Summarizing Magnetic Fields Produced by Currents

Summary

The strength of the magnetic field created by current in a long straight wire is given by
\(B=\cfrac{{\mu }_{0}I}{2\mathrm{\pi r}}(\text{long straight wire}),\)
where \(I\) is the current, \(r\) is the shortest distance to the wire, and the constant \({\mu }_{0}=4\pi \phantom{\rule{0.15em}{0ex}}×\phantom{\rule{0.15em}{0ex}}{\text{10}}^{-7}\phantom{\rule{0.25em}{0ex}}\text{T}\cdot \text{m/A}\) is the permeability of free space.
The direction of the magnetic field created by a long straight wire is given by right hand rule 2 (RHR-2): Point the thumb of the right hand in the direction of current, and the fingers curl in the direction of the magnetic field loops created by it.
The magnetic field created by current following any path is the sum (or integral) of the fields due to segments along the path (magnitude and direction as for a straight wire), resulting in a general relationship between current and field known as Ampere’s law.
The magnetic field strength at the center of a circular loop is given by
\(B=\cfrac{{\mu }_{0}I}{2R}\text{}(\text{at center of loop}),\)
where \(R\) is the radius of the loop. This equation becomes \(B={\mu }_{0}\text{nI}/(2R)\) for a flat coil of \(N\) loops. RHR-2 gives the direction of the field about the loop. A long coil is called a solenoid.
The magnetic field strength inside a solenoid is
\(B={\mu }_{0}\text{nI}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}(\text{inside a solenoid}),\)
where \(n\) is the number of loops per unit length of the solenoid. The field inside is very uniform in magnitude and direction.

Glossary

right hand rule 2 (RHR-2)

a rule to determine the direction of the magnetic field induced by a current-carrying wire: Point the thumb of the right hand in the direction of current, and the fingers curl in the direction of the magnetic field loops

magnetic field strength (magnitude) produced by a long straight current-carrying wire

defined as \(B=\cfrac{{\mu }_{0}I}{2\mathrm{\pi r}}\), where \(I\) is the current, \(r\) is the shortest distance to the wire, and \({\mu }_{0}\) is the permeability of free space

permeability of free space

the measure of the ability of a material, in this case free space, to support a magnetic field; the constant \({\mu }_{0}=4\pi ×{\text{10}}^{-7}\phantom{\rule{0.25em}{0ex}}T\cdot \text{m/A}\)

magnetic field strength at the center of a circular loop

defined as \(B=\cfrac{{\mu }_{0}I}{2R}\) where \(R\) is the radius of the loop

solenoid

a thin wire wound into a coil that produces a magnetic field when an electric current is passed through it

magnetic field strength inside a solenoid

defined as \(B={\mu }_{0}\text{nI}\) 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)

Biot-Savart law

a physical law that describes the magnetic field generated by an electric current in terms of a specific equation

Ampere’s law

the physical law that states that the magnetic field around an electric current is proportional to the current; 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

Maxwell’s equations

a set of four equations that describe electromagnetic phenomena