# 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.