Faraday's law as field circulation

To take a circulation, break the closed path into little vectors dl running along it in a chosen sense, dot each one with the field there, and add: ∮E⋅dl = ∮E dl cos φ, with φ the angle between E and the path. The units are V m⁻¹ × m = V, and the physical reading is the work done per unit charge on a test charge carried once round. For a field made by charges at rest that total is always zero. Take a uniform static field of 300 V m⁻¹ pointing along +x and walk a 0.20 m by 0.10 m rectangle: the two sides along x contribute +60 V and −60 V, the two sides along y contribute nothing because E⋅dl = 0 there, and the sum is zero. Circulation is a property of the whole loop, not of any point on it, and that is precisely why it can be set equal to something happening on a surface.

Faraday's law says ∮E⋅dl = −dΦ_B/dt, where Φ_B = ∫B⋅dA is taken over a surface whose rim is the loop. Two questions arise at once. Which surface? Any of them — a flat disc, a bowl, a long sock all return the same Φ_B, because the flux out of the closed surface formed by any two of them is zero. That is Gauss's law for magnetism doing quiet structural work here. And what are the units? A weber per second is a volt, so the two sides really are the same kind of quantity. A flat disc of radius 5.0 cm sitting square to a uniform field growing at 0.40 T s⁻¹ has dΦ_B/dt = π(0.050 m)²(0.40 T s⁻¹) = 3.1 × 10⁻³ Wb s⁻¹, so the circulation of E round its 31 cm rim is 3.1 mV. Notice what was never needed: no wire, no resistance, no current.

Both sides of the law are signed, and one convention locks the signs together. Choose a sense in which to traverse the loop; curl the right hand that way and the thumb gives the normal n̂ that orients the surface, fixing the sign of Φ_B. Reverse the traversal and both sides change sign at once, so the law is convention-free although neither side is. Work an example. A loop with B out of the page through it, and B growing: with n̂ out of the page, Φ_B > 0 and dΦ_B/dt > 0, so ∮E⋅dl < 0 in the anticlockwise sense — the induced E runs clockwise. A positive charge free to move goes clockwise too, and the current it makes produces a field into the page inside the loop, opposing the growth that caused it. Lenz's law is not an extra rule bolted on; it is this minus sign read aloud. A plus sign would let a loop amplify its own flux without limit, so energy conservation insists on it.

The integral is only useful when symmetry lets you do it, and the long solenoid is the case where it does. Far from the ends nothing changes under rotation about the axis or a slide along it, so the induced E lines must be circles about the axis, with E tangential and of one magnitude everywhere on a given circle. Then ∮E⋅dl collapses to E(2πr). Inside the windings the enclosed flux is πr²B, so E(2πr) = πr²(dB/dt) and E = (r/2)(dB/dt): zero on the axis, rising linearly outwards. Outside, the enclosed flux stops growing with r, because it is πR²B whatever the loop's radius, so E = (R²/2r)(dB/dt) and the field falls as 1/r. Take R = 4.0 cm and dB/dt = 0.50 T s⁻¹: E is 5.0 mV m⁻¹ at r = 2.0 cm, peaks at 10 mV m⁻¹ at the windings, and is still 4.0 mV m⁻¹ at r = 10 cm, where B itself is zero. That last number is the whole point of the topic.

∮E⋅dl ≠ 0 is fatal to the idea of potential. If the work per unit charge round a closed path is not zero, then ∫E⋅dl from P to Q depends on the route taken, and no function V can be attached to the points themselves. The consequence is testable. Put a ring round a solenoid whose flux changes fast enough to give 1.0 V of EMF, and build the ring from a 100 Ω arc and a 400 Ω arc joined at P and Q. The current is 1.0 V ÷ 500 Ω = 2.0 mA everywhere round it. An ideal voltmeter clipped to P and Q reads 0.20 V when its leads run along the 100 Ω side, and 0.80 V of the opposite sign when the same clips are kept and the leads are routed along the 400 Ω side. Neither reading is an error: the meter reports the circulation of E along whatever path its own loop takes. Kirchhoff's loop rule survives only if you enter −dΦ_B/dt explicitly as a source and keep changing flux out of every mesh you write down.

Three limits are worth naming. First, this is the law for a loop fixed in space. Let the loop move or deform and a second contribution appears, ∮(v × B)⋅dl, from the magnetic force on charges carried along with it; the familiar flux rule ℰ = −dΦ_B/dt bundles the two together, which is why a rod sliding on rails and a solenoid ramping up share a bookkeeping while differing in physics. Second, the law constrains only the circulation of E, never E itself: add any field made by static charges and every circulation is unchanged, so Gauss's law for electricity must be solved alongside it before E is pinned down. Third, with ∂B/∂t = 0 the whole statement collapses to ∮E⋅dl = 0, which is why a first course in electrostatics never needed it. What it does supply is half of a coupling — a changing B making a circulation of E — and the partner half, taken up next, is what closes the pair into a wave.