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