Three ideas, one law

What happens when a lamp lights up? The chemical energy in a battery flows along a wire as the motion of electrons, and inside the lamp's filament it runs into resistance and produces heat and light. A single equation puts this simple fact into mathematics:

V=I⋅RV = I \cdot R

The three ideas:

  • V (voltage, potential difference): the "electrical pressure" that pushes the electrons. Unit: volt (V)
  • I (current): the amount of charge passing each second. Unit: ampere (A) = 6.24 × 10¹⁸ electrons per second
  • R (resistance): the opposition the current meets along its path. Unit: ohm (Ω)

Georg Ohm: a doubted discovery

Georg Simon Ohm (1789–1854) was a modest secondary school teacher in Cologne. In the early 1820s he began testing wires with primitive batteries (voltaic piles). He patiently measured how the length and thickness of the wires and the voltage were related.

In 1827 he published his findings: current is directly proportional to voltage and inversely proportional to resistance. In modern form, V = I · R.

The scientific world's reaction: dismissal. Germany's education establishment called his work "speculative", Ohm lost his chance at a university post and fell on hard times. Only in 1841, when he received the Royal Society's Copley Medal, did he gain international recognition. The unit of resistance carries his name: a late but well-deserved honour.

Power: how much energy per second?

The power of an electrical component (the energy it converts each second) is simply voltage × current:

P=V⋅I=I2⋅R=V2RP = V \cdot I = I^{2} \cdot R = \frac{V^{2}}{R}

Unit: watt (W).

  • A 60 W lamp on a 230 V household supply draws 0.26 A (I = P/V)
  • A 2000 W kettle on the same supply draws 8.7 A; that is why running two heaters from one socket trips the fuse (total current > the 16 A fuse limit)
  • A phone charges at 5 V × 2 A = 10 W

Series vs parallel: two basic layouts

Series connection

The components are lined up one after another:

Rtotal=R1+R2+R3+…R_{total} = R_1 + R_2 + R_3 + \dots

There is only one path: the same current flows through every component (ItotalI_{total} is constant), but the voltage drops across each resistor in proportion to its resistance:

V1=I⋅R1,V2=I⋅R2,V1+V2=VbatteryV_1 = I \cdot R_1, \quad V_2 = I \cdot R_2, \quad V_1 + V_2 = V_{battery}

Advantage: simple, little wire. Disadvantage: if one component fails, the whole circuit stops.

A classic example: In old Christmas lights, when one bulb burned out the whole string went dark. That is why modern strings use parallel connections.

Parallel connection

Each component is connected directly to the battery, with branches between them:

1Rtotal=1R1+1R2+1R3+…\frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \dots

Every branch has the same voltage (VV is constant), but the current is shared out in inverse proportion to the resistance:

I1=VR1,I2=VR2,Itotal=I1+I2I_1 = \frac{V}{R_1}, \quad I_2 = \frac{V}{R_2}, \quad I_{total} = I_1 + I_2

Advantage: the components work independently; if one fails, the others are not affected. Disadvantage: more wire, more complexity.

Household wiring is parallel. All sockets and lamps are connected in parallel to 230 V. If the living room lamp burns out, the kitchen lamp keeps working. The total current is the sum of the branch currents, which is why connecting many appliances to the same fused circuit pushes the limits.

Kirchhoff's two laws: for more complex circuits

Gustav Kirchhoff set out two basic rules in 1845, while he was still a student:

  1. Current law (KCL): The sum of the currents entering a junction equals the sum of the currents leaving it. (Conservation of charge.)
  2. Voltage law (KVL): The sum of the voltage changes around any closed loop of a circuit is zero. (Conservation of energy.)

These two rules plus Ohm's law can solve any electronic circuit. Modern circuit simulators (SPICE) solve exactly these equations numerically.

Overcurrent and fuses

If the current through a circuit exceeds the safe limit of a cable or component, it produces heat (P=I2RP = I^2 R). Too much heat means a fire risk. That is why we have:

  • Fuse: a thin wire that physically melts if the limit is exceeded, breaking the circuit
  • Circuit breaker: a modern fuse with an automatic switch that can be reset
  • Earthing (grounding): a safety wire that carries current into the ground if there is a fault

If you plug a heater, a kettle and a dishwasher into a circuit rated for 16 A, the total current exceeds 16 A and the fuse trips. It is a design that saves lives.

One step further

These laws are for direct current (DC), where the current flows steadily in one direction. Household mains supply is alternating current (AC), which changes direction 100 times a second (50 cycles per second). In AC circuits:

  • Inductance (L) and capacitance (C) behave like resistance
  • Impedance (Z) is the AC counterpart of resistance: V=I⋅ZV = I \cdot Z
  • Phase angle: the voltage and current do not rise and fall at the same time

In modern electronics, transistors, op-amps and digital logic gates are all built on the foundations of Ohm and Kirchhoff.

Sources