0.10 Ampere's law  (Page 5/6)

Application of ampere’s law

Ampere’s law is a powerful tool for calculating magnetic field for certain geometric forms of conductors carrying current. It was, however, pointed out that this law may be limited as well for many other situations where left hand side integral can not be evaluated easily. Though there are no specific rules for selecting a closed Ampere loop, but there are certain guidelines which can be helpful in applying this law. These guidelines are :

• Draw closed loop such that the point of observation lies on the loop.
• If required, draw closed loop such that magnetic field is constant along the path of integration.
• If required, draw closed loop such that magnetic field and line vectors are along the same direction or are perpendicular to each other.
• If required, draw closed loop such that there is no magnetic field. This may appear bizarre but we draw such segment of Ampere loop as in the case of solenoid (we shall see this consideration subsequently in this module).
• If required, draw closed loop as a combination of segments (like a rectangular path with four arms) in a manner which takes advantages of the situations enumerated at 2, 3 and 4.

Magnetic field due to a long cylindrical conductor

We consider three points of observation (i) A, inside the conductor (ii) C, just outside the conductor and (iii) D, outside conductor for applying Ampere’s law. One important consideration here is that magnetic field due to infinite conductor is independent of the elevations of observation points with respect to the straight cylindrical conductor. The magnetic field only depends on the perpendicular linear distance (r) of the observation point from the axis of cylindrical conductor. This situation is approximately valid for long conductor as well. If the conductor is not long enough then also we can meet the requirement of independence for observation points at those points, which are close to the conductor and the ones which are not near the ends of the conductor.

In order to apply Ampere’s law, we consider three imaginary circles containing these points separately with their centers lying on the axis of cylinder such that their planes are at right angles to the cylinder. Let the current through the conductor is I. We note here that current in the conductor is confined only to the surface of cylinder of radius R.

Magnetic field due to current in cylindrical conductor

For the point A inside the conductor, the current inside the loop is zero.

$$\oint \mathbf{B}.đ\mathbf{l}={\mu }_{0}I=0$$ $$\Rightarrow BX2\pi r_1=0$$ $$\Rightarrow B=0$$

Note that absence of current here is used to deduce that magnetic field is also absent. We can do this with the circular symmetry having constant magnetic field along the path as circle is a continuous curve without any possibility that integral values in different segments of imaginary loop cancel out along the circular path. Thus, if I = 0, then B=0.

Now, for the point B just outside the conductor, the current inside the loop is I.