Ampère's Law

It is an experimentally demonstrable fact that magnetic fields, like electric fields, are completely superposable. So, if a magnetic field ${\bf B}_1({\bf r})$ is generated by an electric current $I_1$ flowing through some circuit, and a field ${\bf B}_2({\bf r})$ is generated by a current $I_2$ flowing through another circuit, then when the currents $I_1$ and $I_2$ flow through both circuits simultaneously the generated magnetic field is ${\bf B}_1({\bf r})+{\bf B}_2({\bf r})$.

Figure 2.15: Two parallel current-carrying wires.

Consider two parallel wires separated by a perpendicular distance $r$, and carrying electric currents $I_1$ and $I_2$, respectively. The magnetic field-strength at the second wire due to the current flowing in the first wire is $B = \mu_0 \,I_1/2\pi\, r$. [See Equation (2.206).] This field is orientated perpendicular to the second wire, so the force per unit length exerted on the second wire is

$$F= \frac{\mu_0\, I_1 \,I_2}{2\pi\, r}.$$ (2.208)

The previous expression follows from Equation (2.205), which is valid for continuous wires as well as short test wires. The force acting on the second wire is directed radially inward toward the first wire. The magnetic field-strength at the first wire due to the current flowing in the second wire is $B= \mu_0 \,I_2/2\pi\, r$. This field is orientated perpendicular to the first wire, so the force per unit length acting on the first wire is equal and opposite to that acting on the second wire, according to Equation (2.205). Equation (2.208) is known as Ampère's law.

Equation (2.208) is the basis of the official (prior to 2019) SI definition of the ampere, which is:

One ampere is the magnitude of the current which, when flowing in each of two long parallel wires one meter apart, results in a force between the wires of $2\times 10^{-7}$ N per meter of length.

We can see that it is no accident that the constant $\mu_0$ has the numerical value of exactly $4\pi\times 10^{-7}$. (Incidentally, this rather strange definition arose because electromagnetism was originally formulated in the cgs system of units. In the cgs system, the force per unit length exerted by two parallel wires, one centimeter apart, both carrying a current of 1 abampere (i.e., 10 amperes), is $2$ dynes per centimeter.)