Question

Two infinitely long straight wires A1 and A2 carrying currents I and 2I flowing in the same directions are kept ‘d’ distance apart. Where should a third straight wire A3 carrying current 1.5 I be placed between A1 and A2 so that it experiences no net force due to A1 and A2 ? Does the net force acting on A3 depend on the current flowing through it ?

Answer 1

Given :

• A₁ , A₂ and A₃ are three wires carrying currents I₁, I₂ and I₃ respectively.
• I₁ = I , I₂= 2 I , I₃ = 1.5 I
• Distance between A₁ and A₂ is d.

To find :

• Where wire A₃ should be placed so that it experiences no net force

Solution :

• We know that, Force per unit length between two infinite parallel current carrying conductors is given by :

        F = (μ₀/4π)(2 I₁ I₂/d)

• Let wire A₃ be placed at a distance of x from A₁.

• As the current is flowing in the same direction, there will be a force of attraction between the wires. Hence both A₁ and A₂ will attract A₃.
• Force experienced by A₃ due to A₁ will be,

        F₁ = (μ₀/4π)[2 I₁ I₃/(x) ]

            = (μ₀/4π)[3 I² / x ]    ....(1)

• Force experienced by A₃ due to A₂ will be,

        F₂ = (μ₀/4π)[2 I₂ I₃/(d - x) ]

             = (μ₀/4π)[6 I²/(d - x) ]    ....(2)

• As no net force acts on wire A₃, F₁ = F₂.
• From equations (1) and (2),

         (μ₀/4π)[3 I² / x ] = (μ₀/4π)[6 I²/(d - x) ]  

        ∴ 3 ( d - x ) = 6 x

        ∴ d - x = 2x

        ∴ d = 3x

        x = d/3

Answer :

• The wire A₃ should be kept at a distance of d/3 from the wire A₁ .
• The net force acting on the wire A₃ does depend on the magnitude of the current flowing through it. This can be proved by the formula which we have used above, which shows the force depends on the current.
• The direction of the current in A₃ does not affect the net force as either A₁ or A₂ will cancel the effect produced by  A₃.

Answer 2

The third straight wire should be placed at a distance of d/3.

Yes, the net force acting on A3 depend on the current flowing through it.

Explanation:

The force per unit length on two infinitely long straight wires is given by the formula:

F = μ₀$I_aI_b$/2πd

Where,

$I_a$ and $I_b$ are current flowing through the wire

d is the distance between two wires

The force of wire A3 due to wire A1 = (μ₀ × I × 1.5I)/2πx

The force of wire A3 due to wire A2 = (μ₀ × 2I × 1.5I)/2π(d - x)

Since, there are no net force on wire A3, then both equation should be equal.

(μ₀ × I × 1.5I)/2πx = (μ₀ × 2I × 1.5I)/2π(d - x)

2x = d - x

2x + x = d

3x = d

∴ x = d/3