Amperes Circuital Law

The    observation    that    magnetic    field strength varied with distance from the wire led to the following statement:

'If  the  magnetic  field  H  is  integrated along a closed path, the result is equal to the current enclosed'.

This is the basis with which we can relate the strength of a magnetic field to the current producing it. It is an extremely important statement. Note that only the component of H that lies along the path is to be considered, so the Law may be stated mathematically as:

∫ H     dl     ∑ I

where the  vector  dot  product  is used  (= H.dl.cos  ).  Note that  both H  and  dl are vectors.  The  direction  of  H  around  a current carrying conductor is given by the 'right hand corkscrew rule' , attributed to Maxwell.Ampere's  Circuital  Law  applies  for  any path chosen for the integration. If the chosen path does not enclose any current, the result of the integration will be zero.

To apply this law to the case of a long, thin current carrying conductor, it is convenient to choose a circular path of constant radius centred on the wire. Since any path could be chosen, the circular path is chosen simply to make the integration easy, H being  both constant  and  tangential  for  a given radius from the wire.

Hence

∫ H     dl     ∑ I

H.2  r = I
 
H  = I/2  r

For I1  = I2  =1 amp and r = 1 metre, the force/metre   in  a   vacuum  =   2   x  10-7
Newtons   by  definition  of  the  ampere. Hence    =    0 = 4   x 10-7

Had Ampere been able to conduct his experiments involving the force between current carrying conductors in materials other than air, he would have found that the material involved also affected the force. This can be taken into account by introducing a property called the 'relative magnetic permeability' of a material and is given the symbol   r.

Different materials have different values of . These are related to that for a vacuum (or   for practical  purposes, air)  by introducing  the   concept  of         relative permeability  r  which expresses  for the material relative to that for a vacuum, i.e.

=   0   r.

Finally, the quantity .H is defined as the flux density B (i.e. the number of flux lines/m2) so the flux density B is:

B  =    0   rH

so    the    force    on    a    current    carrying conductor in a magnetic field H  may be expressed in terms of the resulting flux density to be:

force/metre = B. I2

or

force = B.I.L

1.    when  we  draw  flux  lines  on  a diagram, the density of those lines (number/m2)  depicts  the  product 0   rH rather than just H.

2.  flux lines are always continuous.They do not start or stop in space -.i.e. they are always loops, even though they may not always be shown as such in some diagrams