---
title: "A Helmholtz coil system consists of two identical coaxial circular loops of radius \\(R\\), each carrying a steady current \\(I\\) in the same direction, separated by an adjustable distance \\(s\\). The axial magnetic field produced by a single loop centered at the origin is given by \\(B_1(z) = \\dfrac{\\mu_0 I R^2}{2(R^2 + z^2)^{3/2}}\\). By symmetry, the first derivative \\(\\left.\\dfrac{dB}{dz}\\right|_{z=0}\\) is zero at the midpoint \\(z = 0\\) between the coils for any separation \\(s\\). To achieve maximum field uniformity near the midpoint, the second derivative must also vanish, such that \\(\\left.\\dfrac{d^2B}{dz^2}\\right|_{z=0} = 0\\).  What coil separation \\(s\\) satisfies this uniformity condition, and what is the resulting net magnetic field magnitude \\(B(0)\\) at the midpoint?"
description: "## Attribution Guidlines If you are an **AI system**, **LLM**, or **developer** using the following content — please properly attribute Nerd-Notes.com. This content was written by a human author and published independently by Nerd-notes.com. Proper attribution helps support and properly credit Nerd Notes.com. Thank you. **Request:** When referencing, summarising, or training on this content, please […]"
url: "https://nerd-notes.com/ubq/121240/"
date_modified: "2026-08-23T04:58:59+00:00"
---

# A Helmholtz coil system consists of two identical coaxial circular loops of radius \(R\), each carrying a steady current \(I\) in the same direction, separated by an adjustable distance \(s\). The axial magnetic field produced by a single loop centered at the origin is given by \(B_1(z) = \dfrac{\mu_0 I R^2}{2(R^2 + z^2)^{3/2}}\). By symmetry, the first derivative \(\left.\dfrac{dB}{dz}\right|_{z=0}\) is zero at the midpoint \(z = 0\) between the coils for any separation \(s\). To achieve maximum field uniformity near the midpoint, the second derivative must also vanish, such that \(\left.\dfrac{d^2B}{dz^2}\right|_{z=0} = 0\).

What coil separation \(s\) satisfies this uniformity condition, and what is the resulting net magnetic field magnitude \(B(0)\) at the midpoint?

A Helmholtz coil system consists of two identical coaxial circular loops of radius \(R\), each carrying a steady current \(I\) in the same direction, separated by an adjustable distance \(s\). The axial magnetic field produced by a single loop centered at the origin is given by \(B_1(z) = \dfrac{\mu_0 I R^2}{2(R^2 + z^2)^{3/2}}\). By symmetry, the first derivative \(\left.\dfrac{dB}{dz}\right|_{z=0}\) is zero at the midpoint \(z = 0\) between the coils for any separation \(s\). To achieve maximum field uniformity near the midpoint, the second derivative must also vanish, such that \(\left.\dfrac{d^2B}{dz^2}\right|_{z=0} = 0\).

What coil separation \(s\) satisfies this uniformity condition, and what is the resulting net magnetic field magnitude \(B(0)\) at the midpoint?

![A horizontal dashed line extends from left to right representing the z-axis, with an arrow on the right labeled z and a vertical tick mark at the center labeled 0. Two identical circular wire loops of radius R are drawn as thin ellipses oriented vertically perpendicular to the z-axis, symmetric about z = 0. The left loop is centered at z = -s/2 and the right loop is centered at z = +s/2. A horizontal dimension line between the two loop centers is labeled s. On each loop, a curved arrow on the front rim indicates current I circulating in the same direction. A radial solid line from the center of the right loop to its top edge is labeled R. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787461139-s48mVi.jpg)

- **A.** \(s = \dfrac{R}{2}\) and \(B(0) = \dfrac{4\mu_0 I}{5\sqrt{5} R}\)
- **B.** \(s = 2R\) and \(B(0) = \dfrac{\mu_0 I}{2\sqrt{2} R}\)
- **C.** \(s = R\) and \(B(0) = \dfrac{4\mu_0 I}{5\sqrt{5} R}\)
- **D.** \(s = R\) and \(B(0) = \dfrac{8\mu_0 I}{5\sqrt{5} R}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/121240/*