© All Rights reserved @ LearnWithDash
Step-by-Step Solution
Step 1: Determine the side length of the regular hexagon
The wire has a total length of 4\pi meters, which is bent into a regular hexagon (6 equal sides). Let s be the length of each side. Hence,
s = \frac{4\pi}{6} = \frac{2\pi}{3}.
Step 2: Find the perpendicular distance from the center to a side (apothem)
For a regular hexagon, the apothem (distance from the center to any side) is
d = \frac{\sqrt{3}}{2} \times s.
Substituting s = \frac{2\pi}{3} , we get
d = \frac{\sqrt{3}}{2} \times \frac{2\pi}{3}
= \frac{\pi \sqrt{3}}{3}.
Step 3: Magnetic field at the center due to one side
Consider a straight conductor of length s carrying current I , and we want the magnetic field at a point that is a distance d from the midpoint of the conductor on its perpendicular bisector. The formula for the magnetic field due to a finite wire is:
B_{\text{one side}} = \frac{\mu_0 I}{4\pi d}\,(\sin \alpha_1 + \sin \alpha_2),
where \alpha_1 and \alpha_2 are the angles subtended by each half of the wire at the point. For a regular hexagon, each side subtends 60^\circ at the center, so each half subtends 30^\circ . Hence,
\sin \alpha_1 = \sin 30^\circ = \frac{1}{2}, \quad
\sin \alpha_2 = \sin 30^\circ = \frac{1}{2}.
Therefore,
\sin \alpha_1 + \sin \alpha_2 = \frac{1}{2} + \frac{1}{2} = 1.
Thus,
B_{\text{one side}} = \frac{\mu_0 I}{4\pi d}.
Step 4: Substitute the values
Given:
\mu_0 = 4\pi \times 10^{-7}\,\mathrm{T\cdot m/A},\;
I = 4\pi \sqrt{3}\,\mathrm{A},\;
d = \frac{\pi \sqrt{3}}{3}.
First, compute \mu_0 I :
\mu_0 I
= \left(4\pi \times 10^{-7}\right)\,\left(4\pi \sqrt{3}\right)
= 16\,\pi^2 \sqrt{3}\times 10^{-7}.
The denominator 4\pi d is:
4\pi d = 4\pi \times \frac{\pi \sqrt{3}}{3}
= \frac{4\pi^2 \sqrt{3}}{3}.
Hence,
B_{\text{one side}}
= \frac{16\,\pi^2 \sqrt{3} \times 10^{-7}}
{\frac{4\,\pi^2 \sqrt{3}}{3}}
= \frac{16\,\pi^2 \sqrt{3}\,\times 10^{-7}\,\times 3}
{4\,\pi^2 \sqrt{3}}
= 12 \times 10^{-7}\,\mathrm{T}.
Step 5: Total magnetic field due to all six sides
Since the hexagon has 6 identical sides and each side contributes an equal magnetic field in the same direction at the center,
B_{\text{total}} = 6 \times B_{\text{one side}}
= 6 \times \left(12 \times 10^{-7}\right)
= 72 \times 10^{-7}\,\mathrm{T}.
Therefore, if B_{\text{total}} = x \times 10^{-7}\,\mathrm{T} , then x = 72.
Final Answer
The value of x is 72.
