Electricity

Electric charge is a basic property of matter, like mass. It is of two kinds, positive and negative (the names were given by Benjamin Franklin). Like charges repel and unlike charges attract. The SI unit is the coulomb (C), and charge is written Q. Every atom holds positive protons in the nucleus and negative electrons around it; the charge of one electron is 1.6 × 10⁻¹⁹ C, so 1 coulomb equals about 6.25 × 10¹⁸ electrons. Only electrons move from one body to another, so a body becomes positive by losing electrons and negative by gaining them.

Three properties of charge: it is conserved (never created or destroyed, only transferred), quantised (always a whole-number multiple of the electron charge, Q = ne) and additive (total charge is the algebraic sum). J. J. Thomson discovered the electron (1897) and R. A. Millikan measured its charge with his oil-drop experiment (1909).

Coulomb's law. The force between two point charges is F = k q₁q₂ / r², where k = 9 × 10⁹ N m²/C². It is attractive for unlike and repulsive for like charges, and it follows an inverse-square law. The space around a charge where its force can be felt is the electric field, with unit N/C. A gold-leaf electroscope detects charge: its leaves spread apart when charged. A capacitor stores charge: C = Q/V, unit farad (F), and the energy stored is U = ½ C V².

Lightning is a huge natural spark. In a thundercloud, friction between ice crystals and water droplets separates charge, and when the charge difference between cloud and ground (or between clouds) becomes very large, air breaks down and a giant discharge takes place. Benjamin Franklin proved in 1752, with his famous kite experiment, that lightning and the small sparks from clothes are the same kind of electric phenomenon. Thunder is the sound of air heated suddenly to extreme temperature; we see the flash first because light travels faster than sound. Lightning tends to strike tall objects, and it can strike telephone cords and electrical wires, so these should be avoided in a storm. A lightning conductor (also devised by Franklin) is a metal rod on top of a tall building, earthed through a thick wire, that carries the discharge safely to the ground.

Electric current is the rate of flow of electric charge through a cross-section of a conductor: I = Q / t. Its SI unit is the ampere (A); 1 A means 1 coulomb of charge passing a point every second. Smaller units: 1 mA = 10⁻³ A, 1 μA = 10⁻⁶ A. Current is a scalar quantity. By convention the direction of current is taken as the direction in which positive charges would move, from the positive terminal of the cell to the negative terminal outside the cell. In a metal the actual flow is of free electrons, in the opposite direction.

Drift velocity. Free electrons in a metal move randomly at high speed, but when an electric field is applied they acquire a small average velocity in one direction, called the drift velocity (of the order of millimetres per second). The current starts almost instantly in the whole wire because the electric field travels at nearly the speed of light, just as water starts flowing in a full pipe the moment the tap is opened.

Potential difference (voltage). The potential difference between two points is the work done to move a unit positive charge from one point to the other: V = W / Q. The SI unit is the volt (V): 1 V = 1 joule per coulomb. It is the 'electrical pressure' that pushes charge through a circuit, just as a height difference makes water flow. A cell or battery maintains this potential difference. The EMF of a cell is the potential difference between its terminals when no current is drawn; when current flows, the terminal voltage is less (V = E − Ir) because of the internal resistance. A voltmeter measures potential difference and is connected in parallel; an ammeter measures current and is connected in series.

Ohm's law. George Simon Ohm (1827) found that, at a constant temperature, the current through a conductor is directly proportional to the potential difference across its ends: V = I R, where R is the resistance. If V is kept fixed, current is inversely proportional to resistance; double the resistance and the current halves. The V–I graph of an ohmic conductor is a straight line through the origin and its slope is R. Materials that do not obey this (filament lamp, diode, transistor, electrolytes) are non-ohmic.

Quick numericals. (1) I = V/R: a 12 V battery across a 6 Ω resistor gives 2 A. (2) Q = I t: 3 A for 10 s moves 30 C. (3) V is fixed and R doubled gives I halved. (4) Same V = 10 V: conductor X takes 2 A (R = 5 Ω) and Y takes 0.5 A (R = 20 Ω), so Y has four times the resistance of X.

Resistance is the opposition a material offers to the flow of current. Free electrons keep colliding with vibrating atoms and lose energy, and these collisions are the origin of resistance. The SI unit is the ohm (Ω): 1 Ω = 1 volt per ampere. Its reciprocal is conductance, with unit siemens (S) (earlier called mho). A device made to give a known resistance is a resistor; a rheostat is a resistor whose value can be changed.

Factors affecting the resistance of a wire. For a uniform wire, R = ρ L / A, where L is the length, A the cross-sectional area and ρ (rho) the resistivity of the material. So resistance is (1) directly proportional to length, (2) inversely proportional to the area of cross-section (a thick wire has less resistance), (3) dependent on the nature of the material through ρ, and (4) dependent on temperature. It does not depend on things like the viscosity of a surrounding medium or the voltage applied.

Resistivity is the resistance of a wire of unit length and unit area. It depends only on the material and temperature, not on the shape or size. Its SI unit is ohm-metre (Ω m). Good conductors have very low resistivity (silver, copper, aluminium); insulators have enormous resistivity (glass, rubber). Silver is the best conductor of electricity, followed by copper, gold and aluminium.

Effect of temperature. In metals resistance increases with temperature, because the atoms vibrate more and collide more often with the electrons. In semiconductors (silicon, germanium), carbon, insulators and electrolytes resistance decreases when temperature rises, since more charge carriers become available. Alloys such as manganin, constantan and nichrome have higher resistivity than their metals and a very small change of resistance with temperature. At very low temperatures some materials become superconductors with zero resistance (Kamerlingh Onnes, 1911, in mercury cooled to about 4 K).

Resistors can be joined end-to-end in a single path (series) or side by side across the same two points (parallel). The two arrangements behave very differently, and almost every SSC numerical in this chapter is built on them. Think of a series circuit as a single road with toll gates one after another: every vehicle (charge) must cross all of them. A parallel circuit is like a road that splits into branches: the traffic divides and the branch with the least obstruction takes the most.

Current division. In a parallel circuit V is the same, so I = V/R means that the branch with the smaller resistance carries more current. Two wires of 3 Ω and 6 Ω across the same battery carry currents in the ratio 2 : 1. In a series circuit I is the same, so the resistor with the larger resistance has the larger voltage drop.

Worked examples. (a) 4 Ω and 12 Ω in series with 32 V: R = 16 Ω, I = 32/16 = 2 A, voltage across 12 Ω = 2 × 12 = 24 V. (b) 2 Ω and 4 Ω in parallel across 6 V: I₁ = 3 A, I₂ = 1.5 A, total I = 4.5 A (or R = 8/6 = 4/3 Ω, I = 6 ÷ 4/3 = 4.5 A). (c) A wire of resistance R is cut into two equal halves and the halves are connected in parallel: each half is R/2, so the combination is (R/2)/2 = R/4.

Kirchhoff's laws (Gustav Kirchhoff, 1845) handle complicated networks: the junction (current) law says the total current entering a junction equals the total current leaving it (conservation of charge); the loop (voltage) law says the algebraic sum of potential changes around any closed loop is zero (conservation of energy). A Wheatstone bridge of four resistors P, Q, R, S is balanced when P/Q = R/S, and then no current flows through the galvanometer in its middle arm. Cells can be connected too: in series their EMFs add; in parallel the EMF stays that of one cell but the arrangement can supply current for longer.

When current flows through a conductor, the moving electrons keep colliding with atoms and give them energy, so the conductor heats up. This is the heating effect of current, studied by James Prescott Joule. Joule's law of heating: H = I² R t joules. Because V = IR, the same heat can be written as H = V I t = (V²/R) t. Heat is proportional to the square of the current, the resistance and the time. Heater, electric iron, toaster, geyser, kettle and the bulb all use this effect.

Electric power is the rate at which electrical energy is used: P = V I = I² R = V² / R. Its SI unit is the watt (W): 1 W = 1 joule per second. Bigger units: 1 kW = 1000 W; 1 horsepower = 746 W. Electrical energy used is E = P × t. The SI unit is the joule, but the commercial unit is the kilowatt-hour (kWh), commonly called one 'unit' on the electricity bill. 1 kWh = 1000 W × 3600 s = 3.6 × 10⁶ J. Bill = units × rate per unit.

Rated values. A bulb marked '40 W, 240 V' means it uses 40 W when connected to 240 V. Its filament resistance at working temperature is R = V²/P = 240²/40 = 1440 Ω, and the current is I = P/V = 1/6 A. A '100 W, 200 V' bulb has R = 200²/100 = 400 Ω. An appliance of 2.2 kW on 220 V draws 10 A, and running it for 30 minutes uses 2.2 × 0.5 = 1.1 kWh, costing ₹4.4 at ₹4 per unit.

Fuse. A fuse is a safety device, a short piece of wire of an alloy with a low melting point and high resistance (usually tin and lead), connected in series in the live wire. If the current exceeds the rated value because of overloading or a short circuit, the wire heats up by Joule heating, melts and breaks the circuit, protecting the appliance and preventing fire. A fuse must be replaced after it blows; a modern MCB (miniature circuit breaker) does the same job by tripping and can be switched back on. Fuse wire is never replaced with a thick copper wire, because copper would not melt in time.

Materials are divided by how easily they let charge flow. Conductors have a huge number of free charge carriers: metals (silver, copper, aluminium), graphite, and also salt solutions (electrolytes) and ionised gas. Insulators (bad conductors) such as rubber, glass, plastic, dry wood, porcelain and mica have almost no free carriers and very high resistivity. Semiconductors lie between the two. The explanation comes from energy bands: in a conductor the bands overlap, in an insulator the gap between the valence and conduction bands is very large, and in a semiconductor the gap is small enough that heat or light can push electrons across.

Mica is an insulator with very high dielectric strength and low power loss, so it is used in high-voltage electrical equipment, capacitors and heating appliances. It also splits into thin, flexible, transparent sheets. Superconductors are materials that lose all resistance below a critical temperature; they allow current to flow without loss and are used for powerful magnets.

Doping and the p-n junction. Pure silicon or germanium is an intrinsic semiconductor. Adding a tiny amount of impurity (doping) changes its conductivity. Adding a pentavalent impurity (phosphorus, arsenic, antimony: 5 valence electrons) gives extra free electrons and makes an n-type semiconductor, where electrons are the majority carriers. Adding a trivalent impurity (boron, aluminium, gallium, indium: 3 valence electrons) creates vacancies called holes and makes a p-type semiconductor, where holes are the majority carriers. Joining p and n type crystals makes a p-n junction diode, which allows current in only one direction (forward bias) and blocks it in the other (reverse bias).

In a transistor the current gain is β = I_C / I_B (collector current over base current), so I_C = β × I_B. For β = 150 and I_B = 20 μA, I_C = 150 × 20 μA = 3000 μA = 3 mA. A printed circuit board (PCB) is the board on which electronic components are mounted and joined by thin copper tracks.

Household wiring. Electricity reaches a house through two wires from the pole: the live (phase) wire at high potential and the neutral wire at zero potential, with a third wire, the earth wire, buried in the ground near the house. In India the supply is alternating current at 220–240 V (nominal 230 V) and 50 Hz. The older colour code is live = red, neutral = black, earth = green; the newer standard uses live = brown, neutral = blue, earth = green-yellow. In both, the earth wire has the green colour.

Why earthing? The metal body of a geyser, iron or refrigerator is joined to the earth wire through the longer, thicker top pin of the three-pin plug. If the live wire touches the body by fault, a large current flows safely to the earth, the fuse blows, and the user does not get a shock. Switches and fuses are always placed on the live wire so that the appliance is fully disconnected when switched off. An ELCB/RCCB detects leakage current and cuts the supply.

Cells and batteries. A cell converts chemical energy into electrical energy; a battery is a group of cells. Primary cells cannot be recharged: the dry (Leclanché) cell with a zinc case and carbon rod gives 1.5 V; alkaline cells are a better version. Secondary cells can be recharged: the lead-acid accumulator (lead and lead dioxide plates in dilute sulphuric acid) gives 2 V per cell, so a car battery of six cells gives 12 V; the nickel-cadmium cell gives 1.2 V; the lithium-ion cell (about 3.6–3.7 V) powers phones, laptops and electric vehicles (Nobel Prize in Chemistry, 2019, to Goodenough, Whittingham and Yoshino). Alessandro Volta made the first electric cell in 1800. A solar cell converts sunlight directly to electricity, and a fuel cell produces electricity from hydrogen and oxygen with water as the by-product.