Let us consider a particularly simple transformer in which the primary and secondary
coils are *solenoids* sharing the same air-filled core. Suppose that
is the length of the core, and is its cross-sectional area. Let be
the total number of turns in the primary coil, and let be the
total number of turns
in the secondary coil. Suppose that an alternating voltage

where is the peak current. This current generates a changing magnetic flux, in the core of the solenoid, which links the secondary coil, and, thereby, inductively generates the alternating emf

in the secondary circuit, where is the peak voltage. Suppose that this emf drives an alternating current

around the secondary circuit, where is the peak current.

The circuit equation for the primary circuit is written

since

The alternating emf generated in the secondary circuit consists of the
emf generated by the self inductance of the secondary coil, plus the
emf generated by the mutual inductance of the primary and secondary coils.
Thus,

Now, the instantaneous power output of the external AC power source which drives the
primary circuit is

(290) |

(291) |

(292) |

Equations (286), (289), and (293) yield

(294) |

(295) |

Equations (293) and (296) can be combined to give

(297) |

Now, from Sect. 10.2, the self inductances of the primary and
secondary coils are given by
and
, respectively. It follows
that

(298) |

In other words, the ratio of the peak voltages and peak currents in the primary and secondary circuits is determined by the ratio of the number of turns in the primary and secondary coils. This latter ratio is usually called the

AC electricity is generated in power stations at a fairly low peak voltage
(*i.e.*, something like 440V), and is consumed by the domestic
user at a peak voltage of 110V (in the U.S.). However, AC electricity
is transmitted from the power station to the location where it is consumed
at a very high peak voltage (typically 50kV). In fact, as soon as an AC signal
comes out of a generator in a power station it is fed into a step-up
transformer which boosts its peak voltage from a few hundred volts to many tens
of kilovolts. The output from the step-up transformer is fed into a
high tension transmission line, which typically transports the electricity over
many tens of kilometers, and, once the electricity has reached its
point of consumption, it is fed through a series of step-down transformers
until, by the time it emerges from a domestic power socket, its peak voltage is
only 110V. But, if AC electricity is both generated and consumed at
comparatively low peak voltages, why go to the trouble of
stepping up the peak voltage to a very high value at the
power station, and then stepping down the voltage again once the electricity
has reached its point of consumption? Why not generate, transmit, and
distribute the electricity at a peak voltage of 110V?
Well, consider an electric
power line which transmits a peak electric power between a power station
and a city. We can think of , which
depends on the number of consumers in the city, and the nature of the
electrical devices which they operate, as essentially a fixed number.
Suppose that and are the peak voltage and peak current
of the AC signal transmitted along the line,
respectively. We can think of these numbers as being variable, since we can change
them using a transformer. However, since , the product of the peak
voltage and the peak current must remain constant. Suppose that the resistance
of the line is . The peak rate at which electrical energy is lost due
to ohmic heating in the line is , which can be written

(300) |

Of course, transformers do not work for DC electricity, because the magnetic flux generated by the primary coil does not vary in time, and, therefore, does not induce an emf in the secondary coil. In fact, there is no efficient method of stepping-up or stepping-down the voltage of a DC electric signal. Thus, it is impossible to efficiently transmit DC electric power over larger distances. This is the main reason why commercially generated electricity is AC, rather than DC.