Calculation of a magnetic field inside Closed and Hollow Conductor

© Eugene A. Grigor'ev

Calculation of a magnetic field inside of system of conductors,
equivalent by electromagnetic properties
to the Closed and Hollow Conductor.

The superficial current  Is current on a hollow closed conductor between its poles A and B, it is possible to present as " strings of a current "  i  [Y.E.Tamm, Bases of the theory of an electricity, page 140], i.e. the currents which current on mathematical meridians of sphere from B to A.

It considerably simplifies the program of calculation of a magnetic field (MF) inside of the system of conductors equivalent on electromagnetic properties (in necessary approach) to spherical Closed and Hollow Conductor (CHC).

The Biot-Savart-Laplas law.

Or in the vector form   
Calculation of a field for two-dimensional case (plane XZ) is made under formulas:

The module of a radius-vector from an element of a current in a point of supervision

It is visible, that at a conclusion of the equation of a vector field, there is an elliptic integral of the second sort [ ds = f (j) ]. It cannot be expressed in elementary functions for simple numerical calculations.

For simplification of calculations we shall accept:
- System factor k = 1;
- Radius of approximated sphere R = 1 (further Rs = 1).
Δl we shall designate, as Δss = R•Δj = Δj ).

The initial data:

Radius of approximated CHC

Rs = 1

Quantity of strings of a current

NL = 400

n = 0.. NL - 1

An angular step between strings

Δq = 2 × p / NL

Quantity of elements of a current

l = 200

m = 0.. NΔl - 1

An angular step centre to centre elements

Δj = p / NΔl

Number of points of supervision

Na = 100

a = 0.. Na - 1

Coordinate of "X" points of supervision

s(a) = a×( R / Na )

A superficial current

Is = 1

A current in a string

i = Is / NL

Because of axial symmetry of system pays off only tangential, q-component of MF.
In a considered case it is an Y-component, that is designated in Mathcad, as an index ( 1) at vector product.


Accuracy of approximation of CHC can be increased by more quantity of strings of a current.

NL = 1000;  NΔl = 1000

NL = 2000;  NΔl = 2000

NL = 5000;  NΔl = 5000

In this case, accuracy of the account is sufficient to believe result authentic.
Check by  BASIC  confirms calculation. File sphere.bas

In practice as more as possible exact approximation of electromagnetic properties of CHC on distance (0 ÷ 0.85) R from the center is interesting.
On the schedule it is visible, that the initial data satisfy to this condition.


Calculation of a magnetic field from bringing conductors 1 and 1a  along axis . Is = I LC .



     Number of elements of splitting of a bringing conductor      z:= 0..NDL - 1




Comparing with results of two calculations it is possible to draw following conclusions:

1. The Magnetic field inside CHC is defined only by a current of bringing conductors (a difference in values of intensity - 14 orders);

2. Within the limits of accuracy of the machine account, the magnetic field created inside CHC by a current, current on its surface and between its poles, is equal to zero.



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Qbasic, unpack and work, see sphere.bas

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