Symbolic Analysis of Magnetic Field with Symbolic Analysis of Magnetic Field with the Biot-Savart-Laplace Method

Daniel Ioan and Irina Munteanu

1  General presentation of the method

The Biot-Savart-Laplace (BSL) method allows the computation of the magnetic field produced by an arbitrary -current distribution, in any point in space. The method can be applied only if the space is magnetically homogeneous.

The BSL formula for the magnetic field is:

H(P) = [(J(Qx R)/(k |  R  | 3)] dv (1)

in which x represents the vector product, k = 4 p and

H [A/m] is the vector of the magnetic field strength in point P;

J [A/m^2] is the vector of the current density in point Q;

R [m] is the vector which links the ßource point" Q with the "field point" P.

The integral (1) is taken over the current-carrying conducting domain.

2  The magnetic flux density

Since the medium is homogeneous, the magnetic flux density B can be computed with the relation:

B = m H (2)

in which m is the absolute magnetic permeability, measured in H/m.

Since the two fields B and H are collinear and proportional, they have the same field lines.

3  The magnetic field lines

Since the magnetic field is solenoidal (div B = 0), its field lines are closed curves.

These turn around the current which produces them and are oriented according to the right screw with respect to the orientation of the current.

The conductor's volume element produces a field with circular lines, placed in planes perpendicular to the current's direction. This elementary field is zero on the direction of the current and decreases inversely proportional to the square of the distance to the source. The BSL integral is improper, because the integrand is infinite if P=Q. However, the field in the interior of massive conductors is bounded because the integral is convergent if the conductor is finite. The integral (1) is not necessarily bounded when the integration domain is unbounded.

At great distance to the source, the magnetic field behaves like the field of a small coil (or small permanent magnet):

H = 1/(4 p) ( 3 (mR)R/ R5 - m/ R3)

in which m = (1/2) r x J dv is the magnetic moment of the current distribution (m =iA in the case of a filamentary one-turn coil of area A), relation which shows the decrease of the field inversely proportional to the cube of the distance to the source.

4  The magnetic potential

Since the magnetic flux density is solenoidal (div B = 0), it admits a vector potential A, such that:

B = rot A (3)

The relation (3) does not yield a unique vector potential A: a boundary condition and a supplementary gauge condition for div A are necessary to ensure uniqueness. Without the gauge condition, adding the gradient of an arbitrary function to A does not modify the magnetic flux density.

A possible expression for the magnetic potential is given by the BSL formula for the potential:

A(P) = m [(J(Q))/(k |  R  | )] dv (4)

The vector potential given by (4) is solenoidal (has div A = 0), if the divergence of the current density is zero in the whole conducting domain (respectively, if the normal component of the current density, Jn, is zero on the surface of the conductor).

It can be noticed that the vector potential has the tendency to be oriented in the direction of the current.

The potential produced by an elementary volume decreases inversely proportional to the distance to the source. Outside the conductor, the potential decreases inversely proportional to the square of the distance to the source, according to the relation:

A = 1/(4 p) (m x R) / R3.

The formula (4) is the magnetic equivalent of the Coulomb integral for the electrostatic potential.

5  Advantages of using the magnetic vector potential

The magnetic flux on a surface S is defined as the following integral on S:

f = B dS (5)

After expressing the flux density B as a function of the vector potential and by applying the Stokes relation, the following expression of the magnetic flux is obtained:

f = A dl (6)

Unlike relation (5) which contains a double integral, the integral (6) is simple, since the integration domain is a closed curve, the boundary of the surface S.

In the 2D case, the magnetic flux can be calculated in an even simpler manner, as a potential difference. In this case, the vector potential has only one component, normal to the problem's plane (A is represented by a scalar rather than a vector), and the magnetic field lines coincide with the equipotential lines. This observation allows the generation of a simple procedure for representing the field lines spectrum.

6  The case of degenerated conductors

In the case of degenerated conductors, it is possible that some of the BSL integrals are divergent. For instance, the field on a filamentary conductor tends to infinity, but it is finite at non-zero distance from the conductor.

7  Algorithm of the metod

Variant

Instead of relation (1) use relation (4), determine the vector potential, and then calculate the magnetic flux density and the field strength using:

B = rot A and H = B/ m.

8  Symbolic analysis

Even in the case of simple geometris, the BSL integrals have quite complicated forms, often with no analyitic expression. For this reason, it is necessary to use computers in their evaluation. The computer can be used in two complementary ways:

In this last case, it is necessary to use a symbolic analysis package, such as MapleV. This package is capable to operate with mathematical expressions and to perform algebraic or calculus operations (e.g. derivatives, integrals) with this expressions. The obtained results are mathematical formulae and not numbers. Subsequently, these formulae can be numerically evaluated, or plotted. The determination of an analytic expression is of maximal importance in engineering, even if this expression only applies to a simplified model.

The symbolic analysis packages assist the user in solving mathematically formulated problems, but they do not "think" for the user. They are rather databases of mathematical formulae and procedures. For this reason, the user is not exempt from the necessity of perfectly understanding the problem, from both physical and mathematical points of view.

The following instructions represent an example of symbolic solution of a simple problem, namely the computation of an integral and its graphical representation.

> f:=exp(-x)*sin(x); # the expression f


f : = e( - x) sin(x)

> Int(f,x=0..z); # the inert definite integral of the expression f



z
0 
e( - x) sin(xdx

> g:=value("); # g (an expression) is the value of the integral of f


g : = ( - 1
2
 cos(z) - 1
2
 sin(z)) e( - z) + 1
2

> plot(f,x=0..2*Pi); # the plot of f versus the variable x

pictures/bsl_en01.gif
> plot(g,z=0..2*Pi); # the plot of g versus the variable z

pictures/bsl_en02.gif
> subs(z=2,g); # the exact value of the expression g for z=2


( - 1
2
 cos(2) - 1
2
 sin(2)) e(-2) + 1
2

> evalf("); # the numerical approximate value of the expression g


.4666296626

>


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On 16 Feb 2000, 02:04.