Electric Charge Distribution and Quantisation

Electric charge is a fundamental property of matter that comes in two kinds, positive and negative. Like charges repel and unlike charges attract. Three properties define how charge behaves: it is additive (total charge is the algebraic sum), conserved (it can be transferred but never created or destroyed in an isolated system), and quantised (any observable charge is an integral multiple of the elementary charge e = 1.6 × 10⁻¹⁹ C, so q = ne).

At the macroscopic scale, the charge on an object is so large compared with e that quantisation goes unnoticed and charge appears continuous. This is why we can describe a charged body using smooth charge densities. When charge is spread along a line, over a surface, or through a volume, we use linear (λ), surface (σ) and volume (ρ) charge density respectively. For non-uniform distributions the density varies with position and charge is found by integration, for example Q = ∫λ dl.

Ways of charging

• Friction: electrons transfer from one body to another (glass rubbed with silk becomes positive).
• Conduction: contact with a charged body shares charge; both end up with the same sign.
• Induction: a charged body brought near a conductor redistributes its charge; earthing then leaves the conductor with a charge of opposite sign.

Coulomb’s Law in Multi-Charge Systems

Coulomb’s law states that the electrostatic force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. The force acts along the line joining the charges, is attractive for unlike charges and repulsive for like charges, and obeys Newton’s third law.

When more than two charges are present, the principle of superposition applies: the force on any charge is the vector sum of the forces due to each of the other charges taken one at a time. The presence of other charges does not change the pairwise force. Vector addition therefore becomes the key skill: resolve each force into components, add components, then find magnitude and direction.

In a medium of relative permittivity (dielectric constant) K, the force reduces by the factor K. Compared with gravity, the electrostatic force between an electron and a proton is about 10³⁹ times stronger, which is why electrical forces dominate atomic and molecular structure.

Electric Field Due to Continuous Charge Distributions

The electric field at a point is the force per unit positive test charge placed there, E = F/q₀. For a point charge, E = kq/r² directed radially outward for positive charge and inward for negative. Field lines begin on positive charges, end on negative charges, never intersect, and are denser where the field is stronger.

For a continuous distribution, divide the body into small elements dq, write the field dE of each element, and integrate. The skill lies in choosing dq well and exploiting symmetry so that components cancel. For a rod use dq = λ dx, for a ring use dq = λ R dθ, and for a disc use thin concentric rings of radius r and width dr, with dq = σ 2πr dr.

Electric Dipoles in External Electric Fields

An electric dipole is a pair of equal and opposite charges ±q separated by a small distance 2a. Its strength is the dipole moment p = q × 2a, a vector pointing from the negative to the positive charge. Many molecules such as H₂O and HCl are natural dipoles.

In a uniform field, the net force on a dipole is zero because the forces on the two charges are equal and opposite, but a torque acts that tends to rotate it into alignment with the field. In a non-uniform field there is also a net force, which draws the dipole toward the region of stronger field. This explains why small pieces of paper are attracted to a charged comb.

Electric Flux and Applications of Gauss’s Law

Electric flux measures how much electric field passes through a surface: Φ = ∮E·dA. It is a scalar, with sign taken positive for field lines leaving a closed surface and negative for entering. Gauss’s law states that the total flux through any closed surface equals the net enclosed charge divided by ε₀, regardless of the surface’s shape or where inside the charge sits.

The law is always true, but it is useful for finding E only when symmetry (spherical, cylindrical or planar) lets us pull E out of the integral. The method: choose a Gaussian surface on which E is constant and perpendicular to the surface (or parallel, contributing zero), calculate the flux, and equate to q_enc/ε₀. Only enclosed charge matters for the total flux, but E itself at the surface is due to all charges, inside and outside.

Electric Potential and Equipotential Surfaces

Electric potential at a point is the work done per unit positive charge in bringing it from infinity to that point, V = W/q. Potential difference between two points is the work per unit charge moving between them. Potential is a scalar, so contributions from several charges simply add algebraically, which is often much easier than adding vector fields.

Field and potential are linked by E = −dV/dr: the field points in the direction of fastest decrease of potential and its magnitude equals the potential gradient. An equipotential surface is one on which V is constant. No work is done along it, so the field is always perpendicular to equipotential surfaces. The surface of a conductor in electrostatic equilibrium is an equipotential, and so is the entire interior.

Electrostatic Potential Energy of Charge Systems

Electrostatic potential energy is the work required to assemble a configuration of charges by bringing them in from infinity. It is stored in the arrangement of charges and is independent of the path taken to build it. For two charges the energy is positive if they are alike (work must be done against repulsion) and negative if unlike (the system is bound).

For three or more charges, add the interaction energy of every distinct pair exactly once. A charge q in an external potential V has energy U = qV. The work done by an external agent in moving a charge slowly equals the change in potential energy, while the work done by the field is its negative.

Capacitors and Dielectric Materials

A capacitor consists of two conductors separated by an insulator and stores energy in the electric field between them. Its capacitance C = Q/V depends only on the geometry of the conductors and the medium between them, never on the charge or voltage. The SI unit is the farad (F); practical values are in μF, nF and pF.

Inserting a dielectric places polarisable molecules in the field. They align (or are distorted) so that induced surface charges produce an opposing field, reducing the net field within the dielectric to E₀/K. For the same charge, the potential difference falls by K and capacitance rises by K. A conducting slab behaves like K → ∞.

Battery connected vs disconnected

Combination and Energy Storage of Capacitors

In a series combination the same charge flows onto every capacitor and the voltages add, so reciprocals of capacitance add. The equivalent capacitance is smaller than the smallest individual one. In a parallel combination every capacitor has the same potential difference and charges add, so capacitances add directly.

The energy stored equals the work done in charging, which is the area under the V–q graph. It resides in the electric field itself, with energy density u = ½ε₀E² (or ½Kε₀E² in a dielectric). When a charged capacitor is connected to an uncharged one, charge redistributes until potentials are equal, and some energy is always lost as heat and radiation even though charge is conserved.

Drift Velocity and Microscopic Theory of Current

Inside a metal, free electrons move randomly at high thermal speeds (about 10⁵ m s⁻¹) but with zero net displacement. When an electric field is applied, each electron is accelerated between collisions with lattice ions, giving a very small average drift superimposed on the random motion. The mean time between collisions is the relaxation time τ.

The drift speed is surprisingly small (a fraction of a millimetre per second) yet a bulb lights almost instantly because the electric field is established throughout the wire at nearly the speed of light, setting all electrons in motion together. Mobility μ = v_d/E measures how easily carriers drift, and microscopic Ohm’s law links current density to field: J = σE.

Electrical Resistivity and Temperature Dependence

Ohm’s law states that for a conductor at constant temperature V = IR. Resistance R depends on length, area and material: R = ρL/A. The property ρ is the resistivity, a characteristic of the material. Microscopically, ρ = m/(ne²τ), so it increases when collisions become more frequent (smaller τ).

In metals, heating increases lattice vibrations, reducing τ, so resistivity rises roughly linearly with temperature. In semiconductors and insulators, heating releases many more carriers, so resistivity decreases (negative temperature coefficient). Alloys like manganin and constantan have very small temperature coefficients and are used in standard resistors. Some materials become superconducting, with zero resistance below a critical temperature.

Kirchhoff’s Rules and Complex Electrical Circuits

Some circuits cannot be simplified using series and parallel rules. Kirchhoff’s two rules solve any network. The junction rule says that total current entering a junction equals total leaving it (conservation of charge). The loop rule says that the algebraic sum of potential changes around any closed loop is zero (conservation of energy).

Procedure: assign current directions arbitrarily, mark each element’s polarity, pick loops, and write the loop rule with a consistent sign convention. Crossing a resistor in the direction of current is a drop (−IR); against the current is a rise (+IR). Crossing a cell from − to + is a rise (+ε). Write as many independent equations as there are unknowns. Remember also that a real cell has internal resistance r, so its terminal voltage is V = ε − Ir while discharging.

Wheatstone Bridge and Electrical Measurement

A Wheatstone bridge has four resistors P, Q, R, S in a diamond with a galvanometer across one diagonal and a cell across the other. When the bridge is balanced, the two junctions connected to the galvanometer are at the same potential, so no current flows through it. This gives P/Q = R/S, which lets an unknown resistance be found without needing the cell’s emf or internal resistance, a strong advantage for accurate measurement.

The metre bridge is a practical form: a uniform wire of length 100 cm acts as the ratio arm. The potentiometer is another null-method device. It compares emfs (ε₁/ε₂ = l₁/l₂) and measures internal resistance without drawing current from the cell at balance, making it more accurate than a voltmeter.

Motion of Charged Particles in Magnetic Fields

A magnetic field exerts the Lorentz force F = qv × B on a moving charge. Because this force is always perpendicular to velocity, it changes the direction but never the speed or kinetic energy of the particle; the field does no work. A charge at rest feels no magnetic force.

If v is perpendicular to B, the particle moves in a circle because the magnetic force supplies the centripetal force. If v makes an angle with B, it follows a helix: circular motion due to the perpendicular component and uniform motion along the field due to the parallel component. If v is parallel to B, the particle goes undeflected. The time period of circular motion is independent of speed, the principle behind the cyclotron. When both fields exist, F = q(E + v × B), and a velocity selector passes only particles with v = E/B.

Magnetic Force on Current-Carrying Conductors

A conductor carrying current in a magnetic field feels a force because the field acts on each moving charge carrier and the force is transmitted to the lattice. For a straight wire, F = IL × B. Two parallel conductors exert forces on each other: parallel currents attract and antiparallel currents repel. This effect defines the SI unit of current, the ampere.

A current-carrying loop in a uniform field experiences zero net force but a torque τ = m × B with magnetic moment m = NIA. This is the working principle of the moving-coil galvanometer, where a restoring spring torque balances magnetic torque so that deflection is proportional to current. A galvanometer becomes an ammeter when a small shunt resistance is placed in parallel, and a voltmeter when a large resistance is placed in series.

Fields produced by currents

Long straight wire: B = μ₀I/2πr. Centre of a circular loop: B = μ₀I/2R (for N turns, multiply by N). Solenoid: B = μ₀nI inside, where n is turns per unit length. Here μ₀ = 4π × 10⁻⁷ T m A⁻¹.

Magnetic Dipoles and Properties of Magnetic Materials

A current loop is equivalent to a magnetic dipole with moment m = NIA, directed by the right-hand rule. A bar magnet behaves similarly, and magnetic field lines form continuous closed loops, so no isolated magnetic monopole exists and the net magnetic flux through a closed surface is always zero (Gauss’s law for magnetism). In a uniform field a dipole has torque τ = mB sinθ and energy U = −m·B.

Materials respond to an applied field through their magnetisation M and magnetic susceptibility χ. Diamagnetic substances have small negative χ and are slightly repelled (bismuth, copper, water). Paramagnetic substances have small positive χ and are weakly attracted (aluminium, sodium); their susceptibility follows Curie’s law χ ∝ 1/T. Ferromagnetic substances (iron, cobalt, nickel) have very large positive χ due to domains, and become paramagnetic above the Curie temperature. Their magnetisation traces a hysteresis loop, which defines retentivity and coercivity.

Electromagnetic Induction and Faraday’s Law

Faraday discovered that a changing magnetic flux through a circuit induces an emf. Magnetic flux is Φ = B·A = BA cosθ, with unit weber (Wb). Flux can change in three ways: by changing B, by changing the area A, or by changing the orientation angle θ. A steady flux, however large, induces nothing.

Faraday’s law states that the induced emf equals the rate of change of flux linkage: ε = −N dΦ/dt. A conductor of length l moving with speed v perpendicular to a uniform field has motional emf ε = Blv. The induced current through a closed circuit of resistance R is I = ε/R, and the charge that flows depends only on the total flux change, q = ΔΦ/R, not on how quickly it happens. Practical applications include AC generators, transformers, induction cooktops and magnetic braking by eddy currents.

Lenz’s Law and Induced Electric Fields

Lenz’s law gives the direction of the induced current: it flows so as to oppose the change in magnetic flux that caused it. This is not an extra rule but a consequence of conservation of energy. If the induced current aided the change, flux would grow without limit and energy would be created from nothing. The minus sign in Faraday’s law expresses exactly this opposition.

To use it: (1) decide whether flux through the loop is increasing or decreasing, (2) state the direction of the induced field that would oppose that change, (3) use the right-hand rule to find the current producing that field. Mechanical work done against the opposing force is converted into electrical energy and then heat. When flux changes in a region without a circuit, it induces a non-conservative electric field with ∮E·dl = −dΦ/dt, whose field lines are closed loops.

Self-Inductance and Mutual Inductance

When current in a coil changes, its own flux changes and induces an emf opposing that change. This property is self-inductance L, defined by Φ = LI, with unit henry (H). It plays the role of electrical “inertia”. Two nearby coils show mutual inductance M: a changing current in one induces an emf in the other, ε₂ = −M dI₁/dt.

An inductor stores energy in its magnetic field. In an LR circuit, the current does not jump instantly but grows exponentially with time constant τ = L/R, reaching about 63% of its final value after one time constant. On breaking such a circuit, a large back emf can produce sparks across the switch.

Alternating Current and Phasor Analysis

Alternating current (AC) varies sinusoidally with time, i = I₀ sin ωt, where ω = 2πf. Its average over a full cycle is zero, so the useful measure is the RMS value, defined so that it produces the same heating as an equal DC value. Household mains of 220 V is an RMS value; its peak is about 311 V.

Different components shift the phase between voltage and current. In a resistor, they are in phase. In an inductor, voltage leads current by 90° and the opposition is inductive reactance X_L = ωL. In a capacitor, current leads voltage by 90° and the opposition is X_C = 1/ωC. In a series LCR circuit a phasor diagram gives the impedance Z and phase angle φ. Average power depends on the power factor cosφ, so a pure inductor or capacitor consumes no power (wattless current).

LCR Circuits and Electrical Resonance

In a series LCR circuit, X_L increases with frequency while X_C decreases. At one special frequency they are equal and cancel, so impedance becomes purely resistive and minimal (Z = R), and current is at its maximum. This is resonance, occurring at ω₀ = 1/√(LC). Voltage and current are then in phase and the power factor is 1.

The sharpness of resonance is described by the quality factor Q. A high Q (low resistance) gives a tall, narrow current peak and good frequency selectivity, while a low Q gives a broad, flat response. Radio tuning uses this: varying a capacitor changes the resonant frequency to select one station out of many. Note that individual voltages across L and C at resonance can far exceed the supply voltage, each being Q times larger.

Transformers and AC Power Transmission

A transformer consists of two coils wound on a common soft-iron core. An alternating current in the primary produces a changing flux in the core that links the secondary and induces an emf by mutual induction. With turns Np and Ns, the voltage ratio equals the turns ratio. A step-up transformer (Ns > Np) raises voltage and lowers current; a step-down transformer does the reverse. Energy is conserved, so an ideal transformer does not increase power.

Real transformers lose energy by copper loss (I²R heating of windings), eddy currents (reduced by laminating the core), hysteresis (reduced by using soft iron), and flux leakage. Efficiencies of 95–99% are common. Power is transmitted at very high voltage because for fixed power P = VI, higher V means lower current, and the line loss I²R falls as the square of the current.

Electromagnetic Waves and Their Applications

Maxwell realised that Ampere’s law was incomplete: a changing electric field also produces a magnetic field, as though it were a current. This displacement current I_d = ε₀ dΦ_E/dt completes the four Maxwell equations and predicts self-sustaining electromagnetic waves. In these waves E and B oscillate in phase, perpendicular to each other and to the direction of propagation (transverse), and need no medium. In vacuum all travel at c = 1/√(μ₀ε₀) ≈ 3 × 10⁸ m s⁻¹, and E₀/B₀ = c.

The waves carry energy and momentum. They are produced by accelerating charges, with radio waves generated by oscillating circuits. The spectrum is ordered by frequency, all being the same kind of wave.

Ray Optics and Advanced Optical Instruments

Ray optics treats light as straight rays that obey the laws of reflection and refraction (Snell’s law n₁ sinθ₁ = n₂ sinθ₂). Use the Cartesian sign convention: distances are measured from the pole or optical centre, positive along the direction of incident light. When light travels from a denser to a rarer medium at an angle above the critical angle (sin i_c = 1/n), it is totally reflected. This effect drives optical fibres, prisms in binoculars, and the sparkle of diamonds.

Thin lenses form images by refraction, with focal length given by the lens maker’s formula. The power of a lens is the reciprocal of focal length in metres, measured in dioptres. In a compound microscope, a short-focus objective forms a magnified real image that the eyepiece magnifies further. In an astronomical telescope, a large-aperture objective collects light from a distant object and the eyepiece magnifies the image. Eye defects are corrected with lenses: myopia with a concave lens, hypermetropia with a convex lens.

Wave Optics, Interference and Diffraction

Huygens’ principle treats every point on a wavefront as a source of secondary wavelets; the new wavefront is their common tangent. It explains reflection and refraction and shows that light slows down in denser media. Interference occurs when two coherent waves (constant phase difference) superpose, giving bright fringes where path difference is nλ and dark fringes where it is (n + ½)λ.

In Young’s double-slit experiment, fringes of equal width β = λD/d appear on a screen. Diffraction is the bending of light around obstacles or through narrow openings of size comparable to the wavelength. In single-slit diffraction a broad central maximum is flanked by weaker, narrower maxima, with minima at a sinθ = nλ. Unlike interference fringes, diffraction fringes have unequal widths and intensities. Polarisation demonstrates that light is transverse: unpolarised light passing through a polariser emerges with half its intensity, and Malus’s law governs a second polariser.

Photoelectric Effect and Photon Behaviour

When light of sufficiently high frequency strikes a metal surface, electrons are ejected. Experiments show that (1) there is a threshold frequency below which no emission occurs, however intense the light, (2) the maximum kinetic energy of electrons depends on frequency, not intensity, (3) the photocurrent is proportional to intensity, and (4) emission is essentially instantaneous. The wave theory cannot explain these results.

Einstein explained them by proposing that light consists of quanta, called photons, each of energy E = hν. One photon gives all its energy to one electron. Part of this, the work function φ, is needed to escape; the remainder becomes kinetic energy. Increasing intensity increases the number of photons, hence the number of electrons, while increasing frequency increases the energy per photon. The stopping potential V₀ is the retarding voltage needed to stop even the fastest electrons.

Matter Waves and Wave-Particle Duality

De Broglie extended the dual nature of light to matter: every moving particle has an associated wave with wavelength λ = h/p. For everyday objects this wavelength is far too small to observe, but for electrons, neutrons and atoms it is comparable to atomic spacings, giving measurable diffraction. Davisson and Germer confirmed this by scattering electrons from a nickel crystal, observing diffraction peaks that matched λ calculated from de Broglie’s relation.

Wave-particle duality means that radiation and matter show wave-like properties in propagation (interference, diffraction) and particle-like properties in interaction (photoelectric effect, collisions). De Broglie’s idea also explains Bohr’s quantisation: a stable orbit must hold a whole number of wavelengths, 2πr = nλ. Matter waves are not mechanical or electromagnetic waves; their intensity gives the probability of finding the particle.

Atomic Energy Levels and Spectral Transitions

Rutherford’s alpha-scattering experiment showed that an atom has a tiny, dense, positive nucleus with electrons around it. Classical physics predicts such an atom should collapse and emit a continuous spectrum, neither of which happens. Bohr’s model for hydrogen-like atoms fixes this with postulates: electrons move in certain stationary orbits without radiating, with angular momentum quantised as mvr = nh/2π, and radiation is emitted or absorbed only when an electron jumps between orbits, with photon energy equal to the energy difference.

The energy of level n is negative, indicating a bound state; the lowest level (n = 1) is the ground state. Spectral lines fall in series according to the lower level: Lyman (to n = 1, ultraviolet), Balmer (n = 2, visible), Paschen (n = 3, infrared), then Brackett and Pfund. Bohr’s model works well for hydrogen but cannot explain multi-electron atoms, fine structure or relative intensities, which require quantum mechanics.

Nuclear Binding Energy and Radioactive Processes

The nucleus consists of protons and neutrons (nucleons) bound by the short-range, charge-independent strong nuclear force. Nuclear radius follows R = R₀A^1/3 with R₀ ≈ 1.2 fm, giving a density of about 2.3 × 10¹⁷ kg m⁻³ that is nearly the same for all nuclei. The mass of a nucleus is less than the sum of its nucleons; this mass defect corresponds to the binding energy, via E = mc². The binding energy per nucleon peaks near 8.75 MeV around iron (A ≈ 56), so energy is released both when heavy nuclei split (fission) and when light nuclei combine (fusion).

Unstable nuclei decay spontaneously. In alpha decay the nucleus loses ⁴He (A − 4, Z − 2). In beta-minus decay a neutron converts to a proton (Z + 1, A unchanged), emitting an electron and an antineutrino. Gamma decay releases a photon with no change in A or Z. Decay is random for individual nuclei but follows an exponential law for large numbers.

Semiconductor Devices and Modern Electronic Circuits

In solids, atomic levels merge into energy bands. In metals the valence and conduction bands overlap; in insulators a wide gap (> 3 eV) separates them; in semiconductors the gap is small (Si ≈ 1.1 eV, Ge ≈ 0.7 eV), so thermal energy promotes some electrons. An intrinsic semiconductor has equal electron and hole concentrations. Doping with pentavalent atoms (P, As) adds donor levels and makes n-type material; trivalent atoms (B, Al, In) create acceptor levels and make p-type material. In equilibrium n_en_h = n_i².

Joining p and n regions creates a p-n junction with a depletion layer and a barrier potential (about 0.7 V in silicon, 0.3 V in germanium). In forward bias the barrier shrinks and current flows readily; in reverse bias the barrier grows and only a tiny leakage current flows, so a diode conducts in one direction and works as a rectifier. Special diodes include the Zener (voltage regulation in reverse breakdown), LED (emits light in forward bias), photodiode (light detector in reverse bias) and solar cell (converts light to electrical energy). Logic gates (AND, OR, NOT, NAND, NOR) form the basis of digital electronics; NAND and NOR are universal gates.

Important Constants

Exam strategy

• Start with a diagram. Ray diagrams, circuit diagrams and force vectors earn marks and prevent sign errors.
• Write the formula first, then substitute. Marks are awarded for the correct formula even when arithmetic slips.
• Check units and powers of ten. Convert μC, cm, mm and nF to SI before calculating.
• Use limiting cases to test your answer. A formula that gives nonsense at x → 0 or x → ∞ is likely wrong.
• Learn derivations with their diagrams. Typical long-answer questions include Gauss’s law applications, capacitor with dielectric, Biot–Savart results, resonance in LCR, Young’s experiment and Bohr’s model.
• Revise definitions precisely. Terms such as “work function”, “mass defect” and “drift velocity” need exact wording.