1. Given data

• Current, $I = 1.5\,\text{A}$
• Side of equilateral triangle, $a = 9\,\text{cm} = 0.09\,\text{m}$
• We need magnetic field at the centroid.

For an equilateral triangle, the centroid, incentre and circumcentre coincide. So the perpendicular distance from centroid to each side is the inradius:

$r = \frac{a\sqrt{3}}{6}$

Thus,

$r = \frac{0.09\sqrt{3}}{6} = 0.015\sqrt{3}\,\text{m}$

2. Magnetic field due to one side of the triangle

For a finite straight wire, magnetic field at a point at perpendicular distance $r$ is

$B = \frac{\mu_0 I}{4\pi r}(\sin\theta_1 + \sin\theta_2)$

At the centroid of an equilateral triangle, for each side the point lies symmetrically, so

$\theta_1 = \theta_2 = \theta$

From geometry,

• Half side $= a/2$
• Perpendicular distance $= r = a\sqrt{3}/6$

So,

$\tan\theta = \frac{a/2}{r} = \frac{a/2}{a\sqrt{3}/6} = \sqrt{3}$

Hence,

$\theta = 60^\circ$

Therefore field due to one side:

$B_1 = \frac{\mu_0 I}{4\pi r}(\sin60^\circ + \sin60^\circ)$

$B_1 = \frac{\mu_0 I}{4\pi r}(2\cdot \frac{\sqrt{3}}{2}) = \frac{\mu_0 I\sqrt{3}}{4\pi r}$

Now substitute $r = a\sqrt{3}/6$:

$B_1 = \frac{\mu_0 I\sqrt{3}}{4\pi \left(a\sqrt{3}/6\right)} = \frac{6\mu_0 I}{4\pi a} = \frac{3\mu_0 I}{2\pi a}$

3. Total magnetic field due to all three sides

All three sides produce magnetic field at the centroid in the same direction, so

$B = 3B_1 = 3\cdot \frac{3\mu_0 I}{2\pi a} = \frac{9\mu_0 I}{2\pi a}$

Using $\mu_0 = 4\pi \times 10^{-7}\,\text{T m/A}$,

$B = \frac{9(4\pi \times 10^{-7})(1.5)}{2\pi(0.09)}$

Cancel $\pi$:

$B = \frac{9\cdot 4 \cdot 1.5 \times 10^{-7}}{2\cdot 0.09}$

$B = \frac{54\times 10^{-7}}{0.18}$

$B = 300 \times 10^{-7} = 3\times 10^{-5}\,\text{T}$

4. Direction of magnetic field

The current is flowing clockwise in the plane of the triangle. By the right-hand rule, magnetic field at the centroid is into the plane.

So the direction is inside the plane of the triangle.

5. Matching with options

We found:

$B = 3\times 10^{-5}\,\text{T}$

Direction: inside the plane

This matches:

• Option D: $3 \times 10^{-5}\,\text{T}$, inside the plane of triangle

6. Comparison with stored correct answer

Stored correct answer: D

My derived answer: D

They agree.