The wire in the figure below carries a current of 13 A. Suppose that a second long, straight wire is placed right next to this wire. The current in the second wire is 26 A. Use Ampère's law to find the magnitude of the magnetic field at a distance of r = 0.72 m from the wires when the currents are in the same direction and in opposite directions.
To find the magnetic field at a distance of r = 0.72 m from the wires using Ampère's law, we need to consider both cases when the currents are in the same direction and in opposite directions.
1) When the currents are in the same direction:
Ampère's law states that the line integral of the magnetic field around a closed path is equal to μ₀ times the total current passing through the surface bound by the closed path.
To apply Ampère's law to this case, we need to consider a circular path of radius r = 0.72 m centered around the wire. We assume that the wire is infinitely long and the circular path is perpendicular to the wire.
The line integral of the magnetic field around the circular path is equal to the product of the magnetic field magnitude and the circumference of the circle, which is 2πr.
According to Ampère's law, this line integral is also equal to μ₀ times the total current passing through the surface bound by the circular path. In this case, the total current passing through the surface is the sum of the currents in the two wires, which is 13 A + 26 A = 39 A.
So, the line integral of the magnetic field around the circular path is 2πr * B = μ₀ * 39 A.
Solving for the magnetic field (B), we get:
B = μ₀ * 39 A / (2πr)
Substituting the given value of r = 0.72 m and the permeability of free space (μ₀) ≈ 4π × 10⁻⁷ T*m/A, we can calculate the magnetic field:
B = (4π × 10⁻⁷ T*m/A) * 39 A / (2π * 0.72 m)
B ≈ 6.80 × 10⁻⁵ T
Therefore, the magnitude of the magnetic field at a distance of r = 0.72 m from the wires when the currents are in the same direction is approximately 6.80 × 10⁻⁵ T.
2) When the currents are in opposite directions:
The only difference in this case is the total current passing through the surface. Since the currents in the wires are in opposite directions, the total current passing through the surface is the difference between the currents in the two wires: 26 A - 13 A = 13 A.
Therefore, we can use the same formula as before to calculate the magnetic field:
B = μ₀ * 13 A / (2π * 0.72 m)
B ≈ 2.90 × 10⁻⁵ T
So, the magnitude of the magnetic field at a distance of r = 0.72 m from the wires when the currents are in opposite directions is approximately 2.90 × 10⁻⁵ T.
To apply Ampère's law, which relates the magnetic field around a closed loop to the current passing through the loop, we need to choose a closed loop that encloses the wires. In this case, we can choose a circular loop centered between the two wires and with a radius of 0.72 m.
Let's calculate the magnetic field for each configuration:
1. When the currents are in the same direction:
Using Ampère's law, we have:
∮B · dl = μ₀I_enclosed,
where B is the magnetic field, dl is an element of the circular loop, I_enclosed is the total current passing through the loop, and μ₀ is the permeability of free space.
Since the currents in the wires are in the same direction, the total current passing through the loop is the sum of the currents in both wires:
I_enclosed = 13 A + 26 A = 39 A
Assuming the magnetic field is constant along the circular loop, we can take it out of the integral:
B ∮dl = μ₀I_enclosed
The right-hand side of the equation represents the product of the permeability of free space and the total current passing through the loop. We can calculate this value:
B × (2πr) = μ₀ × 39 A
B × (2π × 0.72) = μ₀ × 39
B = (μ₀ × 39) / (2π × 0.72)
Now we can calculate the magnitude of the magnetic field when the currents are in the same direction.
2. When the currents are in opposite directions:
Using Ampère's law again, the total current passing through the loop is now the difference between the currents in the wires:
I_enclosed = 26 A - 13 A = 13 A
Using the same process as before, we can calculate the magnitude of the magnetic field when the currents are in opposite directions.
Remember to use the value of μ₀, which is equal to 4π × 10⁻⁷ T · m/A, in the calculations.
I hope this explanation helps in understanding how to apply Ampère's law to find the magnitude of the magnetic field at a given distance from the wires in both configurations.