The sum of the quantities of the dot product of B and vector length ∆L (B. ∆L) for all path elements of the complete Loop is equal to μ。times the current (I) flowing through the conductor enclosed by the loop.

**N**

**Σ (B. ∆L)r = μ。I**

**r = 1**

### EXPLANATION

Consider a closed Circular Path shown around a current-carrying wire. This closed path is referred to as the Amperean path. Divide this path into large no. of elements each of vector length ĀL for each element remains the same. The direction of B at each point is tangent to the curved path. Then

**B11 Δ L = B Cos θ Δ L = B Δ L Cos θ**

**= B. Δ L …(1)**

**Where B11 = B Cos θ = Component of B parallel to ΔL**

Where θ is the angle between B and ΔL. The sum of all the quantities B. Al. for all path elements in a closed path is equal to μo times the current enclosed by the Loop It can be represented as

**(B. ΔL)1 + (B. Δ L)2 + ——— + (B. Δ L)N = μ。I**

In summation form,

**N**

**Σ (Β. Δ L)r = μο1 … (2)**

**r=1**

This is known as Ampere’s circuital law Aho is a constant known as the permeability of free space. In SI units its value is

**μο = 47 ⊼ ✕ 10-7 Wb A-1m-1**

Where, N= Total no. of length elements in a closed loop.

r = Variable changes from 1 to N.