Question

A current $I$ flows through the sides of an equilateral triangle of side $a$. The magnetic field at the centroid is

• A. $\dfrac{9\mu_{0}I}{2\pi a}$
• B. $\dfrac{\mu_{0}I}{\pi a}$
• C. $\dfrac{3\mu_{0}I}{2\pi a}$
• D. $\dfrac{\mu_{0}I}{\pi a}$

Hint:

For symmetric current-carrying polygons, compute the field from one side and multiply by the number of sides.

Answer 1

The Correct Option is A

Step 1: Use magnetic field from a finite wire.
For each side of the triangle:
$B = \frac{\mu_0 I}{4\pi d}(\sin\theta_1 + \sin\theta_2)$.

Step 2: Geometry of equilateral triangle.
Distance from centroid to each side = $\frac{\sqrt{3}}{6}a$.
Angles are $60^\circ$ each: $\sin 60^\circ = \frac{\sqrt{3}}{2}$.

Step 3: Field of one side.
$B_{1} = \frac{\mu_0 I}{4\pi (\frac{\sqrt{3}}{6}a)} \left( \frac{\sqrt{3}}{2} + \frac{\sqrt{3}}{2} \right)$
$B_{1} = \frac{\mu_0 I}{4\pi a} \cdot 3$.

Step 4: Total field (three sides).
$B_{\text{total}} = 3\cdot \frac{\mu_0 I}{4\pi a} \cdot 3 = \frac{3\mu_0 I}{2\pi a}$.
Conclusion. The magnetic field at the centroid is $\dfrac{3\mu_{0}I}{2\pi a}$.