# 2.7.9 Magnetic field of rectangular conductor with current

Flash model

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.

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