Magnetic Flux

Both conventions appear in textbooks, and they differ by 90°. In the SI and most physics conventions, θ is the angle between the magnetic field B and the surface normal n (the vector perpendicular to the loop), and the flux is Φ = B·A·cos θ. In some engineering texts and older treatments, θ is the angle between B and the plane of the loop itself, in which case the flux is Φ = B·A·sin θ. The two expressions are mathematically identical if you substitute θ_plane = 90° − θ_normal. This calculator uses the normal convention, which is the standard in the SI system and in Faraday's law as usually written.

Yes. Flux is technically a signed quantity because it depends on the direction of the surface normal. Once you choose a positive normal direction, field lines that pass through the surface in the same direction as the normal count as positive flux; lines passing in the opposite direction count as negative flux. When a loop rotates past 90°, the flux smoothly passes through zero and then becomes negative — this is exactly what happens in a rotating generator coil and is why the induced EMF reverses direction once per half-turn. This calculator returns the magnitude, |Φ|; the sign depends on which face of the loop you define as positive.

B is the magnetic flux density — the actual magnetic field that exerts forces on moving charges and defines the flux through a surface. H is the magnetic field strength — an auxiliary field that describes the applied magnetising force, ignoring the response of the material. They are related by B = μ₀·(H + M) in SI, where M is the magnetisation of the material, or more simply B = μ·H in linear media, where μ = μ₀·μᵣ is the permeability. In vacuum B = μ₀·H, and the two differ only by a constant factor. In ferromagnetic materials such as iron, B can be hundreds or thousands of times larger than μ₀·H because the material itself contributes magnetisation. The flux Φ is always defined via B, never via H.

Faraday's law says the induced electromotive force around a loop equals the negative rate of change of magnetic flux through it: ε = −dΦ/dt for a single turn, or ε = −N·dΦ/dt for a coil of N turns. The minus sign is Lenz's law: the induced EMF always drives a current whose own magnetic field opposes the change in flux that produced it. This is the single most important equation in electromagnetism — it underlies every generator, transformer, motor, induction cooker, metal detector, wireless charger, and magnetic brake. The flux itself does not cause an EMF; it is the change in flux that does.

One weber is the flux that, when reduced to zero in one second, induces one volt in a single-turn loop — that is, 1 Wb = 1 V·s. This makes the weber equal to 1 T·m², which is a fairly large flux for everyday magnets. A typical neodymium magnet with a surface field of 0.5 T threading a 100 cm² loop produces only 5 mWb, and Earth's field through the same loop produces about 50 µWb. Only in large machines — transformers, MRI bores, industrial magnets — do you see flux values of a weber or more. That's why the calculator displays results in mWb, µWb, and maxwells for everyday magnitudes.

Flux linkage is the total flux passing through all turns of a coil, equal to N·Φ where N is the number of turns and Φ is the flux through a single turn. It is the quantity that appears in Faraday's law as ε = −d(NΦ)/dt. If a coil has 100 turns and each turn links 1 mWb of flux, the total flux linkage is 100 mWb-turns, and a change of 1 mWb per turn per second induces 100 × 10⁻³ = 0.1 V. Flux linkage is measured in weber-turns (equivalent to volt-seconds) and is often written λ instead of NΦ. The self-inductance of a coil is defined as L = NΦ / I, which connects flux linkage directly to the inductor's ability to store energy.

Gauss's law for magnetism states that the net magnetic flux through any closed surface is zero: ∮ B·dA = 0. The physical meaning is that magnetic field lines never begin or end — they always form closed loops, and for every field line that enters a closed surface another must leave it. This is why there are no magnetic monopoles. The flux relation Φ = B·A·cos θ applies to an open surface such as a loop; Gauss's law extends it to closed surfaces and gives the fundamental constraint. In magnetic circuit analysis, Gauss's law becomes Kirchhoff's flux law: the sum of fluxes into any node of a magnetic circuit is zero, exactly analogous to Kirchhoff's current law in electrical circuits.