Ampère's next series of experiments involved bringing a short test wire, carrying a current , close to the original wire, and investigating the force exerted on the test wire. This experiment is not quite as clear cut as Coulomb's experiment because, unlike electric charges, electric currents cannot exist as point entities. They have to flow in complete circuits. We must imagine that the circuit which connects with the central wire is sufficiently far away that it has no appreciable influence on the outcome of the experiment. The circuit which connects with the test wire is more problematic. Fortunately, if the feed wires are twisted around each other, as indicated in Fig. 21, then they effectively cancel one another out, and also do not influence the outcome of the experiment.
Ampère discovered that the force exerted on the test wire is directly proportional to its length. He also made the following observations. If the current in the test wire (i.e., the test current) flows parallel to the current in the central wire then the two wires attract one another. If the current in the test wire is reversed then the two wires repel one another. If the test current points radially towards the central wire (and the current in the central wire flows upward) then the test wire is subject to a downward force. If the test current is reversed then the force is upward. If the test current is rotated in a single plane, so that it starts parallel to the central current and ends up pointing radially towards it, then the force on the test wire is of constant magnitude, and is always at right-angles to the test current. If the test current is parallel to a magnetic loop then there is no force exerted on the test wire. If the test current is rotated in a single plane, so that it starts parallel to the central current, and ends up pointing along a magnetic loop, then the magnitude of the force on the test wire attenuates like (where is the angle the current is turned through, and corresponds to the case where the test current is parallel to the central current), and its direction is again always at right-angles to the test current. Finally, Ampère was able to establish that the attractive force between two parallel current carrying wires is proportional to the product of the two currents, and falls off like one over the perpendicular distance between the wires.
This rather complicated force law can be summed up succinctly in vector notation
provided that we define a vector field , called the magnetic field,
which fills space, and
whose direction is everywhere tangential to the
magnetic loops mapped out by the north
pole of a small
compass. The dependence of the force per unit length, , acting on a
test wire with the different
possible orientations of the test current is described by
The variation of the force per unit length acting on
a test wire with the strength of the
central current, and the perpendicular distance to the central wire, is
accounted for by saying that the magnetic field-strength is proportional to , and
inversely proportional to . Thus, we can write
The concept of a magnetic field which fills the space around a current carrying wire allows the calculation of the force on a test wire to be conveniently split into two parts. In the first part, we calculate the magnetic field generated by the current flowing in the central wire. This field circulates in the plane normal to the wire. Its magnitude is proportional to the central current, and inversely proportional to the perpendicular distance from the wire. In the second part, we use Eq. (152) to calculate the force per unit length acting on a short current carrying wire placed in the magnetic field generated by the central current. This force is perpendicular to both the direction of the magnetic field and the direction of the test current. Note that, at this stage, we have no reason to suppose that the magnetic field has any real existence. It is introduced merely to facilitate the calculation of the force exerted on the test wire by the central wire. It turns out, however, that the magnetic field does have a real existence, since, as we shall see, there is an energy associated with a magnetic field which fills space.