In order to find the magnetic field at the centroid of the equilateral triangle, we can use the Biot-Savart Law. The Biot-Savart Law states that the magnetic field at a point due to a current-carrying wire is directly proportional to the current and inversely proportional to the distance from the wire.

In this case, we have three sides of the triangle carrying a current of 1 ampere each. The magnetic field at the centroid will be the vector sum of the magnetic fields due to each side.

Let's assume that the triangle lies in the xy-plane, with its centroid at the origin (0, 0). We can also assume that the current flows counterclockwise in each side of the triangle.

The formula for the magnetic field due to a straight wire is given by:

B = (μ₀/4π) * (I * dl × r) / r³

Where:
- B is the magnetic field vector
- μ₀ is the permeability of free space (4π x 10⁻⁷ T m/A)
- I is the current in the wire
- dl is an infinitesimal length element in the wire
- r is the position vector from the element dl to the point where we want to calculate the magnetic field

Since the triangle is equilateral, all sides have the same length of 4.5 x 10⁻² m. The centroid of an equilateral triangle lies at a distance of 2/3 times the height of the triangle from the base.

The height of the equilateral triangle can be found using the Pythagorean theorem:

h = √(a² - (a/2)²) = √(4.5 x 10⁻² m)² - (4.5 x 10⁻² m/2)²) = √(4.5 x 10⁻² m)² - (4.5 x 10⁻² m/2)²) ≈ 3.897 x 10⁻² m

The distance from the centroid to each side can be found using the ratio of the side length to the height:

d = (2/3) * h ≈ 2.598 x 10⁻² m

Now we can calculate the magnetic field due to each side of the triangle at the centroid.

B₁ = (μ₀/4π) * (I * dl × r) / r³
B₁ = (μ₀/4π) * (1 A * dl × r) / r³

Since the current is flowing counterclockwise, the dl vector is in the positive z-direction (out of the xy-plane). The position vector r will have components in the x and y-directions.

Let's consider one side of the triangle. We can parameterize the line segment with a variable t that goes from 0 to the side length of the triangle.

r(t) = (4.5 x 10⁻² m) * t * i + (4.5 x 10⁻² m) * t * √3/2 * j + 0 k
dl = (4.5 x 10⁻² m) * dt * k

Substituting these values into the formula for B₁:

B₁ = (μ₀/4π) *