Induced Electric Field
Changing B induces E circulating around it. E ∝ r inside, ∝ 1/r outside.
Key Notes
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).
Formulas
Maxwell–Faraday (integral)
Always holds — a wire is not required.
Inside cylindrical region (r ≤ R)
Linear rise to peak at r = R.
Outside (r ≥ R)
Inverse-r falloff.
Field at r = R (matching)
Common formula; same value from inside and outside expressions.
Important Points
Induced E exists in regions where B is changing — it does NOT require a conducting loop. The loop just samples it to make current.
Non-conservative: no potential V can describe it. Always quote ∮E·dl, not V_a − V_b, when dealing with induced fields.
Field lines of induced E are CLOSED loops (just like B around currents). They do not start or end on charges.
Induced E is what accelerates electrons in a betatron — a particle accelerator that uses changing B to do the accelerating, no scalar potential needed.
Even in vacuum (no charges, no currents) a changing B creates a real E — fundamental to EM-wave propagation.
Tricky sign: if B is into the page and increasing, induced E circulates COUNTER-CLOCKWISE viewed from the side B points to.
Induced Electric Field notes from sciphylab (also known as SciPhy, SciPhy Lab, SciPhy Labs, Physics Lab). Class 12 physics revision for JEE Mains, JEE Advanced, NEET UG, AP Physics 1/2/C, SAT, and CUET-UG.