---
title: "A thin wire carrying a steady current \\(I\\) is bent into the shape of a regular planar polygon with \\(n\\) sides inscribed in a circle of radius \\(R\\). The magnitude of the magnetic field at the center of the polygon is given by the expression  \\[ B(n) = \\dfrac{\\mu_0 n I}{2\\pi R} \\tan\\left(\\dfrac{\\pi}{n}\\right) \\]  Which of the following statements correctly describes the behavior of \\(B(n)\\) in the limit as \\(n \\to \\infty\\), and provides the correct physical justification?"
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url: "https://nerd-notes.com/ubq/124803/"
date_modified: "2026-09-28T14:11:22+00:00"
---

# A thin wire carrying a steady current \(I\) is bent into the shape of a regular planar polygon with \(n\) sides inscribed in a circle of radius \(R\). The magnitude of the magnetic field at the center of the polygon is given by the expression

\[ B(n) = \dfrac{\mu_0 n I}{2\pi R} \tan\left(\dfrac{\pi}{n}\right) \]

Which of the following statements correctly describes the behavior of \(B(n)\) in the limit as \(n \to \infty\), and provides the correct physical justification?

A thin wire carrying a steady current \(I\) is bent into the shape of a regular planar polygon with \(n\) sides inscribed in a circle of radius \(R\). The magnitude of the magnetic field at the center of the polygon is given by the expression

\[ B(n) = \dfrac{\mu_0 n I}{2\pi R} \tan\left(\dfrac{\pi}{n}\right) \]

Which of the following statements correctly describes the behavior of \(B(n)\) in the limit as \(n \to \infty\), and provides the correct physical justification?

![A circle drawn with a thin dashed black line centered at a central point dot. Inscribed inside the circle is a regular hexagon drawn with a solid black line. On the top horizontal segment of the hexagon, an arrow points rightward and is labeled \(I\). From the central dot, a straight dashed radial line extends upward and to the right at an angle of 30 degrees to meet one of the vertices of the hexagon on the dashed circle, labeled \(R\). At the center of the hexagon, the central dot is labeled \(C\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604681-GWrbAi.jpg)

- **A.** \(B(n)\) increases monotonically toward \(\dfrac{\mu_0 I}{2R}\), because the total perimeter of the inscribed polygon increases toward the circumference of the circle.
- **B.** \(B(n)\) decreases monotonically toward \(\dfrac{\mu_0 I}{2R}\), because the perpendicular distance from the center to each wire segment increases toward \(R\).
- **C.** \(B(n)\) increases monotonically toward \(\dfrac{\mu_0 I}{2\pi R}\), because the polygon approaches a circle but retains the geometric factor characteristic of straight wire segments.
- **D.** \(B(n)\) decreases monotonically toward \(\dfrac{\mu_0 I}{2\pi R}\), because the individual contribution of each straight segment diminishes as its length approaches zero.

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124803/*