Vortex Electric Field

A changing magnetic field creates an electric field of a special kind — a vortex field. Its lines do not start on charges: they close into rings around the region where B changes. Change the rate dB/dt, move the probe loop and watch the E-rings turn around the moment B starts to fall.

What is what on the scene

Blue arrows and the cylinder — the magnetic field B: it exists only inside a cylinder of radius R = 10 cm and points up or down. The arrow length is proportional to |B|, and when B = 0 the arrows vanish.
Orange rings with running lights — the lines of the vortex field E: this is an electric field without charges: the lines are closed and wrap the region where B changes. The lights run where the field pushes a positive charge; the brighter and faster they are, the stronger E is.
Red arrows — the vector E at a point: an arrow touches the ring, its length is proportional to |E|, and its direction flips together with the sign of dB/dt.
Green ring with “+” — your probe loop: a wire loop whose radius r is set by the slider. The field E drives positive charges around it; the work of the field over one lap is the EMF ℰ.
Glow on the floor — a map of the strength |E|: brighter where the field is stronger: a dark centre, a maximum at the blue boundary R and a gradual fade outwards.

dB/dt — rate of change of B 0,34 T/s

How B changes

B — induction (move it yourself) 0,000 T

r — radius of the probe loop 14,0 cm (1,40 R)

Playback speed (slow motion) 1,00×

Show

Magnetic field B

Vectors E

Probe loop

|E| map on the floor

Auto-rotate camera

The scale is real: R = 10 cm, B ≤ 0.2 T, so E is measured in millivolts per metre. The model is a long cylinder: the field is the same at every height.

The idea in 30 seconds

B changes — E is born

An electric field does not only belong to charges. If a magnetic field changes (grows or falls), a vortex electric field appears around it — even in empty space, without a single charge. While B stays constant, however strong, E = 0: what matters is not the size of B but how fast it changes, dB/dt.

The lines of E are closed rings

The lines of a charge's field start on “+” and end on “−”. A vortex field has no beginning: its lines circle the region where B changes. The direction comes from Lenz's law: B grows — E runs clockwise (seen from above), B falls — counter-clockwise. B turns around — the whole ring turns with it.

There is no B outside, yet there is E

Inside the cylinder E grows from zero on the axis to a maximum at the boundary: E = r/2·|dB/dt|. Beyond it there is no magnetic field, but E does not vanish — it falls as 1/r: E = R²/(2r)·|dB/dt|. Any ring around the cylinder “feels” the change of flux inside — this is how a transformer works.

Try it yourself

Press “dB/dt ×2”: the field E becomes twice as strong — the rings run faster, the red arrows grow longer and the peak on the |E|(r) graph doubles.
Press “dB/dt = 0”: B stays, but the vortex field disappears — the rings freeze and go dark. A constant magnetic field creates no vortex E.
Watch the moment the blue arrows turn around: exactly then E changes direction (a hint pops up on the scene). Switch on “Sinusoidal” — when B is at its maximum, E = 0 (there is B, but no E!).
Move the probe loop inside (R/2), onto the boundary (R) and outside (2R): the EMF grows as r² while the loop is inside, and stops growing once it encloses the whole field. The strength E on the wire itself is greatest at the boundary.
Choose “By hand” and drag the B slider: fast — E is strong, slow — weak, stop — E is gone. This is how a hand pushing a magnet into a coil works.

Graphs: how E depends on place and time

Left — the strength of the field along the radius: zero on the axis, a peak at the cylinder boundary, then a 1/r fall-off. Middle — B(t) and its rate of change dB/dt(t): E follows the second curve, not the first. Right — E at three points: they all have the same sign and flip together, because they all react to the same change of flux.

|E| at different distances r from the axis (the green dot is your loop)

B(t) and its rate of change dB/dt(t)

E at three points over time

Where it works in real life

TransformerThe alternating current in the primary winding keeps changing B in the core. The vortex E drives a current around the secondary winding — with no wires between them.

Induction hobA coil under the glass creates a rapidly changing B. The vortex E drives the electrons in the pan bottom round and round — the metal heats itself while the glass stays cool.

BetatronAn electron accelerator in which the vortex electric field itself speeds the particle up while the magnetic field grows — exactly like the red arrows on the scene.

Wireless chargerA coil in the charger creates an alternating B, and in the phone's coil the vortex E induces an EMF that charges the battery.

Metal detectorThe coil's alternating B induces eddy currents in a buried coin, and the detector picks up their own field.

What to explore next

Magnet and coilThe same Faraday's law in a real experiment: a magnet passes through a coil and the galvanometer shows the current. Full circuit with an EMFHow an EMF drives a current around a closed circuit: here you saw the field itself, there — its result. Electromagnetic waveA changing B creates E, and a changing E creates B again: this is how light is born.

KEY FORMULAS

Faraday's law: circulation of the vortex field

$$\oint \vec{E}\, d\vec{l} = -\frac{d\Phi_B}{dt}$$

The work of E along a closed loop equals the induced EMF and is not zero — the field is not conservative

▶ Detailed Explanation

The left-hand side is the work of the electric field when a unit positive charge is carried once around the loop; the right-hand side is the rate of change of the magnetic flux through it. The minus sign is Lenz's law: the induced field opposes the change of flux. For a loop of radius R = 0.1 m and dB/dt = 1 T/s the circulation is πR²·dB/dt ≈ 31 mV. This is a non-zero work along a closed path — exactly what an electrostatic field can never produce, since its lines start and end on charges and ∮E·dl = 0 always.

Field strength inside the magnetic field region

$$E = \frac{r}{2}\left|\frac{dB}{dt}\right|, \qquad r < R$$

Linear growth from zero on the axis to a maximum at the boundary r = R

▶ Detailed Explanation

For a loop of radius r inside the cylinder the magnetic flux is Φ = B·πr², hence E·2πr = πr²·|dB/dt| and E = (r/2)·|dB/dt|. On the axis (r = 0) there is no vortex field at all; at the boundary r = R it is maximal: with R = 0.1 m and dB/dt = 1 T/s we get E = 50 mV/m. Note that the strength is set by how fast B changes, not by the value of the field itself — for a constant B (dB/dt = 0) it is zero even though the magnetic field is still there.

Field strength outside the magnetic field region

$$E = \frac{R^{2}}{2r}\left|\frac{dB}{dt}\right|, \qquad r > R$$

The flux through the loop stops growing — the field falls off as 1/r

▶ Detailed Explanation

Outside the cylinder the changing magnetic induction lies entirely inside the loop: Φ = B·πR² does not depend on the radius of the circle, so E·2πr = πR²·|dB/dt| and E = (R²/2r)·|dB/dt|. For R = 0.1 m and dB/dt = 1 T/s, at r = 2R = 0.2 m only 25 mV/m remains, at r = 10R just 5 mV/m. That is why the glow on the floor fades towards the edge, and why the |E|(r) curve drops as a hyperbola after the peak at the boundary.

Induced EMF of a loop enclosing the whole field

$$\mathcal{E} = -\pi R^{2}\,\frac{dB}{dt}, \qquad r \ge R$$

Outside the cylinder the EMF is the same for every loop: their flux no longer depends on the radius

▶ Detailed Explanation

As soon as the loop encloses the whole cylinder, making its radius bigger is pointless: the extra area lies outside B, and the flux stays Φ = B·πR². So ℰ = −πR²·dB/dt: with R = 0.1 m and dB/dt = 1 T/s we get ℰ ≈ −31 mV, and the value does not change whether the loop radius is 2R or 10R. This is exactly what transformer windings and induction sensors rely on: what matters is not the size of the loop but how fast the flux inside it changes.

Maxwell's equation: the curl of E and the change of B

$$\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}$$

The differential form of Faraday's law: the curl of E equals minus the rate of change of B

▶ Detailed Explanation

Faraday's integral law and this local equality are the same equation written differently. A non-zero curl of E means the field lines are rolled into rings and the field cannot be conservative: ∮E·dl ≠ 0. Notice the link: the curl of E is produced by the change of B itself. Let dB/dt vanish — the curl becomes zero, and the electric field either disappears altogether or stays Coulomb-like, with ∇×E = 0 and field lines starting on charges.

Force on a charge and the EMF as the work of the field

$$\vec F = q\vec E, \qquad \mathcal{E} = \frac{A}{q} = \oint \vec E\, d\vec l$$

The vortex field pushes a charge like any electric field; the work per unit charge over one lap is the EMF

▶ Detailed Explanation

The vortex field acts on a charge q with the ordinary force F = qE: a positive charge is pushed along E, a negative one against it. That is why the free electrons in a wire ring start to move and an induced current appears. The work of the field on a charge over one full lap of the ring, divided by that charge, is the EMF: ℰ = A/q = ∮E·dl. On the scene the green “+” signs run around your loop precisely because the force qE acts on them; the longer the red arrows, the stronger this “push”.

A vortex (induced) electric field appears whenever the magnetic field changes in time. Faraday's law in integral form states that the circulation of the electric field around a closed loop equals minus the rate of change of the magnetic flux through that loop: ∮E·dl = −dΦ/dt. The lines of this field neither start nor end on charges — they are closed rings around the region where B changes. That is why the field has no potential: the work of E along a closed path is not zero, and the strength at a point is set not by the distance to a charge but by how fast the magnetic field changes.

The sense of the rings follows Lenz's law: the induced field opposes the change of flux. If B points up and grows, then seen from above E runs clockwise; the moment B starts to fall, every ring flips the other way. Hence the sign of E on the graph follows the sign of dB/dt, and the direction row in the measurements panel changes exactly when dB/dt crosses zero. For a sinusoidal B this happens at the peaks of B itself — at those instants the magnetic field is greatest while the vortex electric field is zero.

Inside the cylinder (r < R) Faraday's law for a circle of radius r gives E·2πr = πr²·|dB/dt|, that is E = (r/2)·|dB/dt|: the strength grows linearly from zero on the axis to its maximum at the boundary. Outside (r > R) the flux through the loop no longer grows — all the coupling with B is locked inside the cylinder — so E = (R²/2r)·|dB/dt| falls off as 1/r. For R = 0.1 m and dB/dt = 1 T/s the maximum is 50 mV/m, and the EMF of any loop enclosing the cylinder is about 31 mV.

In the 3D model the blue arrows are B inside the transparent cylinder, the orange rings with running lights are the closed lines of E, the red arrows are the field strength vector, and the green ring with “+” is the probe loop that you move with the slider. The glow on the floor shows how the strength of the field depends on the distance from the axis. The dB/dt slider sets how fast B changes: at zero the field E disappears, even though the magnetic field itself is still there.

Look at the graphs: on the left the profile |E|(r) peaking at the boundary r = R, in the middle B(t) and dB/dt(t), on the right E at three points — inside (R/2), on your loop and outside (2R); their signs flip together, because every point reacts to the same change of flux. Compare it with the magnet and coil simulator, where the same law works in a real device, and with the full circuit simulator, where the induced EMF drives the current around a circuit; here you see the very field that produces that EMF.