propagation.

Constructive interference occurs when path difference is an integral multiple of wavelength; destructive interference occurs for odd half multiples.
β=λD/d for Young’s double-slit fringe width.

Diffraction is the spreading of waves around obstacles or apertures comparable with wavelength.
a sinθ=nλ for single-slit minima.

Polarisation demonstrates the transverse nature of light.
I=I₀cos²θ (Malus law) tan iₚ=μ (Brewster law)

The electrostatic force between two point charges varies directly with the product of charges and inversely with square of separation.
F=(1/4πε₀)|q₁q₂|/r²

Electric field is force per unit positive test charge; electric flux measures field passing through an area.
E=F/q Φₑ=EA cosθ

The total electric flux through a closed surface is proportional to the enclosed charge.
∮E·dA=Q/ε₀

Potential is work done per unit charge. Capacitance measures charge stored per unit potential difference.
V=W/q C=Q/V C=ε₀A/d

Current is the rate of flow of electric charge.
I=Q/t I=nAev_d

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

Electrical power is the rate at which electrical energy is transferred.
P=VI=I²R=V²/R

Kirchhoff's junction and loop rules allow currents and potential differences in complex circuits to be calculated.
Wheatstone bridge balance: P/Q=R/S.

A moving charge in a magnetic field experiences a force perpendicular to its velocity and the field.
F=q(v×B)

A current-carrying conductor experiences magnetic force in an external magnetic field.
F=I(L×B)

Biot–Savart law calculates field from current elements; Ampere's law relates circulation of B to enclosed current.
B=μ₀I/(2πr) for a long straight wire.

A charged particle entering perpendicular to a uniform B follows circular motion.
r=mv/(qB) f=qB/(2πm)

A current loop behaves like a magnetic dipole.
m=IA

Magnetisation is magnetic moment per unit volume. Susceptibility measures response to magnetising field.
χₘ=M/H

Permeability describes how a material supports magnetic field.
B=μH μ=μ₀μᵣ

Diamagnetic materials are weakly repelled, paramagnetic materials weakly attracted and ferromagnetic materials strongly attracted and capable of retaining magnetisation.
Curie temperature marks loss of ferromagnetic ordering.

Magnetic flux is the amount of magnetic field passing through a surface.
Φ=BA cosθ

A changing magnetic flux produces induced emf. The negative sign represents Lenz's law.
e=−N dΦ/dt

A conductor moving through magnetic field can develop an emf.
e=Blv for mutually perpendicular B, l and v.

Self-induction opposes change of current; mutual induction transfers changing magnetic effects between coils.
e=−L di/dt U=½LI² Vₛ/Vₚ=Nₛ/Nₚ

AC reverses direction periodically and is commonly represented by a sine function.
i=I₀sinωt Iᵣₘₛ=I₀/√2

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

Impedance is the total opposition to AC in an LCR circuit.
Z=√[R²+(X_L−X_C)²] Iᵣₘₛ=Vᵣₘₛ/Z

Average power depends on power factor. Resonance occurs when inductive and capacitive reactances are equal.
P=VᵣₘₛIᵣₘₛcosφ ω₀=1/√(LC)

Light exchanges energy in discrete packets called photons.
E=hν=hc/λ p=h/λ

Electrons are emitted from a suitable surface when incident radiation has sufficient frequency.
hν=φ+Kₘₐₓ Kₘₐₓ=eV₀

Minimum frequency needed for photoemission from a material.
φ=hν₀

Particles such as electrons have wave nature described by de Broglie wavelength.
λ=h/p λ=h/√(2meV) for an electron accelerated through V.

Bohr proposed quantised angular momentum for hydrogen-like atoms.
mvr=nℏ

The allowed hydrogen energy levels become more negative for lower n.
Eₙ=−13.6/n² eV

Unstable nuclei spontaneously decay. The decay law gives the number of undecayed nuclei after time t.
N=N₀e^(−λt) T₁/₂=ln2/λ

Mass defect is converted into binding energy.
BE=Δmc² BE≈Δm(u)×931.5 MeV

A semiconductor has conductivity between that of a good conductor and an insulator. Its conductivity changes strongly with temperature and doping.
This concept should be understood qualitatively and connected to the molecular picture.

Joining p-type and n-type material creates a depletion region and a built-in potential barrier.
This concept should be understood qualitatively and connected to the molecular picture.

A diode conducts preferentially in forward bias and blocks strongly in reverse bias. Rectifiers use this property to convert AC to pulsating DC.
I=I₀(e^(qV/ηkT)−1)

A transistor can amplify or switch signals. Zener diodes, LEDs, photodiodes and solar cells have specialised applications.
I_E=I_C+I_B β=I_C/I_B