---
title: "A long, ideal cylindrical solenoid of radius \\(R\\) has \\(n\\) turns per unit length and carries a time-dependent current \\(I(t)\\) that decreases at a constant rate \\(\\left|\\dfrac{dI}{dt}\\right| = C\\). Points \\(P_1\\) and \\(P_2\\) lie in a plane perpendicular to the solenoid’s central axis at radial distances \\(r_1 = \\dfrac{R}{2}\\) and \\(r_2 = 2R\\) from the center, respectively. Which of the following correctly compares the magnitudes of the induced electric fields \\(E_1\\) at \\(P_1\\) and \\(E_2\\) at \\(P_2\\), and provides the correct physical justification?"
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url: "https://nerd-notes.com/ubq/125036/"
date_modified: "2026-09-28T14:13:28+00:00"
---

# A long, ideal cylindrical solenoid of radius \(R\) has \(n\) turns per unit length and carries a time-dependent current \(I(t)\) that decreases at a constant rate \(\left|\dfrac{dI}{dt}\right| = C\). Points \(P_1\) and \(P_2\) lie in a plane perpendicular to the solenoid’s central axis at radial distances \(r_1 = \dfrac{R}{2}\) and \(r_2 = 2R\) from the center, respectively. Which of the following correctly compares the magnitudes of the induced electric fields \(E_1\) at \(P_1\) and \(E_2\) at \(P_2\), and provides the correct physical justification?

A long, ideal cylindrical solenoid of radius \(R\) has \(n\) turns per unit length and carries a time-dependent current \(I(t)\) that decreases at a constant rate \(\left|\dfrac{dI}{dt}\right| = C\). Points \(P_1\) and \(P_2\) lie in a plane perpendicular to the solenoid's central axis at radial distances \(r_1 = \dfrac{R}{2}\) and \(r_2 = 2R\) from the center, respectively. Which of the following correctly compares the magnitudes of the induced electric fields \(E_1\) at \(P_1\) and \(E_2\) at \(P_2\), and provides the correct physical justification?

![A circular cross section of a solenoid with radius R centered at the origin, drawn with a solid circle. The interior of the circle contains a uniform pattern of exactly six dots indicating magnetic field directed perpendicular out of the page. A horizontal dashed reference line extends from the center origin to the right. Along this dashed line, point P_1 is marked with a solid dot at distance R/2 from the center, and point P_2 is marked with a solid dot at distance 2R from the center. Two concentric dashed circles centered at the origin pass through P_1 and P_2, representing circular paths of radius R/2 and 2R, respectively. A single radial dimension arrow extends from the center to the edge of the solid circle, labeled R. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/solenoid-cross-section-1790604807-O0YOtq.jpg)

- **A.** \(E_2 < E_1\) because the magnetic field outside an ideal solenoid is zero, resulting in zero induced electric field at point \(P_2\).
- **B.** \(E_2 > E_1\) because the circular path through \(P_2\) encloses four times the rate of change of magnetic flux as the path through \(P_1\).
- **C.** \(E_2 = E_1\) because the path through \(P_2\) encloses four times as much changing magnetic flux as the path through \(P_1\), which is exactly offset by its circumference being four times as large.
- **D.** \(E_2 < E_1\) because the induced electric field decreases inversely with radial distance outside the solenoid (\(E \propto \dfrac{1}{r}\)), making the field weaker at \(r = 2R\) than at any interior point.

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/125036/*