---
title: "A long solenoid of length \\(L_0\\) has a total of \\(N\\) turns non-uniformly distributed along its length such that the turn density is given by \\(n(x) = c x\\) for \\(0 \\le x \\le L_0\\), where \\(c\\) is a constant and \\(x\\) is measured from one end. The solenoid carries a constant current \\(I\\). The solenoid is then uniformly compressed along its length to a new length of \\(\\dfrac{L_0}{2}\\) while maintaining the same total current \\(I\\). Which of the following correctly gives the magnetic field magnitude \\(B\\) at the midpoint of the solenoid before compression and the factor by which the magnetic field magnitude at the midpoint changes after compression?"
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url: "https://nerd-notes.com/ubq/118557/"
date_modified: "2026-08-04T08:11:21+00:00"
---

# A long solenoid of length \(L_0\) has a total of \(N\) turns non-uniformly distributed along its length such that the turn density is given by \(n(x) = c x\) for \(0 \le x \le L_0\), where \(c\) is a constant and \(x\) is measured from one end. The solenoid carries a constant current \(I\). The solenoid is then uniformly compressed along its length to a new length of \(\dfrac{L_0}{2}\) while maintaining the same total current \(I\). Which of the following correctly gives the magnetic field magnitude \(B\) at the midpoint of the solenoid before compression and the factor by which the magnetic field magnitude at the midpoint changes after compression?

A long solenoid of length \(L_0\) has a total of \(N\) turns non-uniformly distributed along its length such that the turn density is given by \(n(x) = c x\) for \(0 \le x \le L_0\), where \(c\) is a constant and \(x\) is measured from one end. The solenoid carries a constant current \(I\). The solenoid is then uniformly compressed along its length to a new length of \(\dfrac{L_0}{2}\) while maintaining the same total current \(I\). Which of the following correctly gives the magnetic field magnitude \(B\) at the midpoint of the solenoid before compression and the factor by which the magnetic field magnitude at the midpoint changes after compression?

![A horizontal solenoid of length L_0 centered along a horizontal dashed axis extending from x = 0 at the left end to x = L_0 at the right end. The coil loops are shown in gray, spaced widely near x = 0 and becoming progressively more closely spaced toward x = L_0. A label x = 0 is below the left end, x = L_0/2 is below the midpoint, and x = L_0 is below the right end. A small arrow labeled I indicates current entering the coil. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785831081-eOX8sB.jpg)

- **A.** Midpoint field \(B = \dfrac{\mu_0 N I}{L_0}\) ; Factor change: \(2\)
- **B.** Midpoint field \(B = \dfrac{\mu_0 N I}{L_0}\) ; Factor change: \(4\)
- **C.** Midpoint field \(B = \dfrac{\mu_0 N I}{2L_0}\) ; Factor change: \(2\)
- **D.** Midpoint field \(B = \dfrac{\mu_0 N I}{2L_0}\) ; Factor change: \(\dfrac{1}{2}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118557/*