22.9 Magnetic fields produced by currents: ampere’s law  (Page 2/13)

The magnetic field strength (magnitude) produced by a long straight current-carrying wire    is found by experiment to be

where $I$ is the current, $r$ is the shortest distance to the wire, and the constant ${\mu }_0=\mathrm{4\pi }\times {\text{10}}^{-7}\mathrm{T}\cdot \text{m/A}$ is the permeability of free space    . $({\mu }_0$ is one of the basic constants in nature. We will see later that ${\mu }_0$ is related to the speed of light.) Since the wire is very long, the magnitude of the field depends only on distance from the wire $r$ , not on position along the wire.

Ampere’s law and others

The magnetic field of a long straight wire has more implications than you might at first suspect. Each segment of current produces a magnetic field like that of a long straight wire, and the total field of any shape current is the vector sum of the fields due to each segment. The formal statement of the direction and magnitude of the field due to each segment is called the Biot-Savart law    . Integral calculus is needed to sum the field for an arbitrary shape current. This results in a more complete law, called Ampere’s law    , which relates magnetic field and current in a general way. Ampere’s law in turn is a part of Maxwell’s equations    , which give a complete theory of all electromagnetic phenomena. Considerations of how Maxwell’s equations appear to different observers led to the modern theory of relativity, and the realization that electric and magnetic fields are different manifestations of the same thing. Most of this is beyond the scope of this text in both mathematical level, requiring calculus, and in the amount of space that can be devoted to it. But for the interested student, and particularly for those who continue in physics, engineering, or similar pursuits, delving into these matters further will reveal descriptions of nature that are elegant as well as profound. In this text, we shall keep the general features in mind, such as RHR-2 and the rules for magnetic field lines listed in Magnetic Fields and Magnetic Field Lines , while concentrating on the fields created in certain important situations.