The wire in the figure below carries a current of 12 A. Suppose that a second long, straight wire is placed right next to this wire. The current in the second wire is 30 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 magnitude of the magnetic field at a distance of r = 0.72 m from the wires using Ampère's Law, you need to follow these steps:

Step 1: Understand Ampère's Law
Ampère's Law states that the magnetic field along a closed loop is directly proportional to the current passing through the loop. It can be mathematically expressed as:

∮ B · dl = μ₀ * I

Where:
- ∮ B · dl is the line integral of the magnetic field B along the closed loop.
- μ₀ is the permeability of free space, which is a constant equal to 4π * 10^(-7) T·m/A.
- I is the current passing through the closed loop.

Step 2: Define the closed loop
In this case, the closed loop is a circle of radius r = 0.72 m centered around the wires.

Step 3: Choose the appropriate closed loop
Since the wires are long, straight wires, the most convenient closed loop to choose is a circle transverse to the wires and centered around them.

Step 4: Calculate the line integral
The line integral in Ampère's Law requires calculating the dot product of the magnetic field B and the differential element dl along the closed loop.

For the current in the same direction:
- The line integral ∮ B · dl is equivalent to B * 2πr, as B is constant along the circular loop.
- The current I in this case is 12 A.

For the current in opposite directions:
- The line integral ∮ B · dl is equivalent to the difference between the two sides of the loop.
- The current I in this case is the difference between 30 A and 12 A, which is 18 A.

Step 5: Rearrange Ampère's Law to solve for the magnetic field
Rewrite Ampère's Law as:

B * 2πr = μ₀ * I

Solving for B, we get:

B = (μ₀ * I) / (2πr)

Step 6: Calculate the magnetic field
Substitute the values of μ₀, I, and r into the equation to calculate the magnetic field.

For the current in the same direction:
B = (4π * 10^(-7) T·m/A * 12 A) / (2π * 0.72 m)

For the current in opposite directions:
B = (4π * 10^(-7) T·m/A * 18 A) / (2π * 0.72 m)

By plugging in the respective values and performing the calculations, you will find the magnitudes of the magnetic fields for both cases.