Physics » Magnetism » Magnetic Fields Produced by Currents: Ampere’s Law

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

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