Question

Two long straight parallel current-carrying conductors are kept ‘a’ distant apart in the air. The direction of current in both the conductors is the same. Find the magnitude of force per unit length and the direction of the force between them. Hence define one ampere.

Answer 1

The magnetic induction B₁ set up by the current I₁ flowing in the first conductor at a point somewhere in the middle of the second conductor is

`B_1 = (mu_0I_1)/(2pia)`  ...(1)

The magnetic force acting on the portion P₂Q₂ of length ℓ₂ of the second conductor is

F₂ = l₂ ℓ₂ B₁ sin 90°   ...(2)

From equation (1) and (2),

F₂ = `(mu_0I_1I_2ℓ_2)/(2pia)`, towards first conductor

`F_2/ℓ_2 = (mu_0I_1I_2)/(2pia)`   ...(3)

The magnetic induction B₂ set up by the current I₂ flowing in the second conductor at a point somewhere in the middle of the first conductor is

`B_2 = (mu_0I_2)/(2pia)`   ...(4)

The magnetic force acting on the portion P₁Q₁ of length ℓ₁ of the first conductor is

F₁ = I₁ℓ₁B₂ sin 90°    ...(5)

From equation (3) and (5)

`F_1 = (mu_0I_1I_2ℓ_1)/(2pia)`, towards second conductor

`F_1/ℓ_1 = (mu_0I_1I_2)/(2pia)`  ...(6)

The standard definition of 1A

If I₁ = I₂ = 1A

ℓ₁ = ℓ₂ = 1m

a = 1m in V/A then `F_1/ℓ_1 = F_2/ℓ_2 = (mu_0 xx 1 xx 1)/(2pi xx 1) = 2 xx 10^-7` N/m

∴ One ampere is that electric current which when flows in each one of the two infinitely long straight parallel conductors placed 1m apart in vacuum causes each one of them to experience a force of 2 × 10^-7 N/m.