The transformer

A spectacular practical application of Maxwell’s equations is the transformer, the device that allows the voltage of the electricity grid to be raised or lowered. It consists of two solenoids wound on the same iron core, so that the magnetic flux generated by the first passes through both.

Transformer law

If the primary solenoid has $N_1$ turns and the secondary $N_2$ turns: $\ev{\frac{\Delta V_2}{\Delta V_1} = \frac{N_2}{N_1}}$

If the system is ideal (with no dissipation), the currents are also in inverse ratio, $i_2/i_1 = N_1/N_2$, i.e. $\Delta V_1\cdot i_1 = \Delta V_2\cdot i_2$ (conservation of power). Raising the voltage on the secondary therefore means proportionally lowering the current, and vice versa.

The operation is pure Faraday induction: the alternating current in the primary creates a varying magnetic flux in the iron core; this flux, linked with the $N_2$ turns of the secondary, induces an EMF there. This is why the transformer works only with alternating current: it needs a flux that varies in time.

Historical context — Tesla, Edison and the War of Currents

At the end of the 19th century the “War of Currents” broke out between Thomas Edison, a supporter of direct current (DC), and Nikola Tesla, who championed alternating current (AC). Tesla won precisely thanks to the transformer: with DC, transporting energy over distance requires large currents and hence enormous losses through the Joule effect ($Ri^2$). With AC, the voltage is raised for transport (reducing current and losses) and lowered again on arrival with a second transformer. That is why household mains electricity is alternating current.

Links

Topics: Electromagnetic waves · Electromagnetic induction Concepts: Faraday-Neumann-Lenz law · Magnetic flux

Related exercises: Worked exercise — Varying B field generates circulation of E · Problem — Ranking EMF of four square loops · EMF from flux B = 10t