2.7.9 Magnetic field of rectangular conductor with current
Let us calculate the spatial distribution of magnetic field generated by density
current passing through a rectangular conductor having length
, width
and thickness
,
and
(Fig. 1).
Fig. 1. Cross-section of rectangular conductor.
Fig. 2. Schematics of infinitely-thin wire which carries constant current
.
According to the Biot-Savart-Laplace law [1,2], magnetic field
from the infinitely-long thin current-carrying wire at distance
(Fig. 2) in Gaussian coordinates is given by
(1)
where
,
– light velocity,
– current in the wire, the magnetic field vector and vector product
being codirectional.
Dividing the conductor cross-section into a infinite number of wires having section
as in Fig. 1, we can write the magnetic field of elementary wire at point
in accordance with formula (1) as follows:
(2)
where
,
– current density,
– smallest distance from elementary wire to point A,
– angle between vector
and axis X, and
,
. We will not calculate further the magnetic field along the Y-axis because at an arbitrary point
it obviously is zero.
The total magnetic field at point
can be calculated by integration of expression (2) over the conductor cross-section:
(3)
where we made the transformation of variable:
. Integrals of the following type
(4)
can be expressed through analytical functions as follows:
(5)
The Z-derivatives of functions
and
in accordance with (5) are given by:
(6)
Similarly, the second derivatives of functions
and
along the Z-axis in accordance with (5) are determined by following expressions:
(7)
Thus, magnetic field
defined by expressions (3) can be written using formulas (5) as follows
(8)
The derivatives of magnetic field
components along the Z-axis, by analogy with (8) and in accordance with (6), are given by:
(9)
The second Z-derivatives of magnetic field
components, by analogy with (8) and in accordance with (7), are given by:
(10)
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.
The analysis of interaction between magnet probe and rectangular wire can be performed using a special Flash application. Using this application which is based on the theory of cantilever small oscillations, the probe amplitude, phase and resonance frequency in standart MFM method can be calculated.
Summary.
- Derived are analytical expressions for spatial distribution of magnetic field, its first and second derivatives over the surface of rectangular conductor with current (see formulas 8-10).
- Theoretical expressions for spatial distribution of magnetic field, its first and second derivatives as a function of conductor parameters can be analyzed using a special Flash application.
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.