Magnetic Fields Produced by Currents: Ampere’s Law

Magnetic fields have both direction and magnitude. As noted before, one way to explore the direction of a magnetic field is with compasses, as shown for a long straight current-carrying wire in Figure 10.8.1. Hall probes can determine the magnitude of the field. The field around a long straight wire is found to be in circular loops. The right hand rule 2 (RHR-2) emerges from this exploration and is valid for any current segment—point the thumb in the direction of the current, and the fingers curl in the direction of the magnetic field loops created by it.

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 ${\mu }_0\approx \mathrm{4\pi }\times {10}^{-7}\mathrm{T}\cdot \text{m/A}$ 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. The displayed numerical value is an approximation; in the current SI this constant is determined experimentally.

Each small segment of a steady current contributes to the magnetic field. The Biot-Savart law gives the size and direction of that contribution; adding the contributions from all segments gives the total field. Ampere’s law provides another relationship: for steady currents, the accumulated component of the magnetic field along a closed path is proportional to the net current passing through a surface bounded by that path. This is especially useful for highly symmetric current arrangements. Maxwell later extended this law to include changing electric fields, as discussed in Maxwell’s Equations: Electromagnetic Waves Predicted and Observed.

The magnetic field near a current-carrying loop of wire is shown in Figure 10.8.2. Both the direction and the magnitude of the magnetic field produced by a current-carrying loop are complex. RHR-2 can be used to give the direction of the field near the loop, but mapping with compasses and the rules about field lines given in “Magnetic Fields and Magnetic Field Lines” are needed for more detail. There is a simple formula for the magnetic field strength at the center of a circular loop. It is

where $R$ is the radius of the loop. This equation is very similar to that for a straight wire, but it is valid only at the center of a circular loop of wire. The similarity of the equations does indicate that similar field strength can be obtained at the center of a loop. One way to get a larger field is to have $N$ loops; then, the field is $B=N{\mu }_{0}I/(2R)$. Note that the larger the loop, the smaller the field at its center, because the current is farther away.

A solenoid is a long coil of wire (with many turns or loops, as opposed to a flat loop). Because of its shape, the field inside a solenoid can be very uniform, and also very strong. The field just outside the coils is nearly zero. Figure 10.8.3 shows how the field looks and how its direction is given by RHR-2.

The magnetic field inside of a current-carrying solenoid is very uniform in direction and magnitude. Only near the ends does it begin to weaken and change direction. The field outside has similar complexities to flat loops and bar magnets, but the magnetic field strength inside a solenoid is simply

where $n$ is the number of loops per unit length of the solenoid ($n=N/l$, with $N$ being the number of loops and $l$ the length). Note that $B$ is the field strength anywhere in the uniform region of the interior and not just at the center. Large uniform fields spread over a large volume are possible with solenoids, as Example 10.8.2 implies.

There are interesting variations of the flat coil and solenoid. A toroid is much like a solenoid bent into a circle, with magnetic field lines forming closed loops inside it. Tokamaks use magnetic fields around a ring-shaped chamber to help confine charged particles for fusion. The particles spiral around magnetic field lines while moving along them. A simple toroidal field does not provide complete confinement: particles can drift across the field, so tokamaks combine magnetic fields in different directions to improve confinement. A whole range of coil shapes are used to produce different magnetic field shapes. Adding ferromagnetic materials can strengthen and reshape the field. These materials can also redirect magnetic fields around a protected region and are used as shields for devices sensitive to external magnetic fields, including Earth’s magnetic field.