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

Daniel Ioan and Irina Munteanu

9.3  Problem 3 - Interaction force between two parallel conductors

Calculate the interaction force between two portions of straight filamentary conductors, parallel and of the same length l, carrying the currents i1 si i2. The conductors are placed in air, at the distance a between each other.

Solution

The magnetic (Laplace) force exerted on a straight conductor has the expression:

F = I (l x B).

According the the results of problem 1, the magnetic flux density produced by a straight segment of the first conductor has (with the notations of the present problem) the expression:

B1 : = [(m0 i1 ([(x)/({x2 + a2})] - [(x - l)/({(x - l)2 + a2})]))/(4 p a)]

where x is the current coordinate along the second conductor.

Since the magnetic flux density B1 is normal on the current's direction, the force exerted on the second conductor can be obtained by integration, with the formula:

F21 = i2 0lB1 dx

Indicate in which conditions for the length l the result thus obtained has physical meaning.

> restart:

> B[1]:= mu[0]/4*i[1]/Pi/a*(x/sqrt(x^2+a^2)-(x-l)/sqrt((x-l)^2+a^2));


B1 : = 1
4
 
m0 i1 ( x
{x2 + a2}
- x - l
{x2 -x l + l2 + a2}
)

p a

> F[21]:=i[2]*Int(B[1],x=0..l);


F21 : = i2 
l
0 
1
4
 
m0 i1 ( x
{x2 + a2}
- x - l
{x2 -x l + l2 + a2}
)

p a
 dx

> f:=value(F[21])/l; # Per-unit-length force


f : = 1
2
 
i2 m0 i1 (

 

l2 + a2
 
-

 

a2
 
)

p  a l

> fA:=limit(f,l=infinity); # Per-unit-length force between infinitely long conductors (Ampere's force)


fA : = 1
2
  i2 m0 i1
p a

Numerical application (the SI definition of the Ampere)

> i[1]:=1; i[2]:=1; a:=1: mu[0]:=4*Pi*10^(-7):


i1 : = 1


i2 : = 1

> evalf(fA,1);


.2 10-6

>

When the conductors are carrying currents of 1 A and are placed at a distance of 1m, the per-unit-length force has therefore the value

2 10( - 7) N/m = 0.2 mN/m.


File translated fromTEX by TTH, version 2.62.
On 16 Feb 2000, 02:04.