0.10 Ampere's law  (Page 2/6)

Further, considering our ability or constraints for integration around any path, we look for a contour which passes through points where magnetic field is same or where certain simplifying relation between magnetic field and line element vectors exists. This issue is important as it renders integration derivable. Clearly, this is where symmetry of object carrying current comes into play.

Thus, symmetry of object carrying current and selection of path for the integration are two important requirements for putting Ampere law to use though the law itself is true for all closed path and any configuration of conductor.

Statement of ampere law

There are few variants of this law. We shall begin with the simplest form. There is one precondition as well. This law in the form discussed here is true for steady current and is not valid for time varying current. In the simplest form, it states that the line integral of scalar product of magnetic field and length element vectors along a closed imaginary line is equal to the product of absolute permeability of free space and the net "free" current through the imaginary closed line. Mathematically,

$$\oint \mathbf{B}.đ\mathbf{l}={\mu }_{0}I$$

The "free" current represents the current owing to moving electrons or ions. This law is modified by Maxwell for time dependent varying current using the concept of "displacement" current. We shall briefly discuss displacement current and the Maxwell modification in the next section.

The sign of current through the loop is determined by the direction in which line integral is executed. We curl fingers of right hand such that it is aligned with the direction of integration along the closed path. The extended thumb, then, points in the direction of positive current. Alternatively, if the direction of integration is counterclockwise, then current coming toward the viewer of closed path is positive and the current going away is negative. The net current through the loop is the algebraic sum of positive and negative currents. See Ampere's law(exercise) : Problem 2 and 4 for illustrations.

Sign of current

In the second form of the law, the right hand side of the equation is substituted with a surface integral as given here :

$$\oint \mathbf{B}.đ\mathbf{l}={\mu }_0\oint \mathbf{J}.đ\mathbf{S}$$

Here J is current density through surface S . The S is the surface for which imaginary closed line serves as boundary. Note that we consider surface area element (đ S ) as a vector. The surface area element vector is normal to the surface and its orientation across the surface is determined in the same manner as we determine the sign of the current. We curl fingers of right hand such that it is aligned with the direction of integration along the closed path. The extended thumb, then, points in the direction of surface area element vector. Alternatively, if the direction of integration along the Ampere loop is anticlockwise, then surface area element vector is directed toward the viewer of closed path and if the direction of integration is clockwise, then surface area element vector is directed away from the viewer of closed path.