(a) Derive an expression for the force per unit length between two long straight parallel current carrying conductors .Hence define SI unit of current (Ampere).

F/L = μ₀I₁I₂/2πr

when a force of 2 x 10⁻⁷ N is produced between two conductors 1 m apart from each other and having a cross-sectional radius 1 m. The current on each conductor in this situation is 1 Ampere

Explanation:

Consider two conductors of same length parallel to each other. The Magnetic field due to current in 1st conductor is given by Ampere's Law as:

B₁ = μ₀I₁/2πr

where,

B₁ = Magnetic Field due to Conductor 1

μ₀ = permeability of free space = 4π x 10⁻⁷ N/A²

I₁ = Current through conductor 1

r = radius of cross section of conductor

Now, the force of the conductors on each other is given by:

F₁₂ = B₁I₂L

where,

F₁₂ = Force of conductor 1 on conductor 2

B₁ = Magnetic Field of Conductor 1

I₂ = Current through conductor 2

L = Length

Therefore,

F₁₂ = μ₀I₁I₂L/2πr

The force of conductor 2 on conductor will also have same magnitude but opposite direction:

F₁₂ = |- F₂₁| = F

F/L = μ₀I₁I₂/2πr

F/L = μ₀I₁I₂L/2πr

Now, to define S.I unit of current (Ampere) we substitute values:

F/(1 m) = (4π x 10⁻⁷ N/A²)(1 A)(1 A)/2π(1 m)

F = 2 x 10⁻⁷ N

So, when a force of 2 x 10⁻⁷ N is produced between two conductors 1 m apart from each other and having a cross-sectional radius 1 m. The current on each conductor in this situation is 1 Ampere

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