Ampere’s circuital law has many applications in calculating magnetic field at any point due to current-carrying wire, current-carrying solenoid, and toroid.

**Application of ampere’s circuital law**

- Magnetic field due to a current-carrying long wire.
- Magnetic field due to the Long current-carrying solenoid.
- Magnetic field due to current carrying Toroid.

**Magnetic field due to a current carrying long wire**

**Magnetic field at a point outside the wire**

A current carrying long wire is cylindrical in shape. Let us consider a straight wire AB of radius R, carrying a current I. The lines of magnetic induction are concentric circles centered on the wire in planes perpendicular to wire.

Hence, magnetic field at point P at a distance r from the axis of coil is given by ampere’s law as

Thus, the magnetic field due to a long cylindrical wire at a point outside the wire is inversely proportional to the distance of the point from the axis of the wire.

**Magnetic field at the surface of wire**

At the surface of wire, r = R. Hence eqn. (1) becomes

**Magnetic field at a point inside the wire**

if r < R i.e. point P is inside the wire, then symmetry suggests that the B vector is tangent to the path of the concentric circle. If the current is steady and distributed uniformly through the cross-section of the wire, then the current surrounded by the path is

Thus from Ampere’s law

*Thus, the magnetic field at a point inside the wire is directly proportional to the distance of the point from the axis of the wire.*

At the axis of wire, r = 0, therefore magnetic field on the axis of the wire is zero.

The variation of magnetic field due to current carrying long wire with the distance (r) from the axis of the wire is shown in fig.

**Magnetic field due to the Long current-carrying solenoid**

Consider a long solenoid having n turns per unit length. If the turns are uniformly spaced and carrying a current I, the distribution of the magnetic field is as shown in fig.

The field is uniform on its cross-section in the central region of the solenoid. For an infinitely long solenoid, the magnetic field outside the solenoid is very small and can be approximately taken as zero.

Let us apply Ampere’s law to a rectangular path abcd, over the solenoid, so that

- Along bc and da, B vector is either zero (for outside parts of bc and da) or perpendicular (for the parts inside the solenoid) to the length.

2. For an ideal solenoid, the field B vector outside the solenoid is zero. Therefore, for the path cd, which lies outside solenoid.

3. Inside the solenoid B vector is parallel to ab (=1)

Therefore, Ampere’s law in this case reduces to

which is the required expression for magnetic field inside a long solenoid.

Note. From equation (1), B vector independent of the position of the point of observation within the solenoid and so the field within the long solenoid is not only parallel to the axis but is uniform as well. This is true of course, only for points far from the ends.

**Magnetic field due to current carrying Toroid or Anchor Ring**

A toroid is just a solenoid of finite length bent into a circle so that it has no ends.

This fig. repersents a toroid, wound uniformly with N turns of wire carrying current I.

**Case 1**.

Consider a closed path 1 of radius r1. If there is any magnetic field at all in this region, then it will be along the tangent to the path at all points.

According to Ampere’s circuital law

**B=0**

Hence magnetic field in this region is zero.

**Case 2.**

Now consider a path 3 outside the toroid.

According to Ampere’s circuital law

Since I = 0 in the path 3

**B=0**

**Case 3.**

Consider a closed path 2 of radius r within the toroid. According to Ampere’s circuital law.

Thus the magnetic field induction (B) at a point within the toroid is inversely proportional to the distance of the point from the centre of the toroid.

If the cross section of the toroid is sufficiently small, then 2πr is the mean circumference or length of the toroid.