“Field is a vector, potential is a number”
Adding contributions from several charges
Add fields with directions and components. Add potentials as plain numbers with their signs. Most electrostatics errors are adding fields like numbers.
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Class 12 Physics formula book
Every chapter on one page, to read straight through or print. 13 chapters, 210 formulas with every symbol defined, 45 memory hooks, and the mistakes that cost the most marks.
1Electrostatics
Electrostatics opens Class 12 and sets the pattern for the whole year: a field, a potential, and an energy, for each of which there is one formula and one sign rule. Gauss's law here is the shortcut that turns page-long integrals into single lines.
Force and field
| What | Formula | Symbols and notes |
|---|---|---|
| Coulomb's law | F = kq₁q₂/r² | k = 1/4πε₀ = 9 × 10⁹ N m² C−2 |
| In a medium | Fmedium = Fvacuum/K | K is the dielectric constant of the medium |
| Electric field | E = F/q = kQ/r² | unit N C−1 or V m−1 |
| Field of a dipole, axial | E = 2kp/r³ | along the dipole axis, for r much greater than the dipole length |
| Field of a dipole, equatorial | E = kp/r³ | half the axial value, and opposite in direction |
| Dipole moment | p = q × 2a | directed from the negative charge to the positive |
| Torque on a dipole | τ = pE sinθ | zero when the dipole lines up with the field |
Gauss's law
| What | Formula | Symbols and notes |
|---|---|---|
| Gauss's law | Φ = qenclosed/ε₀ | the flux depends only on the charge inside |
| Infinite line charge | E = λ/2πε₀r | λ is charge per unit length; E falls as 1/r |
| Infinite sheet of charge | E = σ/2ε₀ | independent of distance from the sheet |
| Charged conducting shell, outside | E = kQ/r² | behaves as if all the charge were at the centre |
| Charged conducting shell, inside | E = 0 | which is why a car is safe in a lightning strike |
Potential and energy
| What | Formula | Symbols and notes |
|---|---|---|
| Potential of a point charge | V = kQ/r | a scalar, so potentials simply add |
| Relation to field | E = −dV/dr | the field points from high potential to low |
| Potential energy of two charges | U = kq₁q₂/r | negative for unlike charges |
| Work done | W = q(VB − VA) | independent of the path taken |
| Dipole potential energy | U = −pE cosθ | minimum when aligned with the field |
How to remember this chapter
“Axial is twice equatorial”
Dipole fields
Along the axis 2kp/r³, on the perpendicular bisector kp/r³, and the two point opposite ways.
“Inside a conductor, nothing”
Fields in conductors and shells
The field inside the material of a conductor and inside a hollow charged shell is zero, and all the charge sits on the outer surface.
“Cube of r for dipoles”
How dipole fields fall off
A single charge falls as 1/r², a dipole as 1/r³, because its two opposite charges nearly cancel at a distance.
The mistakes that cost the most
- Adding electric fields as if they were scalars. They are vectors and need components.
- Forgetting that the field inside a charged conducting shell is zero while the potential inside is not; the potential is constant and equal to the surface value.
- Using 1/r² for a dipole field. It falls as 1/r³.
- Dropping the sign of the charge in potential and potential energy, which flips the answer.
- Thinking Gauss's law only works for a sphere. It always holds; it is only useful when the symmetry is high.
2Capacitance
Capacitance is where circuit questions begin. Only four ideas are needed: how capacitance is defined, what a dielectric does, how capacitors combine, and how much energy is stored. Series and parallel behave opposite to resistors, and that swap is where marks are lost.
Capacitance
| What | Formula | Symbols and notes |
|---|---|---|
| Definition | C = Q/V | unit farad; one farad is very large |
| Parallel plate capacitor | C = ε₀A/d | A plate area, d separation |
| With a dielectric | C = Kε₀A/d | K is the dielectric constant, always greater than 1 |
| Spherical capacitor, isolated sphere | C = 4πε₀R | capacitance of a sphere of radius R |
Combinations
| What | Formula | Symbols and notes |
|---|---|---|
| In series | 1/C = 1/C₁ + 1/C₂ + … | charge is the same on each, total C is smaller than the smallest |
| In parallel | C = C₁ + C₂ + … | voltage is the same across each, capacitances add |
| Two capacitors in series | C = C₁C₂/(C₁ + C₂) | the product over sum shortcut |
Energy
| What | Formula | Symbols and notes |
|---|---|---|
| Energy stored | U = ½CV² = ½QV = Q²/2C | three forms of one result |
| Energy density | u = ½ε₀E² | energy per unit volume of the field |
| Loss on connecting two capacitors | ΔU = C₁C₂(V₁ − V₂)²/2(C₁ + C₂) | energy is always lost, as heat and radiation |
| Common potential | V = (C₁V₁ + C₂V₂)/(C₁ + C₂) | after the two are joined |
Dielectrics
| What | Formula | Symbols and notes |
|---|---|---|
| Battery still connected | V constant, C rises, Q rises, U rises | the battery supplies the extra charge |
| Battery disconnected | Q constant, C rises, V falls, U falls | the field pulls the slab in |
| Field inside the dielectric | E = E₀/K | the induced charges oppose the original field |
How to remember this chapter
“Capacitors do it backwards”
Series and parallel
Resistors add in series; capacitors add in parallel. Whenever you are unsure, remember they are the opposite of resistors.
“Connected or cut, ask first”
Dielectric questions
With the battery still connected the voltage is fixed. With it disconnected the charge is fixed. Decide which is constant before writing anything else.
“Half CV squared”
Energy stored
U = ½CV². The half is there because the capacitor charges up gradually, so the average voltage is half the final one.
The mistakes that cost the most
- Adding capacitors in series the way resistors add in series. In series it is the reciprocals that add.
- Forgetting to decide whether the battery stays connected. Every dielectric numerical turns on that one question.
- Using U = QV without the half. The half comes from the charging being gradual.
- Thinking charge is shared equally between series capacitors of different values. In series the charge is the same on each, and it is the voltages that differ.
- Assuming energy is conserved when two charged capacitors are connected. Some is always lost.
3Current Electricity
Current electricity is the most predictable scoring chapter of Class 12. Ohm's law, the two Kirchhoff rules and the Wheatstone bridge cover almost every question, and the potentiometer results are standard derivations worth memorising outright.
Current and resistance
| What | Formula | Symbols and notes |
|---|---|---|
| Electric current | I = Q/t = neAvd | n is the free electron density, vd the drift velocity |
| Drift velocity | vd = eEτ/m = I/neA | τ is the relaxation time; a few millimetres per second |
| Ohm's law | V = IR | holds for metals at constant temperature |
| Resistance of a wire | R = ρL/A | ρ is resistivity, a property of the material |
| Resistivity and temperature | ρt = ρ₀(1 + αΔT) | α is positive for metals, negative for semiconductors |
| Conductivity | σ = 1/ρ = ne²τ/m | unit siemens per metre |
Circuits
| What | Formula | Symbols and notes |
|---|---|---|
| Resistors in series | R = R₁ + R₂ + … | same current through each |
| Resistors in parallel | 1/R = 1/R₁ + 1/R₂ + … | same voltage across each |
| EMF and terminal voltage | V = E − Ir | r is the internal resistance of the cell |
| Kirchhoff's junction rule | ΣI = 0 at a junction | conservation of charge |
| Kirchhoff's loop rule | ΣV = 0 around a loop | conservation of energy |
| Power | P = VI = I²R = V²/R | choose the form that uses what you are given |
Bridges and the potentiometer
| What | Formula | Symbols and notes |
|---|---|---|
| Wheatstone bridge, balanced | P/Q = R/S | no current flows through the galvanometer |
| Metre bridge | R/S = l/(100 − l) | l is the balancing length in centimetres |
| Potentiometer, comparing EMFs | E₁/E₂ = l₁/l₂ | balancing lengths compared directly |
| Internal resistance by potentiometer | r = R(l₁ − l₂)/l₂ | l₁ open circuit, l₂ with resistance R across the cell |
How to remember this chapter
“Slow electrons, fast signal”
Drift velocity
Electrons drift at a few millimetres per second, yet the bulb lights instantly, because the electric field travels at nearly the speed of light and pushes every electron at once.
“Junction is charge, loop is energy”
Kirchhoff's two rules
The junction rule is conservation of charge; the loop rule is conservation of energy. Naming which one you are using earns the mark in a derivation.
“A potentiometer draws nothing”
Why it beats a voltmeter
At balance no current is drawn from the cell, so it measures true EMF. A voltmeter always draws a little current and reads terminal voltage instead.
The mistakes that cost the most
- Using V = IR across a cell without subtracting Ir. The terminal voltage is less than the EMF whenever current flows.
- Adding parallel resistances directly. It is the reciprocals that add, and the result is smaller than the smallest of them.
- Thinking resistance and resistivity are the same. Cutting a wire in half halves its resistance and changes its resistivity not at all.
- Forgetting that a semiconductor's resistance falls as temperature rises, the opposite of a metal.
- Using the metre bridge formula with the balancing length measured from the wrong end.
4Moving Charges and Magnetism
This chapter is where the right hand earns its marks. Every formula here has a direction attached, and the numerical part is usually easier than getting that direction right, so the hand rules deserve more practice than the algebra.
Force on charges and wires
| What | Formula | Symbols and notes |
|---|---|---|
| Lorentz force | F = qvB sinθ | maximum when the velocity is perpendicular to B, zero when parallel |
| Force on a conductor | F = BIL sinθ | the working principle of the electric motor |
| Radius in a magnetic field | r = mv/qB | the faster or heavier the particle, the wider the circle |
| Time period in a magnetic field | T = 2πm/qB | independent of speed, which is what makes the cyclotron work |
| Cyclotron frequency | f = qB/2πm | the resonance condition |
| Velocity selector | v = E/B | only this speed passes undeflected |
Fields from currents
| What | Formula | Symbols and notes |
|---|---|---|
| Biot-Savart law | dB = (μ₀/4π)(I dl sinθ)/r² | μ₀ = 4π × 10−7 T m A−1 |
| Straight infinite wire | B = μ₀I/2πr | circles around the wire, given by the right-hand grip rule |
| Centre of a circular loop | B = μ₀I/2R | perpendicular to the plane of the loop |
| On the axis of a loop | B = μ₀IR²/2(R² + x²)3/2 | reduces to μ₀I/2R at the centre |
| Long solenoid | B = μ₀nI | n is turns per unit length; the field inside is uniform |
| Toroid | B = μ₀NI/2πr | the field outside is zero |
Current loops and instruments
| What | Formula | Symbols and notes |
|---|---|---|
| Force between parallel wires | F/L = μ₀I₁I₂/2πd | same direction attracts, opposite repels |
| Magnetic moment of a loop | m = NIA | N turns, area A |
| Torque on a loop | τ = NIAB sinθ = mB sinθ | how a moving-coil galvanometer works |
| Galvanometer to ammeter | shunt S = IgG/(I − Ig) | a small resistance in parallel |
| Galvanometer to voltmeter | R = V/Ig − G | a large resistance in series |
How to remember this chapter
“Ammeter parallel and low, voltmeter series and high”
Converting a galvanometer
An ammeter takes a small shunt in parallel; a voltmeter takes a large resistance in series. An ideal ammeter has zero resistance, an ideal voltmeter infinite.
“Same way, they hug”
Force between parallel wires
Parallel currents in the same direction attract; opposite directions repel. This is the opposite of what charges do, and it is how the ampere is defined.
“Magnetic force never does work”
Why speed does not change
The force is always perpendicular to the velocity, so it changes direction but never speed. That is why the time period does not depend on how fast the particle is going.
“Grip the wire”
Direction of the field around a wire
Thumb along the current, curled fingers give the circular field. For a loop or solenoid, curl the fingers along the current and the thumb points to the north face.
The mistakes that cost the most
- Expecting a magnetic force on a charge at rest. The force needs motion, and motion not parallel to the field.
- Thinking a magnetic field speeds a particle up. It cannot; it only bends the path.
- Using B = μ₀nI with n as the total number of turns. It is turns per unit length.
- Reversing the attraction rule for parallel wires. Same direction attracts, unlike charges at rest.
- Forgetting that the field is zero outside a toroid and outside an ideal solenoid.
5Magnetism and Matter
A short, mostly theoretical chapter that reliably supplies one or two marks. The bar magnet formulas mirror the electric dipole exactly, and the three kinds of magnetic material are a classification question you can answer in a table.
Bar magnet as a dipole
| What | Formula | Symbols and notes |
|---|---|---|
| Magnetic dipole moment | m = pole strength × 2l | unit A m² |
| Axial field | B = (μ₀/4π)(2m/r³) | on the axis, far from the magnet |
| Equatorial field | B = (μ₀/4π)(m/r³) | half the axial value, opposite direction |
| Torque | τ = mB sinθ | the compass needle lines up with the field |
| Potential energy | U = −mB cosθ | minimum when aligned |
| Time period of oscillation | T = 2π√(I/mB) | I is the moment of inertia of the magnet |
Earth's magnetism
| What | Formula | Symbols and notes |
|---|---|---|
| Components | BH = B cosδ, BV = B sinδ | δ is the angle of dip |
| Angle of dip | tanδ = BV/BH | zero at the magnetic equator, 90° at the poles |
| Declination | the angle between geographic and magnetic north | varies with place and slowly with time |
Materials
| What | Formula | Symbols and notes |
|---|---|---|
| Magnetic intensity | H = B/μ₀ − M | M is the magnetisation |
| Susceptibility | χ = M/H | a pure number |
| Permeability | μ = μ₀(1 + χ) | relative permeability μr = 1 + χ |
| Diamagnetic | χ small and negative | weakly repelled, for example bismuth, copper, water |
| Paramagnetic | χ small and positive | weakly attracted, for example aluminium, oxygen |
| Ferromagnetic | χ large and positive | strongly attracted, for example iron, cobalt, nickel |
| Curie's law | χ ∝ 1/T | heating destroys the alignment |
How to remember this chapter
“Di runs away, para follows, ferro clings”
The three kinds of material
Diamagnetic is weakly repelled, paramagnetic weakly attracted, ferromagnetic strongly attracted and able to stay magnetised.
“Same shape as the electric dipole”
Bar magnet fields
Axial 2m over r cubed, equatorial m over r cubed, exactly as for an electric dipole with p replaced by m.
“Dip is zero at the equator”
Earth's magnetic field
A dip needle lies flat at the magnetic equator and stands vertical at the magnetic poles.
The mistakes that cost the most
- Saying magnetic field lines start at the north pole and stop at the south. They are closed loops and continue inside the magnet.
- Believing a single magnetic pole can exist. Cutting a magnet in half gives two complete magnets.
- Mixing up dip and declination. Dip is measured vertically, declination horizontally.
- Forgetting that heating a ferromagnet past its Curie temperature makes it paramagnetic.
- Using B where H is asked in a magnetisation question; they have different units.
6Electromagnetic Induction
Induction is one sentence with many consequences: a changing magnetic flux makes an EMF, and the minus sign in Lenz's law is where the physics lives. Almost every question here is a test of whether you can say what the induced current is opposing.
Flux and Faraday's law
| What | Formula | Symbols and notes |
|---|---|---|
| Magnetic flux | Φ = BA cosθ | unit weber; θ is measured from the normal to the area |
| Faraday's law | e = −dΦ/dt | for N turns, e = −N dΦ/dt |
| Motional EMF | e = Blv | a rod of length l moving with speed v across a field |
| Induced current | I = e/R | and the charge that flows is q = ΔΦ/R |
| Rotating coil | e = NABω sin(ωt) | the basis of the AC generator |
Inductance
| What | Formula | Symbols and notes |
|---|---|---|
| Self inductance | e = −L dI/dt | L is in henry |
| Inductance of a solenoid | L = μ₀n²Al | n is turns per unit length |
| Energy stored in an inductor | U = ½LI² | compare with ½CV² for a capacitor |
| Mutual inductance | e₂ = −M dI₁/dt | M depends on the geometry and the coupling |
| Coils in series | L = L₁ + L₂ ± 2M | plus when the fluxes help each other |
Eddy currents and transformers
| What | Formula | Symbols and notes |
|---|---|---|
| Transformer ratio | Vs/Vp = Ns/Np = Ip/Is | step-up means more turns on the secondary and less current |
| Ideal transformer | VpIp = VsIs | power in equals power out |
| Efficiency | η = output power/input power | losses are copper, iron, hysteresis and flux leakage |
| Eddy currents | circulating currents in a solid core | reduced by laminating the core |
How to remember this chapter
“Nature hates the change”
Lenz's law
The induced current always opposes the change that caused it. Push a magnet in and it pushes back; pull it out and it pulls after you. This is conservation of energy wearing a minus sign.
“Transformers change voltage, never power”
Step-up and step-down
More turns means more volts and less current. An ideal transformer gives out exactly what it takes in, so nothing is gained for free.
“Transformers refuse DC”
Why a transformer needs AC
Steady current gives constant flux, so dΦ/dt is zero and no EMF is induced. Only a changing current works.
The mistakes that cost the most
- Dropping the minus sign in Faraday's law. It is Lenz's law and it is usually worth a mark on its own.
- Thinking an EMF needs a strong field. It needs a changing flux; a strong steady field induces nothing.
- Confusing flux with field. Flux includes the area and the angle.
- Expecting a transformer to work on DC.
- Forgetting that the induced charge q = ΔΦ/R depends on the total change in flux, not on how fast it changed.
7Alternating Current
AC takes the DC circuit you already know and adds phase. Reactance replaces resistance for capacitors and inductors, impedance ties them together, and resonance is the one result asked every single year.
RMS and average
| What | Formula | Symbols and notes |
|---|---|---|
| RMS current | Irms = I₀/√2 = 0.707I₀ | what an AC ammeter reads |
| RMS voltage | Vrms = V₀/√2 | 230 V mains means a peak of about 325 V |
| Average over a full cycle | zero | which is why RMS values are used at all |
| Angular frequency | ω = 2πf | f is 50 Hz in India |
Reactance and impedance
| What | Formula | Symbols and notes |
|---|---|---|
| Inductive reactance | XL = ωL = 2πfL | rises with frequency; an inductor blocks high frequency |
| Capacitive reactance | XC = 1/ωC = 1/2πfC | falls with frequency; a capacitor blocks DC |
| Impedance of a series LCR | Z = √(R² + (XL − XC)²) | the AC version of resistance |
| Phase angle | tanφ = (XL − XC)/R | current lags when inductive, leads when capacitive |
| Current | Irms = Vrms/Z | Ohm's law with impedance in place of resistance |
Power and resonance
| What | Formula | Symbols and notes |
|---|---|---|
| Average power | P = VrmsIrms cosφ | cosφ is the power factor |
| Power factor | cosφ = R/Z | one for a pure resistor, zero for a pure inductor or capacitor |
| Wattless current | P = 0 when φ = 90° | a pure inductor or capacitor consumes no average power |
| Resonant frequency | f₀ = 1/2π√(LC) | at resonance XL = XC |
| At resonance | Z = R, current is maximum, φ = 0 | the circuit behaves as a pure resistor |
| Quality factor | Q = (1/R)√(L/C) | sharper resonance means a higher Q |
How to remember this chapter
“ELI the ICE man”
Who leads, voltage or current
In an inductor (L), E leads I: ELI. In a capacitor (C), I leads E: ICE. One phrase settles every phase question in the chapter.
“Inductors hate fast, capacitors hate slow”
How reactance changes with frequency
XL rises with frequency and XC falls. That is why a capacitor blocks DC completely and an inductor passes it freely.
“Resonance is a tie”
What happens at f₀
The two reactances cancel exactly, impedance drops to R, and the current peaks. It is how a radio picks one station out of many.
The mistakes that cost the most
- Using peak values where RMS is meant. Meters read RMS, and the mains 230 V is RMS.
- Adding the reactances to the resistance directly. They are at 90°, so they combine as a right triangle.
- Forgetting the power factor. Without cosφ the power is wrong for every circuit except a pure resistor.
- Thinking a capacitor or inductor consumes power. Over a full cycle they consume none.
- Believing the voltage across L can never exceed the source voltage. At resonance it can, and by a large factor.
8Electromagnetic Waves
A short chapter with few formulas and reliable marks. Most questions ask for the order of the spectrum, the uses of one band, or the relation between the electric and magnetic field amplitudes, so it rewards memory more than method.
The wave
| What | Formula | Symbols and notes |
|---|---|---|
| Speed in vacuum | c = 1/√(μ₀ε₀) = 3 × 10⁸ m s−1 | the same for every wavelength |
| Speed in a medium | v = 1/√(με) = c/n | n is the refractive index |
| Field amplitudes | E₀ = cB₀ | the electric field is far larger in number, not in effect |
| Directions | E, B and the direction of travel are mutually perpendicular | E × B points the way the wave goes |
| Energy density | u = ½ε₀E² + B²/2μ₀ | shared equally between the two fields |
| Intensity | I = ½ε₀cE₀² | power per unit area |
| Displacement current | Id = ε₀ dΦE/dt | Maxwell's addition, which completes Ampere's law |
The spectrum, longest to shortest
| What | Formula | Symbols and notes |
|---|---|---|
| Radio waves | > 0.1 m | radio, television, mobile communication |
| Microwaves | 0.1 m to 1 mm | radar, microwave ovens |
| Infrared | 1 mm to 700 nm | heating, remote controls, night vision |
| Visible light | 700 nm to 400 nm | the only part the eye detects |
| Ultraviolet | 400 nm to 1 nm | absorbed by the ozone layer, causes sunburn |
| X-rays | 1 nm to 10−3 nm | medical imaging |
| Gamma rays | < 10−3 nm | from nuclei, used to treat cancer |
How to remember this chapter
“Radio Men In Very Unusual X-ray Gowns”
Order of the spectrum
Radio, Microwave, Infrared, Visible, Ultraviolet, X-ray, Gamma, in order of decreasing wavelength and increasing frequency and energy.
“E is c times B”
The two field amplitudes
E₀ = cB₀, so the numbers differ by a factor of 3 × 10⁸, even though the two fields carry equal energy.
“ROYGBIV shrinks”
Wavelengths inside visible light
Red is the longest visible wavelength and violet the shortest, so violet carries the most energy per photon.
The mistakes that cost the most
- Thinking an electromagnetic wave needs a medium. It does not, which is how sunlight crosses space.
- Writing the spectrum in the wrong direction. Check against a known pair: radio is longest, gamma shortest.
- Assuming the electric field matters more because E₀ is a much bigger number. The energy is shared equally.
- Forgetting that the speed is the same for all electromagnetic waves in vacuum, whatever the frequency.
- Leaving out displacement current when asked why Ampere's law needed correcting.
9Ray Optics and Optical Instruments
Ray optics carries the most numerical marks in Class 12 physics, and nearly all of them hang on one habit: fixing the sign convention before touching the formula. Learn the convention once, properly, and mirrors, lenses and prisms all follow.
The sign convention
| What | Formula | Symbols and notes |
|---|---|---|
| Origin | all distances are measured from the pole or optical centre | the incident light travels left to right |
| Along the light | distances measured with the light are positive | against it, negative |
| Heights | upwards from the axis is positive | downwards negative, so an inverted image has negative height |
| Result | real object distance u is negative for both mirrors and lenses | this single line prevents most sign errors |
Mirrors
| What | Formula | Symbols and notes |
|---|---|---|
| Mirror formula | 1/v + 1/u = 1/f | f = R/2 for a spherical mirror |
| Magnification | m = −v/u = h'/h | negative m means an inverted, real image |
| Concave mirror | f is negative | can give real or virtual images |
| Convex mirror | f is positive | always a virtual, erect, diminished image |
Refraction and lenses
| What | Formula | Symbols and notes |
|---|---|---|
| Snell's law | n₁ sin i = n₂ sin r | light bends towards the normal entering a denser medium |
| Refractive index | n = c/v = real depth/apparent depth | always greater than 1 |
| Critical angle | sin C = 1/n | total internal reflection beyond this angle, denser to rarer only |
| Lens formula | 1/v − 1/u = 1/f | note the minus, unlike the mirror formula |
| Lens maker's formula | 1/f = (n − 1)(1/R₁ − 1/R₂) | for a thin lens in air |
| Power of a lens | P = 1/f (metres) | unit dioptre; converging lenses are positive |
| Lenses in contact | 1/f = 1/f₁ + 1/f₂, P = P₁ + P₂ | powers simply add |
| Magnification | m = v/u | positive means erect and virtual for a lens |
Prism and instruments
| What | Formula | Symbols and notes |
|---|---|---|
| Prism relation | A + δ = i + e | A is the prism angle, δ the deviation |
| At minimum deviation | n = sin((A + δm)/2)/sin(A/2) | i = e and the ray passes symmetrically |
| Simple microscope | m = 1 + D/f | D = 25 cm, the least distance of distinct vision |
| Compound microscope | m = (v₀/u₀)(1 + D/fe) | the objective has the shorter focal length |
| Astronomical telescope | m = f₀/fe, length = f₀ + fe | the objective has the longer focal length |
How to remember this chapter
“Mirror adds, lens subtracts”
The two formulas
1/v + 1/u = 1/f for mirrors, 1/v − 1/u = 1/f for lenses. The sign in the middle is the only difference, and it is the one most often written wrong.
“Real is inverted, virtual is erect”
Checking an answer
If your working gives a real image that is upright, or a virtual one that is inverted, a sign has gone wrong somewhere. Use it as a check on every numerical.
“Denser to rarer only”
Total internal reflection
TIR needs light going from denser to rarer and an angle beyond the critical angle. Both conditions, every time, which is how optical fibres and diamonds work.
“Microscope objective small, telescope objective large”
Which focal length goes where
A microscope needs a very short objective focal length; a telescope needs a very long one. Their eyepieces are short in both.
The mistakes that cost the most
- Starting a numerical before fixing the sign convention. Almost every wrong answer in this chapter is a sign.
- Using the mirror formula for a lens. The middle sign differs.
- Expecting total internal reflection when light passes from rarer to denser. It cannot happen that way.
- Forgetting that a convex mirror and a concave lens always give a virtual, erect, diminished image.
- Using centimetres for focal length when calculating power. Power needs the focal length in metres.
10Wave Optics
Wave optics is where light stops being a ray and becomes a wave. Young's double slit gives the standard derivation and the standard numerical, and the difference between interference and diffraction is a theory question asked almost every year.
Interference
| What | Formula | Symbols and notes |
|---|---|---|
| Path difference for maxima | Δ = nλ | bright fringe, n = 0, 1, 2, … |
| Path difference for minima | Δ = (2n − 1)λ/2 | dark fringe |
| Fringe width | β = λD/d | D slit-to-screen distance, d slit separation |
| Position of the nth bright fringe | yn = nλD/d | measured from the central maximum |
| Intensity | I = I₁ + I₂ + 2√(I₁I₂)cosφ | maximum when the waves are in step |
| Ratio of maximum to minimum | Imax/Imin = (√I₁ + √I₂)²/(√I₁ − √I₂)² | in terms of amplitudes, ((a₁ + a₂)/(a₁ − a₂))² |
| In a medium | β' = β/n | the fringes crowd together |
Diffraction and polarisation
| What | Formula | Symbols and notes |
|---|---|---|
| First minimum, single slit | a sinθ = λ | a is the slit width |
| Width of the central maximum | 2λD/a | twice as wide as the other maxima |
| Resolving power of a telescope | θ = 1.22λ/D | the smaller the angle, the better |
| Malus's law | I = I₀cos²θ | θ between the polariser and analyser axes |
| Brewster's law | tan ip = n | the reflected light is fully polarised at this angle |
| Unpolarised through one polaroid | I = I₀/2 | half the intensity always gets through |
How to remember this chapter
“Two slits interfere, one slit diffracts”
Telling the two apart
Interference needs two sources and gives fringes of equal width and equal brightness. Diffraction comes from one slit and gives a wide bright centre with fading edges.
“Wider slit, narrower pattern”
Single slit diffraction
The central maximum width is 2λD/a, so a wider slit gives a narrower pattern. Everything in diffraction is inverse to what feels natural.
“Half goes through the first”
Unpolarised light and polaroids
An unpolarised beam loses exactly half its intensity at the first polaroid. Malus's law only applies from the second one onwards.
“Only transverse waves polarise”
What polarisation proves
Sound cannot be polarised and light can, which is the evidence that light is a transverse wave.
The mistakes that cost the most
- Applying Malus's law to the first polaroid with unpolarised light. That one simply halves the intensity.
- Mixing the conditions for maxima and minima. Interference maxima need a whole number of wavelengths of path difference.
- Forgetting that the central diffraction maximum is twice the width of the others.
- Thinking interference creates energy at the bright fringes. It only redistributes it, which is why dark fringes exist.
- Using the air value of wavelength when the apparatus is immersed in a liquid.
11Dual Nature of Radiation and Matter
This chapter is short, conceptual, and nearly free marks if the photoelectric graphs are learned. Its central point is that light arrives in packets, so intensity changes how many electrons come out and frequency changes how fast they leave.
Photons
| What | Formula | Symbols and notes |
|---|---|---|
| Energy of a photon | E = hν = hc/λ | h = 6.626 × 10−34 J s |
| In electron volts | E (eV) = 1240/λ (nm) | the shortcut worth memorising |
| Momentum of a photon | p = h/λ = E/c | a photon has momentum but no rest mass |
| Number of photons per second | n = P/hν | P is the power of the source |
Photoelectric effect
| What | Formula | Symbols and notes |
|---|---|---|
| Einstein's equation | hν = φ₀ + KEmax | φ₀ is the work function |
| Maximum kinetic energy | KEmax = hν − φ₀ = h(ν − ν₀) | zero at the threshold frequency |
| Threshold frequency | ν₀ = φ₀/h | below this no electron comes out, however bright the light |
| Stopping potential | eV₀ = KEmax = hν − φ₀ | V₀ depends on frequency, never on intensity |
| Photocurrent | proportional to intensity | more photons means more electrons |
Matter waves
| What | Formula | Symbols and notes |
|---|---|---|
| de Broglie wavelength | λ = h/p = h/mv | every moving particle has one |
| For an accelerated electron | λ = h/√(2meV) = 12.27/√V Å | V in volts |
| For a particle at temperature T | λ = h/√(3mkT) | thermal de Broglie wavelength |
How to remember this chapter
“Brightness gives number, colour gives energy”
Intensity versus frequency
More intense light ejects more electrons but not faster ones. Higher frequency light ejects faster electrons but not more of them.
“1240 over lambda”
Photon energy in electron volts
E in eV equals 1240 divided by the wavelength in nanometres. A 620 nm photon carries 2 eV, and the arithmetic is done.
“No delay, no threshold, no argument”
Why the wave theory failed
Emission is instantaneous, and below the threshold frequency nothing happens at all, however long you wait. Waves could explain neither.
The mistakes that cost the most
- Thinking brighter light gives faster electrons. It gives more of them at the same maximum speed.
- Using the work function in joules with the frequency and then reporting electron volts, or the reverse. Fix the unit first.
- Expecting emission below the threshold frequency if you wait long enough. It never comes.
- Giving a photon a rest mass. It has energy and momentum but no rest mass.
- Forgetting that the stopping potential depends only on the frequency and the metal, never on the intensity.
12Atoms and Nuclei
Atoms and nuclei supplies the Bohr model results and the decay laws, both of which are pure formula work. The energy level diagram for hydrogen is worth sketching until it is automatic, because half the questions here start from it.
The Bohr model
| What | Formula | Symbols and notes |
|---|---|---|
| Quantisation of angular momentum | mvr = nh/2π | n is the principal quantum number |
| Radius of the nth orbit | rn = 0.529n²/Z Å | for hydrogen, Z = 1 |
| Energy of the nth level | En = −13.6Z²/n² eV | negative, because the electron is bound |
| Energy of a transition | ΔE = 13.6(1/n₁² − 1/n₂²) eV | for hydrogen |
| Rydberg formula | 1/λ = R(1/n₁² − 1/n₂²) | R = 1.097 × 10⁷ m−1 |
| Velocity in the nth orbit | vn = 2.18 × 10⁶ Z/n m s−1 | about c/137 for hydrogen's ground state |
Spectral series of hydrogen
| What | Formula | Symbols and notes |
|---|---|---|
| Lyman | n₁ = 1 | ultraviolet |
| Balmer | n₁ = 2 | visible, the series actually seen |
| Paschen | n₁ = 3 | infrared |
| Brackett and Pfund | n₁ = 4 and 5 | far infrared |
The nucleus
| What | Formula | Symbols and notes |
|---|---|---|
| Nuclear radius | R = R₀A1/3 | R₀ = 1.2 × 10−15 m |
| Nuclear density | about 2.3 × 10¹⁷ kg m−3 | the same for every nucleus |
| Mass defect | Δm = [Zmp + (A − Z)mn] − M | the missing mass became binding energy |
| Binding energy | BE = Δm × 931.5 MeV per u | the conversion worth memorising |
| Binding energy per nucleon | highest near A = 56, iron | which is why both fusion and fission release energy |
Radioactivity
| What | Formula | Symbols and notes |
|---|---|---|
| Decay law | N = N₀e−λt | λ is the decay constant |
| Half-life | T1/2 = 0.693/λ | independent of how much you started with |
| Mean life | τ = 1/λ = T1/2/0.693 | always longer than the half-life |
| After n half-lives | N = N₀/2n | the quickest way to answer most questions |
| Activity | A = λN | unit becquerel; 1 curie = 3.7 × 10¹⁰ Bq |
| Alpha decay | A falls by 4, Z falls by 2 | a helium nucleus leaves |
| Beta minus decay | A unchanged, Z rises by 1 | a neutron becomes a proton |
How to remember this chapter
“Minus 13.6 over n squared”
Hydrogen energy levels
Ground state −13.6 eV, then −3.4, −1.51, −0.85. The gaps shrink quickly, which is why the spectral lines crowd together at the high end.
“Balmer is the one you can see”
Which series is visible
Lyman is ultraviolet, Balmer visible, Paschen infrared. Only Balmer ends on n = 2 and only Balmer reaches the eye.
“Halve it n times”
Decay numericals
After n half-lives, N₀ over 2 to the n. For anything that is a whole number of half-lives, you never need the exponential at all.
“Iron is the bottom of the valley”
Why fusion and fission both release energy
Binding energy per nucleon peaks near iron. Light nuclei release energy by joining, heavy ones by splitting, both moving towards iron.
The mistakes that cost the most
- Forgetting the minus sign on the energy levels, which makes every transition energy come out wrong.
- Mixing up mean life and half-life. Mean life is the longer of the two, by a factor of about 1.44.
- Thinking half-life depends on the amount of the sample or on the temperature. It depends on neither.
- Using A1/3 as A/3 in the nuclear radius formula.
- Saying the Bohr model works for every atom. It works only for hydrogen and hydrogen-like single-electron ions.
13Semiconductor Electronics
The last chapter of the course, and often the most scoring, because the questions repeat. Doping, the p-n junction, the two rectifiers and the logic gates account for nearly every mark, and none of them need heavy algebra.
Semiconductors
| What | Formula | Symbols and notes |
|---|---|---|
| Intrinsic semiconductor | ne = nh = ni | pure silicon or germanium |
| Mass action law | nenh = ni² | holds for doped material too |
| n-type | pentavalent dopant, electrons are the majority carriers | phosphorus, arsenic, antimony |
| p-type | trivalent dopant, holes are the majority carriers | boron, aluminium, indium |
| Energy gap | Si about 1.1 eV, Ge about 0.7 eV | an insulator's gap is far larger |
| Conductivity | σ = e(neμe + nhμh) | μ is the mobility |
The p-n junction
| What | Formula | Symbols and notes |
|---|---|---|
| Barrier potential | about 0.7 V for silicon, 0.3 V for germanium | the knee voltage of the diode |
| Forward bias | p to positive, depletion layer narrows, current flows | resistance is low |
| Reverse bias | p to negative, depletion layer widens, tiny leakage only | resistance is high |
| Half wave rectifier | output frequency = input frequency | uses one diode, efficiency about 40.6% |
| Full wave rectifier | output frequency = 2 × input frequency | uses two or four diodes, efficiency about 81.2% |
| Ripple factor | half wave 1.21, full wave 0.48 | lower is smoother |
| Zener diode | works in reverse breakdown as a voltage regulator | heavily doped, thin depletion layer |
Logic gates
| What | Formula | Symbols and notes |
|---|---|---|
| OR | Y = A + B | output 1 if any input is 1 |
| AND | Y = A·B | output 1 only if all inputs are 1 |
| NOT | Y = Ā | the inverter |
| NAND | Y = (A·B)‾ | AND followed by NOT; a universal gate |
| NOR | Y = (A + B)‾ | OR followed by NOT; also universal |
| XOR | Y = AB̄ + ĀB | output 1 when the inputs differ |
How to remember this chapter
“n gives negatives, p gives positives”
Which dopant makes which type
Pentavalent doping gives spare electrons, so n-type. Trivalent doping leaves holes, so p-type. The letter tells you the majority carrier's sign.
“Forward is narrow, reverse is wide”
The depletion layer
Forward bias narrows the depletion layer and current flows. Reverse bias widens it and the current nearly stops.
“Full wave doubles”
Rectifier output frequency
A half wave rectifier passes 50 Hz through as 50 Hz; a full wave gives 100 Hz, and it is smoother because it uses both halves of the cycle.
“NAND and NOR build everything”
Universal gates
Either one alone can be wired to make every other gate, which is why whole chips are built from them.
The mistakes that cost the most
- Saying a doped semiconductor is charged. Doping adds carriers but the material stays electrically neutral.
- Thinking a diode conducts in reverse bias. Only a tiny leakage current flows, until breakdown.
- Giving the full wave rectifier the same output frequency as the input. It is doubled.
- Forgetting that a semiconductor's resistance falls as temperature rises, unlike a metal's.
- Connecting a Zener diode in forward bias for regulation. It regulates in reverse breakdown.
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