Magnetic Field of a Loop

Claimed by Jeffrey Mullavey

Creation of a Magnetic Loop

Like other magnetic field patterns, A magnetic field can be created through motion of charge through a loop. Ideal loops are considered to be circular. Thus, the conventional current is directed clockwise or counterclockwise through the loop. Formulas have been derived to assist in the calculation of these magnetic fields.

Calculation of Magnetic Field

The magnetic field created by a loop is easiest to calculate on axis. This means drawing a line though the center of the loop perpendicular to the circumference. This axis is commonly referred to as the "z-axis." The magnetic field is calculated by integrating across the bounds of the loop (0 to 2 pi), but can also be approximated with great accuracy using a derived formula. Calculation of magnetic field off of this axis is much more difficult, and usually requires the assistance of computer software. For this reason, only the calculation on axis will be addressed.

Magnitude of Magnetic Field on z-axis

When calculating the magnetic field at a point on the z-axis, one can use the following formula:

[math]\displaystyle{ \vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{(z^2 + R^2)^1.5} \text{ ,where R is the radius of the circular loop, and z is the distance from the center of the loop} }[/math]

This allows for the calculation of the magnitude in units of Teslas.

If the distance from the center of the loop is much greater than the radius of the loop, an approximation can be made. The formula can be simplified to

[math]\displaystyle{ \vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{z^3} \text{ ,where z is much greater than R} }[/math]

Direction of Magnetic Field on z-axis

The right hand rule can be used to find the direction of the magnetic field at a given point. Putting your right fingers in the direction of the conventional current, and curling them over the vector r will allow your thumb to point in the direction of the magnetic field. For example, a conventional current, I, running counterclockwise would produce a field pointing out of the page on the z-axis. This rule holds regardless of where the observational location is on the center axis.

The magnetic field pattern for locations outside the ring points in the opposite direction, in accordance with the right hand rule. The right fingers still point in the direction of the current, however the observational vector perpendicular to the perimeter of the loop is now in the opposite direction. Thus, the magnetic field is in the opposite direction.

References

Matter and Interactions Vol. II