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title: "An ideal solenoid with length \\(\\ell\\), cross-sectional area \\(A\\), and \\(n\\) turns per unit length has negligible internal resistance and is connected in series with an open switch, a resistor of resistance \\(R\\), and a battery of emf \\(\\mathcal{E}\\). A second solenoid has the same length and cross-sectional area, but has \\(2n\\) turns per unit length and negligible internal resistance. If the second solenoid replaces the first in the circuit and the switch is closed until steady state is reached in each case, what is the ratio of the initial maximum magnetic energy stored in the solenoid to the new maximum magnetic energy stored, \\(\\dfrac{U_1}{U_2}\\)?"
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url: "https://nerd-notes.com/ubq/125020/"
date_modified: "2026-09-28T14:13:17+00:00"
---

# An ideal solenoid with length \(\ell\), cross-sectional area \(A\), and \(n\) turns per unit length has negligible internal resistance and is connected in series with an open switch, a resistor of resistance \(R\), and a battery of emf \(\mathcal{E}\). A second solenoid has the same length and cross-sectional area, but has \(2n\) turns per unit length and negligible internal resistance. If the second solenoid replaces the first in the circuit and the switch is closed until steady state is reached in each case, what is the ratio of the initial maximum magnetic energy stored in the solenoid to the new maximum magnetic energy stored, \(\dfrac{U_1}{U_2}\)?

An ideal solenoid with length \(\ell\), cross-sectional area \(A\), and \(n\) turns per unit length has negligible internal resistance and is connected in series with an open switch, a resistor of resistance \(R\), and a battery of emf \(\mathcal{E}\). A second solenoid has the same length and cross-sectional area, but has \(2n\) turns per unit length and negligible internal resistance. If the second solenoid replaces the first in the circuit and the switch is closed until steady state is reached in each case, what is the ratio of the initial maximum magnetic energy stored in the solenoid to the new maximum magnetic energy stored, \(\dfrac{U_1}{U_2}\)?

![A rectangular single-loop circuit schematic oriented horizontally. On the left vertical branch is a battery symbol labeled \(\mathcal{E}\), with the longer horizontal line on top and the shorter line on the bottom. Along the top horizontal branch is an open single-pole switch labeled \(S\). Along the right vertical branch is a resistor symbol drawn as a zigzag line labeled \(R\). Along the bottom horizontal branch is an inductor symbol drawn as four adjacent semicircular loops labeled \(L\). Connecting wires form straight horizontal and vertical lines completing the rectangular loop. No other labels, lines, text, or components appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604797-rWivQS.jpg)

- **A.** \(\dfrac{1}{4}\)
- **B.** \(\dfrac{1}{2}\)
- **C.** \(1\)
- **D.** \(4\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/125020/*