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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 .
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| 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
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(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
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(3) |
Substituting expressions (2) and (3) into formula (1), we get
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(4) |
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(2) |
Elementary vector  as a function of and angle is written as follows:
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:
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(5) |
where  .
Unfortunately, functions of  and  type can not be expressed through elementary analytic functions, therefore, calculation of  and  can be performed by numerical integration.
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:
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(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:
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(7) |
The other components of vector are calculated similarly. In case  ,  (point is on the ring axis) formulas (6,7) are transformed as follows
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(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.
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