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