Current Electricity

Charges on the move — current, resistance, circuits and cells · Class 12 Physics

1The big idea

Electrostatics dealt with charges sitting still. Current electricity is about charges on the move — a steady river of electrons drifting through a wire, pushed along by a battery. Almost everything in this chapter follows from one deceptively simple statement: a potential difference drives a current, and the conductor resists it. That is Ohm's law, V = IR, and the rest of the chapter is really about (a) where the resistance comes from, (b) how currents split and add when you wire components together, and (c) how a real battery falls short of its ideal promise.

Keep one mental picture throughout: a battery is a charge pump. It does not create charge — it lifts charge to a higher potential, and the charge then flows "downhill" through the external circuit, giving up that energy as heat, light or motion.

2Electric current & drift velocity

Electric current is simply the rate at which charge crosses a section of the conductor. If a charge Q passes in time t,

By convention current flows in the direction a positive charge would move — from + to − in the external circuit. Inside a metal it is really the electrons drifting the other way, but the conventional direction is the one we draw arrows for.

Here is the subtle part. When you close a switch the bulb lights instantly, yet the electrons themselves crawl along astonishingly slowly. The electric field is established through the wire at nearly the speed of light, and it nudges every electron at once — but each electron's own average forward speed, its drift velocity v_d, is often less than a millimetre per second.

Let us derive the link between the everyday current I and the microscopic drift speed. Suppose the metal has n free electrons per unit volume, each carrying charge e, and the wire has cross-sectional area A. In a small time Δt every electron moves forward a distance v_d Δt. So all the electrons in a cylinder of length v_d Δt and area A sweep past the shaded face. That cylinder's volume is A v_d Δt, it contains n A v_d Δt electrons, and their total charge is

Rearranged, the drift velocity is

Because n in a metal is enormous (around 10²⁸–10²⁹ m⁻³), even a healthy current gives a drift speed of a fraction of a millimetre per second. The current is large not because the electrons are fast, but because there are so many of them moving together.

3Ohm's law and resistance

Ohm's law states that, for many conductors at constant temperature, the current is directly proportional to the potential difference across them:

The constant of proportionality R is the resistance, measured in ohms (Ω). A material that obeys this straight-line relation is called ohmic; a bulb filament, a diode or an electrolyte need not — their V–I graph curves.

What decides the resistance of a given wire? Experiment shows it grows with length and shrinks with thickness:

A longer wire (L) means the electrons face more scattering on the way through, so more resistance. A fatter wire (larger A) offers more parallel paths, so less resistance — exactly like a wider pipe carrying water more freely. The proportionality constant ρ (rho) is the resistivity, a property of the material itself, independent of its shape. Its unit is the ohm-metre (Ω m). Its reciprocal σ = 1/ρ is the conductivity.

4Resistivity and temperature

Push more heat into a metal and its atoms vibrate harder, so the drifting electrons collide more often and lose their forward momentum faster. The resistivity therefore rises with temperature. Over a modest range it is very nearly linear:

where α is the temperature coefficient of resistivity. For metals α is positive (a few ×10⁻³ per °C) — hotter means more resistive. For semiconductors and electrolytes α is negative: heating frees more charge carriers, so resistivity falls. This opposite behaviour is one of the fingerprints that separates a metal from a semiconductor.

5Resistors in series and parallel

Real circuits chain resistors together. Two rules cover almost everything, and both are worth deriving once so you never confuse them.

Series — same current, voltages add

In series the components sit on a single path, so the same current I flows through each. The battery's voltage is shared out among them, V = V₁ + V₂ + V₃. Writing each drop as V_k = I R_k:

The series total is always bigger than the largest single resistor.

Parallel — same voltage, currents add

In parallel every resistor is connected across the same two nodes, so each feels the same voltage V. The total current splits, I = I₁ + I₂ + I₃. Writing each branch as I_k = V / R_k:

The parallel total is always smaller than the smallest single resistor — adding another path can only make it easier for current to flow. For just two resistors this simplifies to the handy form R_p = R₁R₂ / (R₁ + R₂) ("product over sum").

6EMF vs terminal voltage

An ideal battery would supply a fixed voltage no matter how much current you draw. A real cell cannot — the chemical reactions and the electrolyte inside offer their own small internal resistance r. The electromotive force (EMF) ε is the full voltage the cell would give with no current flowing (an open circuit). The moment current I flows, some of that voltage is used up pushing charge through r itself, and what is left at the terminals — the terminal voltage V — is smaller.

Apply energy conservation round the loop. The EMF supplies ε per coulomb; this is spent partly on the internal resistance (Ir) and partly on the external resistance (V):

So the terminal voltage always sits below the EMF whenever current flows, and the gap Ir grows as you draw more current. Since the external resistor obeys V = IR, substitute to find the current in the complete circuit:

Two limits check out: on open circuit (R → ∞) no current flows and V = ε; on a dead short (R = 0) the current is capped at ε/r, which is why a low-r car battery can deliver a dangerous surge.

7Grouping cells

Cells combine much like resistors do:

• Series — connect + of one to − of the next. The EMFs add and so do the internal resistances: ε_eq = ε₁ + ε₂ and r_eq = r₁ + r₂. Use series when the external resistance is much larger than r — you want the extra voltage.
• Parallel — connect all the + terminals together and all the − terminals together. For n identical cells the EMF stays ε but the internal resistance drops to r/n. Use parallel when the external resistance is small — you want to share the current and reduce the internal loss.

8Kirchhoff's rules

Series and parallel shortcuts fail for tangled networks (like a bridge). Two conservation laws, due to Kirchhoff, always work.

Junction rule (conservation of charge). The total current flowing into any junction equals the total current flowing out — charge does not pile up:

Loop rule (conservation of energy). Around any closed loop the algebraic sum of the potential changes is zero — a charge taken all the way round returns to the same potential it started at:

The whole skill is bookkeeping the signs: a resistor traversed with the assumed current gives a drop (−IR); a cell entered at its − terminal and left at its + gives a rise (+ε). Write one junction equation and enough loop equations to match the number of unknown currents, then solve.

9Wheatstone bridge & potentiometer

The Wheatstone bridge is the classic use of Kirchhoff's rules — four resistors P, Q, R, S arranged in a diamond with a galvanometer bridging the middle. You adjust the resistors until the galvanometer reads zero: the bridge is then balanced, meaning points B and D are at the same potential and no current crosses the bridge.

At balance no current flows through G, so P and R carry one current I₁ while Q and S carry another I₂. Because B and D are at equal potential, the drop across P equals the drop across Q, and the drop across R equals that across S:

Dividing one equation by the other cancels the currents and gives the celebrated balance condition:

An unknown resistance can now be found from three known ones. The metre bridge is a practical version: a 1 m resistance wire replaces two of the arms, and the balance point at length l (from one end) gives the unknown S from R/S = l/(100 − l), since the two wire lengths play the role of P and Q.

The potentiometer takes the null idea further and lets you measure a voltage or EMF without drawing any current from the source under test. A steady current runs through a long uniform wire, setting up a constant potential gradient k (volts per metre). You tap the wire until the galvanometer nulls; then the balancing length l times the gradient equals the unknown EMF, ε = k l. Because at balance no current is drawn, there is no Ir drop — so a potentiometer measures the true EMF, something a voltmeter (which always draws a little current) cannot quite do. Comparing two cells, ε₁ / ε₂ = l₁ / l₂.

10Electrical power

As charge Q falls through a potential difference V it loses energy QV. Dividing by time gives the rate of energy delivery — the power:

Using Ohm's law V = IR you can rewrite this two more ways, and all three are worth knowing because different problems hand you different pairs:

In a resistor this energy appears as heat — Joule heating — at a rate I²R, and over a time t the heat produced is H = I² R t (Joule's law). Power is measured in watts (W); the electricity bill's "unit" is the kilowatt-hour, the energy of 1 kW running for 1 hour = 3.6 × 10⁶ J.

11Common confusions to clear up

• Drift velocity is tiny, but the signal is fast. Electrons drift at well under a mm/s, yet the bulb lights instantly because the electric field spreads through the wire near light speed and nudges every electron at once.
• Conventional current opposes electron flow. We draw current from + to −; the electrons actually move − to +. Both descriptions carry the same charge per second.
• Resistance vs resistivity. Resistivity ρ is a material property fixed by what the wire is made of; resistance R = ρL/A also depends on its length and thickness.
• EMF is not "the battery's voltage" once current flows. EMF is the open-circuit value; the terminal voltage V = ε − Ir is what a load actually receives, and it drops as you draw more current.
• Series total goes up, parallel total goes down. Adding a resistor in series always raises the total; adding one in parallel always lowers it — a parallel combination is smaller than its smallest member.
• Biggest resistor is not always the hottest. In series (same I) the largest R dissipates most; in parallel (same V) the smallest R dissipates most. Ask which quantity is shared first.

12Check yourself

Class 12 Physics · Current Electricity · aligned to NCERT Chapter 3 · SmartStudy.School