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.

72

To find the magnitude of the magnetic field at a distance of 0.72 m from the wires using Ampère's law, we need to use the formula:

B = (μ₀ * I₁) / (2π * r₁) + (μ₀ * I₂) / (2π * r₂)

where B is the magnetic field, μ₀ is the permeability of free space (a constant), I₁ and I₂ are the currents in the two wires, and r₁ and r₂ are the distances from the wires.

First, let's calculate the magnetic field when the currents are in the same direction.

Given:
Current in the first wire (I₁) = 13 A
Current in the second wire (I₂) = 26 A
Distance from the wires (r₁ and r₂) = 0.72 m

We know that the magnetic field due to the first wire is given by:
B₁ = (μ₀ * I₁) / (2π * r₁)

Substituting the values:
B₁ = (4π × 10⁻⁷ T·m/A * 13 A) / (2π * 0.72 m)
B₁ = (4 × 13 × 10⁻⁷) / (2 × 0.72) T
B₁ = 8.06 × 10⁻⁷ T

Similarly, the magnetic field due to the second wire is given by:
B₂ = (μ₀ * I₂) / (2π * r₂)

Substituting the values:
B₂ = (4π × 10⁻⁷ T·m/A * 26 A) / (2π * 0.72 m)
B₂ = (8 × 26 × 10⁻⁷) / (2 × 0.72) T
B₂ = 9.44 × 10⁻⁷ T

Now, let's find the total magnetic field when the currents are in the same direction:
B = B₁ + B₂
B = 8.06 × 10⁻⁷ T + 9.44 × 10⁻⁷ T
B = 17.5 × 10⁻⁷ T
B = 1.75 × 10⁻⁶ T

Therefore, the magnitude of the magnetic field at a distance of 0.72 m from the wires when the currents are in the same direction is 1.75 × 10⁻⁶ T.

Now, let's calculate the magnetic field when the currents are in opposite directions.

Using the same values as before:
B₁ = 8.06 × 10⁻⁷ T
B₂ = -9.44 × 10⁻⁷ T (negative sign indicates opposite direction)

Now, let's find the total magnetic field when the currents are in opposite directions:
B = B₁ + B₂
B = 8.06 × 10⁻⁷ T + (-9.44 × 10⁻⁷ T)
B = -1.38 × 10⁻⁷ T

Therefore, the magnitude of the magnetic field at a distance of 0.72 m from the wires when the currents are in opposite directions is 1.38 × 10⁻⁷ T.