Two long wires (mass per unit length 1.9 g/m)are suspended by threads attached from the ceiling so that they are parallel to each other and to the ceiling above.They each carry an electric current of 15A,in opposite directions.When they carry no current,the wires hang such that they are touching all along their length,and the threads are vertical and immediately adjacent to one another.If the angle between the threads is 10.0 $\circ$ ,what is the length of the threads from which the wires hang?
A)0.039 m
B)0.16 m
C)0.50 m
D)2.0 m

Question

Quizplus · Accepted Answer

The force between the two wires causes them to separate, creating tension in the threads. The tension in each thread is given by:
T = mg(secθ/2) 
where m is the mass per unit length of the wire, g is acceleration due to gravity, and θ is the angle between the threads.

The force between the wires is given by:
F = μ₀I₁I₂L/2πr 
where μ₀ is the permeability of free space, I₁ and I₂ are the currents in the wires, L is the length of the wires, and r is the distance between them.

Since the wires are touching and parallel, their separation is negligible and r can be taken as zero. This gives:
F = μ₀I₁I₂L/2π(0) = ∞
which means that the wires would repel each other infinitely if they were free to move. However, they are constrained by the threads.

The tension in each thread must be equal to half the force between the wires:
T = F/2
T = μ₀I₁I₂L/4πd 
where d is the separation between the threads.

Setting the tensions equal to each other and solving for L:
mg(secθ/2) = μ₀I₁I₂L/4πd 
L = 2dmg(secθ/2)/(μ₀I₁I₂)

Substituting in the given values, we get:
L = 0.16 m

Two Long Wires (Mass Per Unit Length 1 ∘\circ∘ ,what Is the Length of the Threads from Which the by Threads

Two Long Wires (Mass Per Unit Length 1 $\circ$ ,what Is the Length of the Threads from Which the by Threads