Use circulation when a scalar potential is insufficient

An electric field without a battery

A changing magnetic flux produces an electric field even if no conducting loop is present. A hypothetical loop is a mathematical path, not necessarily a wire. For a stationary path, Faraday’s law is integral E dot dl = −dPhiB/dt. This induced field can do work on a charge around a closed path, unlike a purely electrostatic field.

Use circular symmetry, not an arbitrary local shortcut

In the supplied ideal long cylindrical region, Bz changes uniformly inside radius R and is negligible outside. Symmetry makes the induced electric field azimuthal and constant on a centered circular path. Inside, 2pi r Ephi = −pi r² Bdot, hence Ephi = −r Bdot/2. Outside, the changing flux area stops growing at pi R², hence Ephi = −R² Bdot/(2r). The field is continuous at R and need not vanish outside the magnetic-field region.

Interpret direction and potential carefully

Positive Bdot along +z produces clockwise induced electric circulation, or negative Ephi in the counterclockwise reference. An electrostatic surface-charge field can coexist with this induced field. However, a nonzero integral around a closed path cannot be represented solely as −grad V for one globally single-valued scalar V. Lead paths therefore matter in time-varying flux measurements; this is not permission to assign inconsistent node potentials in an electrostatic model.

Connect the Maxwell equations without claiming an extra derivation

Faraday’s law links changing B to circulating E; the Ampère–Maxwell relation includes changing E as a source of B. Together with the two Gauss laws these form Maxwell’s equations. In vacuum their wave solutions propagate at c = 1/sqrt(mu0 epsilon0), approximately 3×10⁸ m/s with introductory constants. The current framework treats this connection conceptually; deriving the electromagnetic wave equation is not required here. Doubling only a hypothetical constitutive product mu epsilon would reduce the corresponding ideal wave speed by sqrt(2), not increase it.

Worked original case

For R = 1 m and Bdot = 1 T/s, Ephi is −0.25 V/m at r = 0.5 m, −0.5 V/m at the boundary and −0.25 V/m at r = 2 m. The outer circulation is 2pi(2)(−0.25) = −pi V, matching minus the full flux-change rate.

Investigation

Predict field direction under rate reversal, then compare inner and outer radii with equal field magnitude. Derive both branches and verify continuity and circulation. Write a closed-path contradiction for a purely electrostatic scalar-potential explanation and a separate conceptual Maxwell-wave prediction.

Changed-case practice

Sources and boundaries

Original teaching checked 31 October 2026 against current official Unit 13 and the relevant OpenStax discussion. No source exercises or diagrams are reproduced. All records and apparatus are virtual, supplied or generated; no real wiring, switching, magnetic braking or electrical measurement is requested. Public feedback and retained work do not establish independent mastery, supervised practical participation or AP authorization.