A = length of side of equilateral triangle

r = perpendicular distance of each side from centroid = (√3) a/6

θ = angle by each end of each side at centroid = 60

Using Biot-savart's law , magnetic field at the centroid by each side is given as

B = (μ/4π) (i/r) (Sinθ + Sinθ)

B = (μ/4π) (i/ (√3) a/6) (Sin60 + Sin60)

B = (μ/4π) (6i/ (√3) a) (2) ((√3)/2)

B = (μ/4π) (6i/a)

total magnetic field by all three sides is given as

B'' = 3 B = 3 (μ/4π) (6i/a)

B'' = (μ/2π) (9i/a)

Introduction:
In this problem, we are given an equilateral triangle with a current flowing through it. We need to find the magnetic induction at the centroid of the triangle.

Given:
- Current flowing in the triangle = i ampere
- Side length of the equilateral triangle = a

Solution:

Step 1: Find the magnetic field due to one side of the triangle:
To find the magnetic induction at the centroid, we need to calculate the magnetic field due to each side of the triangle at the centroid.

- Consider one side of the equilateral triangle. The side length is 'a'.
- The magnetic field due to a current-carrying wire can be calculated using Ampere's Law: B = (μ₀ * i) / (2πd), where B is the magnetic field, μ₀ is the permeability of free space, i is the current, and d is the distance from the wire.

- The distance from the centroid to one side of the triangle is (a/2√3).
- Substituting the values in the formula, we get B = (μ₀ * i) / (2π * a/2√3)

Step 2: Find the magnetic field due to all three sides:
Since the magnetic field is a vector quantity, we need to consider the vector sum of the magnetic fields due to each side of the triangle.

- As the triangle is equilateral, the magnetic fields due to each side will have the same magnitude but different directions.
- Two magnetic fields will cancel each other out, and the resultant magnetic field will be along the median of the triangle.

Step 3: Find the magnetic field at the centroid:
- The centroid of an equilateral triangle is the point of intersection of its medians.
- The magnetic field at the centroid will be the vector sum of the magnetic fields due to each side of the triangle.
- Since two magnetic fields cancel each other out, the resultant magnetic field will be along the median.

Conclusion:
The magnetic induction at the centroid of the equilateral triangle carrying a current of i ampere can be found by calculating the magnetic field due to each side of the triangle and taking the vector sum. The resultant magnetic field will be along the median of the triangle.