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title: "A stationary circular wire loop with fixed electrical resistance \\(R\\) is placed in a uniform magnetic field directed perpendicular to the plane of the loop. The magnitude of the magnetic field changes monotonically from an initial value \\(B_i\\) to a final value \\(B_f\\) over an elapsed time interval \\(\\Delta t\\). An experimenter observes that when the transition occurs over a very short time interval, the peak induced current is substantially greater than when the transition occurs over a long time interval, yet the total electric charge passing through a cross section of the wire is identical in both trials. Which of the following statements provides the physically correct explanation for why the total charge transferred is independent of \\(\\Delta t\\)?"
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url: "https://nerd-notes.com/ubq/124783/"
date_modified: "2026-09-28T14:11:15+00:00"
---

# A stationary circular wire loop with fixed electrical resistance \(R\) is placed in a uniform magnetic field directed perpendicular to the plane of the loop. The magnitude of the magnetic field changes monotonically from an initial value \(B_i\) to a final value \(B_f\) over an elapsed time interval \(\Delta t\). An experimenter observes that when the transition occurs over a very short time interval, the peak induced current is substantially greater than when the transition occurs over a long time interval, yet the total electric charge passing through a cross section of the wire is identical in both trials. Which of the following statements provides the physically correct explanation for why the total charge transferred is independent of \(\Delta t\)?

A stationary circular wire loop with fixed electrical resistance \(R\) is placed in a uniform magnetic field directed perpendicular to the plane of the loop. The magnitude of the magnetic field changes monotonically from an initial value \(B_i\) to a final value \(B_f\) over an elapsed time interval \(\Delta t\). An experimenter observes that when the transition occurs over a very short time interval, the peak induced current is substantially greater than when the transition occurs over a long time interval, yet the total electric charge passing through a cross section of the wire is identical in both trials. Which of the following statements provides the physically correct explanation for why the total charge transferred is independent of \(\Delta t\)?

![A single closed circular loop drawn with a solid line sits in the plane of the page. Near the top edge of the circular loop is the text label R. Across the interior of the loop are exactly six uniformly spaced symbols indicating a magnetic field directed perpendicular into the page, arranged in two horizontal rows of three. Above the loop is an arrow pointing into the page accompanied by the text label \vec{B}. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604674-22bHHX.jpg)

- **A.** The external work performed in changing the magnetic field is strictly conserved between the two trials, so the total electrical energy dissipated as Joule heating must be constant, directly fixing the total net charge that can pass through the loop.
- **B.** The magnetic force acting on the conduction electrons is always directed perpendicular to their instantaneous motion, so the magnetic field does no net work on the charge carriers, leaving their total integrated displacement determined only by the endpoints.
- **C.** The loop develops a self-induced back electromotive force that scales proportionally with the rate of field change, dynamically suppressing rapid currents so that the average current over any duration remains identical.
- **D.** The instantaneous current is proportional to the time derivative of the magnetic flux, so integrating the current over the elapsed duration mathematically reduces the charge to the net change in magnetic flux divided by the loop resistance.

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124783/*