Physics · Electricity and magnetism · Grade 7

Electric circuit, Ohm's law

Why does the circuit breaker trip when a kettle and an iron are on together?

The situation

One evening the kettle is boiling in the kitchen while someone in the other room is ironing a shirt. Suddenly the circuit breaker in the fuse board clicks off and the whole flat is plunged into darkness. When the kettle and the iron were switched on separately, nothing had happened. How are the appliances in the home connected to the mains, and how is the current shared between them?

Topic

Electric circuit, Ohm's law. Household wiring: series and parallel connection

Key ideas

Ohm's law
Series and parallel connection
Electrical power

Formulas

I = \frac{U}{R}, R = ρ \frac{l}{S}

Ohm's law: the current in a section of a circuit is directly proportional to the voltage and inversely proportional to the resistance. The resistance of a wire is directly proportional to its length and inversely proportional to its cross-sectional area ( $ρ$ is the resistivity).

R = R_{1} + R_{2}, I_{1} = I_{2}, U = U_{1} + U_{2}

Connection in series: the current is the same, the voltages add up.

\frac{1}{R} = \frac{1}{R_{1}} + \frac{1}{R_{2}}, U_{1} = U_{2}, I = I_{1} + I_{2}

Connection in parallel: the voltage is the same, the currents add up. All the sockets in a home are in parallel.

P = U I = I^{2} R = \frac{U^{2}}{R}

Electrical power. The “100 W” on an appliance is its power at 220 V: $R = U^{2} / P$ . The greatest total power the breaker can carry is $P_{m a x} = U I_{b r} = 220 \cdot 16 = 3520$ W.

j = \frac{I}{S} = σ E, G = \frac{1}{R}

Current density and the differential form of Ohm's law ( $σ = 1 / ρ$ is the conductivity). $G$ is the electrical conductance, whose unit is the siemens (S).

Theory

Current and Ohm's law

Current is the ordered motion of charges. For a current to flow you need free charges, an electric field to move them (maintained by the power source) and a closed circuit. A current shows itself through its heating (kettle, iron), light (lamp), magnetic (the electromagnet in a breaker) and chemical (charging a battery, electrolysis) effects. The voltage “pushes” the charges and the resistance gets in their way: $I = U / R$ is Ohm's law for a section of circuit with no power source in it. In the home mains $U = 220$ V stays constant, so an appliance with a small resistance draws a large current. The resistance of a 2 kW kettle is only $220^{2} / 2000 \approx 24$ Ω, and the current is ≈ 9.1 A.

Series and parallel

When lamps are connected in series, the current has only one path: the resistances add, the mains voltage is shared between the lamps and they glow dimly. When they are connected in parallel, each lamp is connected directly to 220 V: they do not depend on one another, and the currents add up in the common wire. All the appliances in a home are connected in parallel — so every new appliance increases the total current.

Power and brightness

The brightness of a lamp depends on the power in it: $P = I^{2} R = U^{2} / R$ . If two identical lamps are connected in series, each gets 110 V and the power falls by a factor of 4 ( $P \propto U^{2}$ ). If different lamps are connected in series, the current is the same, so the lamp with the larger resistance — the one with the lower wattage — glows brighter.

The circuit breaker and electrical safety

The copper wires in the wall (usually 2.5 mm²) are designed for a certain current; an excess current can heat them and melt the insulation. The bimetallic strip in a breaker bends from the heat of the current and, in an overload, breaks the circuit after some time; the electromagnet, on the other hand, breaks it instantly in a short circuit (≥ 5·16 A). Safety: do not touch sockets and appliances with wet hands, do not use a damaged cable, switch off the breaker before any repair, and never put a piece of wire in place of a fuse or a breaker. An alternating current of a few tens of milliamperes passing through the human body is dangerous to life; the casing of an appliance is earthed, and a residual-current device (RCD, usually 30 mA) senses the current leaking to earth and breaks the circuit in less than 0.3 s. Model: appliances are treated as simple resistors ( $R = U^{2} / P$ ), the resistance of the lamp filament is given at its working temperature (the resistance of a cold filament is roughly 10–15 times smaller), the values are effective (RMS) values, and the heating of the bimetal is simplified.

Rheostat, potentiometer and I–V curve (laboratory)

A resistor is an element with a known resistance; a rheostat is a coil of wire whose resistance is changed with a slider. A rheostat connected in series with a load controls the current in the circuit; if all three of its terminals are connected and it acts as a voltage divider, it is a potentiometer: it varies the voltage across the load from zero up to the source voltage. An ammeter is connected in series with the load and a voltmeter in parallel with it; the resistance is found from $R = U / I$ (the ammeter–voltmeter method). The $I (U)$ relationship measured at several points is the current–voltage (I–V) characteristic: for a resistor it is a straight line through the origin (Ohm's law, with gradient $1 / R$ ), and for a lamp it is a curve. The unit of resistance is the ohm: 1 Ω = 1 V / 1 A; 1 kΩ = 10³ Ω, 1 MΩ = 10⁶ Ω. The reciprocal quantity is the electrical conductance $G = 1 / R$ , whose unit is the siemens: 1 S = 1/Ω.

Current density and the differential form of Ohm's law

The current density $j = I / S$ is the current passing through each square millimetre of the cross-section: at 16 A in a 2.5 mm² copper wire $j = 6.4$ A/mm² (watch it in “Measurements”). The resistance of a wire is $R = ρ l / S$ ; in metals the resistivity rises with temperature, $ρ \approx ρ_{0} (1 + α Δ t)$ — so the resistance of a glowing lamp filament is roughly 10–15 times that of a cold one (the I–V curve of the lamp in the laboratory). Substituting $U = E l$ and $I = j S$ into Ohm's law gives the differential form of Ohm's law $\vec{j} = σ \vec{E}$ — it holds at every point of a conductor. The conservation of charge is written as the continuity equation $\partial ρ_{q} / \partial t + \nabla \cdot \vec{j} = 0$ : in a direct current charge accumulates nowhere, so the same current passes through every cross-section of a series circuit, and at a junction the sum of the currents flowing in equals the sum of those flowing out (Kirchhoff's rules are in the “Circuits and Kirchhoff's rules” experiment, and the motion of electrons in metals in the “Electric current in metals” experiment).

Did you know?

In Uzbekistan the voltage of the household mains is 220 V and the frequency is 50 Hz: the direction of the current changes 100 times a second.
Ohm's law was published by the German physicist Georg Simon Ohm in 1827 in his book “The Galvanic Circuit Investigated Mathematically”.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

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