32: Force Between Two Infinite, Parallel Wires

INTRO: 
You are given two infinite, parallel wires each carrying current I. The wires are separated by a distance d, and the current in the two wires is flowing in the same direction. This problem concerns the force per unit length between the wires.

PART A:
Is the force between the wires attractive or repulsive?

SOLUTION: 
The second right hand rule assists us in solving this problem. Since we have two parallel wires with equal current going the same direction, that vector represents each of the wires velocity vectors. By pointing our right thumb in this direction and using our pointer finger to represent the magnetic field (pointing back toward ourselves), at any point, the Force of the wire will be represented by our middle finger. Thinking of this for both wires, the force vectors are pointing towards each other, resulting in an attractive relationship. 

PART B:
What is the force per unit length F/L between the two wires?
Express your answer in terms of I, d, and constants such as μ₀ and π.

SOLUTION:
First some relevant formulas:
F_B=qvBsin(θ) = qv^→×B^→ 
on a long straight wire,
F=ILBsin(θ) = I^→L×B^→ where L is the length of the wire
So the magnetic field at wire 2 from the current in wire 1 will be B=μ₀I₁ ⋅ 1/[2πd]
The force on a length ΔL of wire 2 will be F=ΔLI₂×B
The force per unit length in terms of the currents will be,
F/ΔL = μ₀I₁I₂ ⋅ 1/[2πd]
since I₁ = I₂,
F/L = μ₀I² / [2πd]

PART C:
In the SI system, the unit of current, the ampere, is defined by this relationship using an apparatus called an Ampère balance. What would be the force per unit length of two infinitely long wires, separated by a distance 1m, if 1A of current were flowing through each of them?
Express your answer numerically in newtons per meter.

SOLUTION:
If I = 1 A, & d = 1 m, plugging these values into the formula from the previous section,
F/L = μ₀(1 A)² / [2π(1m)]
= μ₀/ [2π] ⋅ A²/m
μ₀ = 4π×10^-7 ⋅ T⋅m/A
→ 4π×10^-7 / [2π] ⋅A²/m ⋅T⋅m/A  →  2×10^-7 ⋅ T⋅A
1 T = 1 N / [A⋅m] →2×10^-7 ⋅ N/m