Question

A regular polygon of 6 sides is formed by bending a wire of length 4 \(\pi\) meter. If an electric current of \(4 \pi \sqrt3 \) A is flowing through the sides of the polygon, the magnetic field at the centre of the polygon would be \(x × 10^{–7}\) T. The value of \(x\) is ____.

Answer 1

Correct Answer: 72

This problem requires calculating the magnetic field at the center of a regular hexagonal wire loop, given the total wire length and the current.

Concept Used:

The magnetic field at a point due to a finite straight current-carrying wire is determined by the Biot-Savart law:

\[B = \frac{\mu_0 I}{4\pi d} (\sin\theta_1 + \sin\theta_2)\]

Here, \( \mu_0 \) is the permeability of free space (\( 4\pi \times 10^{-7} \) T·m/A), \( I \) is the current, \( d \) is the perpendicular distance from the point to the wire, and \( \theta_1 \) and \( \theta_2 \) are the angles subtended by the wire's ends at that point.

For a regular n-sided polygon, the total magnetic field at the center is the sum of the fields from each side. Due to symmetry, the field from each side is identical. Thus, the total field is:

\[B_{\text{center}} = n \times B_{\text{one side}}\]

Step-by-Step Solution:

Step 1: Identify given information and calculate the hexagon's side length.

Number of sides, \( n = 6 \).

Total wire length, \( L_{\text{total}} = 4\pi \) meters.

Current, \( I = 4\pi\sqrt{3} \) A.

The length of one side of the regular hexagon, \( a \), is:

\[a = \frac{L_{\text{total}}}{n} = \frac{4\pi}{6} = \frac{2\pi}{3} \text{ meters}\]

Step 2: Determine geometric parameters for magnetic field calculation.

A regular hexagon comprises 6 equilateral triangles meeting at the center. Each side subtends \( \frac{360^\circ}{6} = 60^\circ \) at the center. The angles \( \theta_1 \) and \( \theta_2 \) for the formula are half of this central angle.

\[\theta_1 = \theta_2 = \frac{60^\circ}{2} = 30^\circ\]

The perpendicular distance, \( d \), from the center to a side is the altitude of an equilateral triangle.

\[d = \frac{a\sqrt{3}}{2} = \left(\frac{2\pi}{3}\right) \frac{\sqrt{3}}{2} = \frac{\pi\sqrt{3}}{3} = \frac{\pi}{\sqrt{3}} \text{ meters}\]

Step 3: Calculate the magnetic field at the center due to one hexagon side.

Using the formula for a finite wire:

\[B_{\text{one side}} = \frac{\mu_0 I}{4\pi d} (\sin\theta_1 + \sin\theta_2)\]

Substituting values:

\[B_{\text{one side}} = \frac{(4\pi \times 10^{-7}) (4\pi\sqrt{3})}{4\pi (\frac{\pi}{\sqrt{3}})} (\sin 30^\circ + \sin 30^\circ)\]

Simplifying:

\[B_{\text{one side}} = \frac{10^{-7} \times 4\pi\sqrt{3}}{\frac{\pi}{\sqrt{3}}} \left(\frac{1}{2} + \frac{1}{2}\right)\]\[B_{\text{one side}} = 10^{-7} \times \frac{4\pi\sqrt{3} \times \sqrt{3}}{\pi} \times (1)\]\[B_{\text{one side}} = 10^{-7} \times 4 \times 3 = 12 \times 10^{-7} \text{ T}\]

Step 4: Calculate the total magnetic field at the hexagon's center.

The total magnetic field is the sum of the fields from all 6 sides. Since all fields are in the same direction (perpendicular to the polygon's plane), we multiply the field from one side by 6.

\[B_{\text{center}} = n \times B_{\text{one side}} = 6 \times (12 \times 10^{-7} \text{ T})\]\[B_{\text{center}} = 72 \times 10^{-7} \text{ T}\]

Step 5: Determine the value of \( x \).

The problem states the magnetic field is \( x \times 10^{-7} \) T. Comparing this to our result:

\[x \times 10^{-7} = 72 \times 10^{-7}\]

Therefore, \( x = \) 72.