# 2.7.10 Magnetic field of ring with current

Let us calculate the magnetic field generated by direct current
passing through a ring of radius
**(Fig. 1)**. Let the width and thickness of a conductor be much less than
.

**Fig. 1. Schematics of ring with current.**

**Fig. 2. Cross-section of the ring.**

According to the Biot-Savart-Laplace law [**1,2**], the magnetic field produced by a current-carrying wire element of length
at distance
from it in Gaussian coordinates is given by

(1)

where , – light velocity.

Placing the right-hand coordinate system *XYZ* into the ring center so that the *XY* plane lies in the ring plane **(Fig. 1, 2)** and noting that the problem is symmetrical relative to the ring center, it is enough to determine the magnetic field distribution in a plane containing vector codirectional with the ring radius and the Z-axis. For mathematical convenience we can choose the plane *XZ* and determine the magnetic field at point
as shown in **Fig. 2**. The radius-vector
from point
to the ring element
as a function of angle
is given by the following expression

(2)

Elementary vector as a function of and angle is written as follows:

(3)

Substituting expressions (2) and (3) into formula (1), we get

(4)

To determine the total magnetic field produced by all the ring at point , one needs to integrate every component of vector with respect to from 0 to 2p. Then, the components X, Y and Z of vector in accordance with (4) are defined as:

(5)

where .

Formulas (5) give the magnetic field distribution in the *XZ* plane. It is clear that due to the problem symmetry, the magnetic field along the Y-axis is zero and at an arbitrary point
it is equal to that at point
in the *XZ*-plane. Accordingly, formulas (5) are rewritten as:

(6)

where .

Because behaves as a parameter in the integrand of functions and , the first and second Z-derivatives of the magnetic field components can be obtained by direct differentiation of functions , with respect to and subsequent numerical integration. For example, the first -derivative of in accordance with (6) is given by:

(7)

The other components of vector are calculated similarly. In case , (point is on the ring axis) formulas (6,7) are transformed as follows

(8)

Using analitical expressions of the magnetic field first and second Z-derivates, one can calculate the interaction force (and its first derivative) between magnet probe and rectangular conductor with current. These calculations for different probe geometry are given in **Appendix**.

## Summary.

- Derived are formulas (6-8) for the spatial distribution of the magnetic field and its derivatives along the Z-axis over a current ring.

## References.

- D.V. Sivukhin. Electricity (General course of physics). Moscow, Nauka 1983. - 688 pp. (in Russian).
- R. Feinman, R. Leitos, M. Sands The Feinman lectures on physics. Electricity and magnetism. Moscow, MIR 1977. - 299 pp.