# Magnetic Field Induced by a Current-Carrying Wire

An infinitely long uniform wire carrying current induces a magnetic field (more precisely, magnetic induction) that varies with the distance from the wire and the amount of current. Ampère's law is used to determine the magnetic field at any point on the imaginary Amperian loop at a given distance from the wire with a given amount of current. Ampère's law is and can be solved for to get , where is the vacuum permeability constant, is the current enclosed by the Amperian loop, and is the radius of the Amperian loop. This Demonstration lets you vary the relative height of the Amperian loop, the Amperian loop radius, current, wire radius, and opacity of the wire to see each variable's effect on the strength of the induced magnetic field, measured in teslas (T).

### DETAILS

Snapshot 1: current running in the opposite direction
Snapshot 2: measuring strength of magnetic field within wire
Snapshot 3: measuring strength of magnetic field within wire with opposite current
As the Amperian loop increases in radius, the loop encloses more current, which increases the integral in Ampère's law. When the Amperian loop is less than the radius of the wire, the loop does not enclose all of the current running through the wire, so the strength of the magnetic field within the wire is directly proportional to the radius of the Amperian loop. When the Amperian loop is greater than the radius of the wire, the loop does enclose all of the current running through the wire, so the strength of the magnetic field within the wire is inversely proportional to the radius of the Amperian loop.
The cylinder represents a current-carrying wire. The circle represents the Amperian loop that determines the strength of the magnetic field through Ampère's law. The arrows within the wire represent current vectors. The arrows on the Amperian loop show the direction of the magnetic field at that point.
Changes in height and opacity have no effect on the strength of the magnetic field. The three variables that affect the strength of the magnetic field are the strength of the current, the radius of the Amperian loop, and the radius of the current-carrying wire. The direction of the magnetic field is shown by the tangent vectors of the Amperian loop. The direction can also be found using the right-hand rule.
Reference
[1] D. J. Griffiths, Introduction to Electrodynamics, Vol. 3, Upper Saddle River, NJ: Prentice Hall, 1999.
Special thanks to the University of Illinois NetMath Program and the Mathematics department at William Fremd High School.

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