A long, straight wire has a 12.1 A current flowing in the positive x-direction, as shown in the figure. Close to the wire is a square loop of copper wire that carries a 2.23 A current in the direction shown. The near side of the loop is d = 57.1 cm away from the wire. The length of each side of the square is a = 1.71 m. Find the net force between the two current-carrying objects.
To find the net force between the two current-carrying objects, we need to consider the magnetic interaction between them.
The force between two current-carrying wires can be calculated using the formula:
F = (μ₀ * I₁ * I₂ * L) / (2π * d)
Where:
F is the force between the wires,
μ₀ is the permeability of free space, which is approximately 4π × 10⁻⁷ T·m/A,
I₁ and I₂ are the currents in the two wires,
L is the length of the wire segment where the magnetic field is acting,
and d is the distance between the wires.
In this case, we have a long straight wire and a square loop. Let's break down the problem step by step:
1. Calculate the force between the straight wire and each side of the square loop individually using the above formula.
2. Since the current in the straight wire is flowing in the positive x-direction and the current in the loop is flowing in the direction shown, the force between the wire and the near side of the loop will be attractive, while the force between the wire and the other three sides of the loop will be repulsive. Thus, we need to consider the directions of the forces when calculating the net force.
3. The net force will be the vector sum of the forces between the straight wire and each side of the square loop.
So, let's calculate the force between the wire and each side of the loop individually:
For the near side of the loop:
F_near = (μ₀ * I₁ * I₂ * L) / (2π * d)
For the other three sides of the loop:
F_other = -(μ₀ * I₁ * I₂ * L) / (2π * d)
Now, to calculate the net force:
Net Force = F_near + 3 * F_other
Plug in the given values:
I₁ = 12.1 A
I₂ = 2.23 A
L = length of each side of the square = 1.71 m
d = 57.1 cm = 0.571 m
After substituting these values into the formulas and performing the necessary calculations, you will find the net force between the two current-carrying objects.