Induced Electric Field

A changing magnetic field produces a non-conservative electric field, even where no charges or wires exist: ∮E·dl = −dΦ_B/dt.

For a cylindrically symmetric region of uniform B(t) of radius R: E inside follows E(r) = (r/2)·|dB/dt| (rises linearly with r).

Outside the region (r ≥ R): E(r) = (R²/(2r))·|dB/dt| — falls off as 1/r.

E is maximum at the boundary r = R and the field lines form closed loops AROUND the axis (just like B around a current — but without any current).

Unlike electrostatic E from charges, this induced E is NOT conservative: ∮E·dl ≠ 0 — there's no scalar potential V for it.

Direction: use the right-hand rule with B(t). If B is INTO the page and INCREASING, induced E circulates COUNTER-CLOCKWISE (so a positive charge would be pushed in the direction that opposes the flux increase).

Induced E is the mechanism behind betatrons (electron accelerators using changing B in a toroidal volume).