Problem — current as flow

Problem

Explain in five lines the analogy between electric current in a wire and the flow of water in a pipe. At which point in the wire is the current “highest” in a series circuit? What do voltage, resistance and power represent in the hydraulic analogy?

Solution

The hydraulic analogy — the electric current $i$ is the amount of charge crossing a cross-section of the wire per unit time, exactly as the flow rate of water is the volume crossing a cross-section of the pipe per second: $i = \frac{\Delta q}{\Delta t} \quad \longleftrightarrow \quad Q = \frac{\Delta V_\text{water}}{\Delta t}$

Electrons act as “water”, the wire as “pipe”. The pump that pushes the water corresponds to the battery.

Where is the current highest in series? — In a series circuit the current is the same at every point: there is no point where it is higher. It is the hydraulic equivalent of a single pipe with no branches: the same water that enters at one end leaves at the other, the flow rate is identical at every cross-section. Charge neither accumulates nor disappears (conservation of charge).

Correspondences:

• Voltage $\leftrightarrow$ pressure difference (or height difference) supplied by the pump: it is what drives the flow.
• Resistance $\leftrightarrow$ constriction of the pipe (narrow cross-section, long or rough pipe): it hinders the flow for a given driving force.
• Power $\leftrightarrow$ energy delivered by the flow per unit time, for example the water that turns a mill; in the wire $P = V\,i$ is the energy dissipated or delivered per second.

$\ev{\text{in series the current is the same at every point; }V\to\text{pressure},\ R\to\text{constriction},\ P\to\text{energy/time}}$

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Topics: Circuiti elettrici Concepts: Corrente elettrica