Magnetic Flux and Gauss's Law for Magnetism

Fig 1 — Field lines of a bar magnet form closed loops: every line leaving a closed surface S re-enters it, so the net magnetic flux is zero

Introduction to Magnetic Flux

In Part 2: Understanding Electric Flux and Gauss's Law we defined electric flux and found that the flux out of a closed surface measures the electric charge inside. The same mathematical tool can be applied to the magnetic field. Magnetic flux measures how much magnetic field passes through a surface — and it turns out to be one of the most practically important quantities in electrical engineering. It is magnetic flux, not the field itself, that a coil "sees": the voltage induced in a transformer winding, the inductance of a coil, and the signal picked up by an NFC antenna or H-field probe are all determined by the magnetic flux through the loop.

When we apply the flux idea to a closed surface, however, magnetism behaves completely differently from electricity. The net magnetic flux out of any closed surface is always zero. That simple result is Gauss's law for magnetism, the second of Maxwell's four equations, and it encodes a basic fact about nature: there are no isolated magnetic charges.

Understanding Magnetic Flux

Definition of Magnetic Flux

Magnetic flux Φ_B through a surface is the total amount of magnetic flux density 𝐁 passing through that surface. Just as for electric flux, only the component of the field perpendicular to the surface contributes. This is why 𝐁 is called the magnetic flux density: it is flux per unit area, measured in webers per square metre (tesla).

Mathematical Expression of Magnetic Flux

Magnetic flux — general and uniform-field forms

Φ_{B} = \int_{S} 𝐁 \cdot d 𝐀 Φ_{B} = B A \cos θ (uniform 𝐁, flat surface)

θ is the angle between 𝐁 and the surface normal n̂. Flux is maximum when the loop faces the field (θ = 0°) and zero when the field lies in the plane of the loop (θ = 90°).

Magnetic Flux Through Different Surfaces

Flat surface in a uniform field: Φ_B = BA cos θ. This is the usual model of a small loop antenna or pickup coil.
Curved surface or non-uniform field: the surface is divided into small patches and the contributions B cos θ dA are summed (integrated). The flux through a loop depends only on the loop's edge, not on which surface spanning that edge is chosen — a direct consequence of Gauss's law for magnetism, shown below.
A coil of N turns: each turn encloses the flux Φ_B, so the coil links a total flux linkage λ = NΦ_B (weber-turns). Flux linkage is what appears in Faraday's law and in the definition of inductance, L = λ/I.
Closed surface: the net flux is always zero, ∮𝐁·d𝐀 = 0.

Units of Magnetic Flux

Quantity	SI unit	Equivalent	CGS unit
Magnetic flux Φ_B	weber (Wb)	T·m² = V·s	maxwell (Mx); 1 Wb = 10⁸ Mx
Flux density B	tesla (T)	Wb/m² = V·s/m²	gauss (G); 1 T = 10⁴ G
Field intensity H	A/m	—	oersted (Oe); 1 A/m = 4π × 10⁻³ Oe
Flux linkage λ	Wb-turn	V·s	—

The equivalence 1 Wb = 1 V·s is not a coincidence. Faraday's law (Part 4) says that the voltage induced in a loop equals the rate of change of the flux through it, so flux is literally "volts × seconds". This is why transformer core flux is calculated from the applied voltage and frequency, as we do in Example 3.

Magnetic Field Lines and Their Properties

We introduced magnetic field lines in Part 1. With the concept of flux, their properties can be stated more precisely:

Magnetic field lines always form closed loops. They never begin or end at a point. This is the geometric content of Gauss's law for magnetism.
Direction: outside a magnet they run from the north pole to the south pole; inside the magnet they return from south to north. Around a current they circle according to the right-hand rule.
Density represents flux density: the number of lines crossing a unit area perpendicular to them is proportional to B. Flux Φ_B is proportional to the total number of lines through a surface.
They never intersect, since 𝐁 has a unique direction at each point.
They prefer high-permeability paths. A ferrite or iron core gathers the lines and guides them around a magnetic circuit, much as a copper wire guides current. This is why transformers and RF chokes use cores: more flux for the same current.
They are continuous across material boundaries. The normal component of 𝐁 is the same on both sides of an interface — a boundary condition derived in Part 9.

Introduction to Gauss's Law for Magnetism

Statement of Gauss's Law for Magnetism

📌 Gauss's law for magnetism

The net magnetic flux through any closed surface is always zero. Every magnetic field line that enters a closed surface must also leave it.

Mathematical Expression

Gauss's law for magnetism

\oint_{S} 𝐁 \cdot d 𝐀 = 0

Compare with Gauss's law for electricity, ∮𝐃·d𝐀 = Q_enc. The magnetic version has zero on the right-hand side because there is no magnetic equivalent of electric charge.

Physical Meaning of Gauss's Law for Magnetism

The electric field has sources: field lines spring from positive charges and terminate on negative ones, so the flux out of a surface counts the charge inside. The magnetic field has no sources of this kind. Its field lines circulate — around currents, and through magnets from pole to pole and back — so whatever flux flows out of a closed surface through one part of it flows back in through another.

A practical consequence: the flux through a loop depends only on the loop, not on the surface used to calculate it. Take two different surfaces S₁ and S₂ sharing the same boundary loop. Together they form a closed surface, which has zero net flux, so the flux through S₁ must equal the flux through S₂. This is what makes "the flux through a coil" a well-defined quantity, and Faraday's law therefore gives a unique induced voltage.

Why Magnetic Monopoles Do Not Exist in Classical Electromagnetism

A magnetic monopole would be an isolated north or south pole — a particle carrying "magnetic charge" from which magnetic field lines radiate outward like electric field lines from a point charge. If one existed inside a closed surface, the net magnetic flux would not be zero.

Everyday experience and experiment agree that no monopole has been found:

Cutting a magnet never isolates a pole. Break a bar magnet in half and each piece becomes a complete magnet with its own north and south poles. Keep dividing down to single atoms and every piece remains a dipole.
Magnetism comes from moving charge. At the microscopic level, the magnetic field of a magnet is produced by electron spin and orbital motion — tiny current loops. A current loop is always a dipole: its field lines pass through the loop and wrap around outside, closing on themselves.
Experimental searches have found none. Dedicated detectors, searches in cosmic rays, lunar rock and particle-collider experiments such as MoEDAL at CERN have set tight limits, with no confirmed detection.

Theoretical physics does allow monopoles: Paul Dirac showed in 1931 that the existence of even one monopole would explain why electric charge is quantised, and some grand unified theories predict very heavy monopoles formed in the early universe. Whether or not they exist, classical electromagnetism — and all of RF engineering — is built on ∇·𝐁 = 0.

📌 Magnetic "charge" as a design tool

Antenna engineers sometimes use fictitious magnetic currents 𝐌 and magnetic charges as a mathematical device — for example to model the radiation from a slot in a ground plane or the aperture of a horn antenna (equivalence principle). These are bookkeeping quantities that make the equations symmetric and simplify calculation; they do not imply that physical monopoles exist.

Integral Form of Gauss's Law for Magnetism

The integral form is the version stated above, ∮𝐁·d𝐀 = 0, valid for any closed surface in any medium and for both static and time-varying fields. It is most useful in magnetic circuit analysis. Apply it to a closed surface surrounding a junction of magnetic paths — for example where the centre leg of an E-core splits into two outer legs — and it gives the magnetic equivalent of Kirchhoff's current law: the flux flowing into the junction equals the flux flowing out.

Flux continuity at a magnetic junction (magnetic "KCL")

\sum_{k} Φ_{k} = 0 e.g. Φ_{centre} = Φ_{left} + Φ_{right}

Fluxes are counted positive out of the junction. Flux is conserved along a core just as current is conserved along a wire, which is why the cross-section area — not the length — sets the flux density B = Φ/A in each leg.

Differential Form of Gauss's Law for Magnetism

Applying the divergence theorem, ∮𝐁·d𝐀 = ∫(∇·𝐁) dV, and requiring the result to vanish for every volume gives the point form:

Gauss's law for magnetism — differential form

\nabla \cdot 𝐁 = \frac{\partial B_{x}}{\partial x} + \frac{\partial B_{y}}{\partial y} + \frac{\partial B_{z}}{\partial z} = 0

The magnetic field is "divergence-free" (solenoidal) at every point in space. Any proposed field with non-zero divergence cannot be a physical magnetic field.

A mathematical identity states that the divergence of a curl is always zero, ∇·(∇×𝐀) = 0. Because ∇·𝐁 = 0, the magnetic field can always be written as the curl of a magnetic vector potential 𝐀:

Magnetic vector potential

𝐁 = \nabla \times 𝐀

The vector potential is the standard route to computing antenna radiation: the currents on the antenna determine 𝐀, and the radiated fields are obtained from 𝐀 (Part 10). Note that this 𝐀 is a potential in Wb/m, not an area.

Practical Applications

Magnetic Flux in Electrical Devices

Every inductor is a device for storing energy in magnetic flux. Its inductance is the flux linkage per ampere:

Inductance and stored energy

L = \frac{λ}{I} = \frac{N Φ_{B}}{I} W_{m} = \frac{1}{2} L I^{2}

More turns, a larger cross-section and a higher-permeability core all increase the flux per ampere and therefore the inductance. Relays, solenoid actuators, motors, Hall-effect current sensors and magnetic recording heads all rely on controlling where magnetic flux flows.

Magnetic Flux in Transformers

In a transformer, the primary winding sets up a time-varying flux in the core and the same flux links the secondary winding. Because the flux is guided by a high-permeability core and is continuous (∇·𝐁 = 0), nearly all of the primary's flux reaches the secondary, and the voltage per turn is the same on both windings. For a sinusoidal supply the peak core flux is fixed by the voltage, frequency and number of turns — the transformer equation:

Transformer EMF equation (sinusoidal excitation)

V_{rms} = 4.44 f N Φ_{peak} B_{peak} = \frac{Φ_{peak}}{A_{core}}

4.44 = 2π/√2. If B_peak exceeds the saturation flux density of the core material (≈ 1.5–1.8 T for silicon steel, ≈ 0.3–0.5 T for ferrite), the permeability collapses, the magnetising current spikes and the transformer overheats.

Magnetic Field Behavior in RF Systems

Ferrite cores and beads. At RF, ferrite toroids are used in baluns, broadband transformers and common-mode chokes. The closed toroidal path keeps the flux inside the core, so the component has little external field and couples weakly to neighbours.
Loop antennas and NFC. A small loop antenna responds to the magnetic flux through it. NFC (13.56 MHz) and RFID tags couple through the shared flux between reader and tag coils.
H-field probes for EMC. A small shielded loop measures the flux of a circuit's magnetic field, locating noisy current loops on a PCB.
Ferrite RF components. Circulators and isolators use magnetised ferrites whose permeability depends on the direction of wave travel, giving one-way transmission.
Return-current paths. Because flux circulates around currents, a signal trace and its return current in the ground plane enclose a loop of flux. Minimising that loop area reduces both radiated emissions and susceptibility.

Numerical Examples

Flux Through a Tilted Loop

Problem: A flat square loop of side 20 cm sits in a uniform magnetic field of 0.5 T. The normal to the loop makes an angle of 60° with the field. Find the magnetic flux. The loop is then wound into a 50-turn coil of the same size — what is the flux linkage?

A = 0.2 × 0.2 = 0.04 m²; Φ_B = BA cos θ

Φ_{B} = 0.5 \times 0.04 \times \cos 60 ° = 0.01 Wb = 10 mWb λ = N Φ_{B} = 50 \times 0.01 = 0.5 Wb-turn

Answer: Φ_B = 10 mWb | λ = 0.5 Wb-turn

Rotating the loop changes cos θ and therefore the flux. In Part 4 we will see that rotating the loop continuously makes the flux vary sinusoidally — the principle of the AC generator.

Net Flux Through a Closed Cube

Problem: A closed cube sits in a non-uniform magnetic field near a magnet. The outward fluxes measured through five of its faces are +1.2, −0.8, +0.5, −2.3 and +0.6 mWb. What is the flux through the sixth face?

Gauss's law for magnetism: the six outward fluxes must sum to zero

Φ_{6} = - (1.2 - 0.8 + 0.5 - 2.3 + 0.6) = - (- 0.8) = + 0.8 mWb

Answer: Φ₆ = +0.8 mWb (flux leaving the cube through the sixth face)

The answer does not depend on the shape of the field, the position of the magnet or whether the magnet is partly inside the cube. Had a measurement given a non-zero total, it would indicate either a measurement error — or the discovery of a magnetic monopole.

Peak Flux Density in a Transformer Core

Problem: A mains transformer has a 600-turn primary connected to 230 V rms, 50 Hz. The core cross-section is 12 cm². Find the peak core flux and flux density. Is the core at risk of saturation if it is made of silicon steel (B_sat ≈ 1.6 T)? What happens if the same transformer is connected to 230 V at 60 Hz, and at 25 Hz?

Φ_peak = V_rms / (4.44 f N)

Φ_{peak} = \frac{230}{4.44 \times 50 \times 600} = \frac{230}{133200} = 1.727 mWb

B_peak = Φ_peak / A

B_{peak} = \frac{1.727 \times 10^{- 3}}{12 \times 10^{- 4}} = 1.44 T

Answer: Φ_peak = 1.73 mWb, B_peak = 1.44 T — below saturation, but with only ~10% margin

Flux is inversely proportional to frequency. At 60 Hz the peak drops to 1.44 × 50/60 = 1.20 T (safe). At 25 Hz it would double to 2.88 T — far beyond saturation, and the transformer would draw a destructive magnetising current. This is why transformers designed for 60 Hz can be unsafe on 50 Hz supplies, and why ferrite cores in switch-mode supplies run at hundreds of kilohertz: high frequency means low flux per turn, so a small core suffices.

Flux and Flux Linkage in an RF Inductor

Problem: A 20-turn air-core RF inductor has L = 10 μH and carries a peak current of 0.5 A. Find (a) the peak flux linkage, (b) the flux per turn, and (c) the peak stored magnetic energy.

λ = LI; Φ = λ/N; W = ½LI²

λ = 10 \times 10^{- 6} \times 0.5 = 5 μWb-turn Φ_{B} = \frac{5 μWb}{20} = 0.25 μWb W_{m} = \frac{1}{2} \times 10 \times 10^{- 6} \times {0.5}^{2} = 1.25 μJ

Answer: λ = 5 μWb-turn | Φ = 0.25 μWb per turn | W = 1.25 μJ

This assumes every turn links the same flux. In a real air-core coil some flux lines escape between turns (leakage), so the outer turns link less flux than the centre turns. That is one reason why measured inductance differs from simple formulas, and why closed toroidal cores — which keep the flux inside the core — give more predictable inductance and less coupling to nearby circuits.

Testing Whether a Field Can Be a Magnetic Field

Problem: Which of these vector fields could be physical magnetic fields? (a) 𝐁₁ = (2x, −3y, z) mT/m. (b) 𝐁₂ = (x, y, z) mT/m. (c) 𝐁₃ = (−y, x, 0) mT/m.

Apply ∇·𝐁 = ∂B_x/∂x + ∂B_y/∂y + ∂B_z/∂z

\nabla \cdot 𝐁_{1} = 2 - 3 + 1 = 0 \nabla \cdot 𝐁_{2} = 1 + 1 + 1 = 3 \nabla \cdot 𝐁_{3} = 0 + 0 + 0 = 0

Answer: 𝐁₁ and 𝐁₃ are allowed; 𝐁₂ is not a valid magnetic field

𝐁₂ points radially outward from the origin — it is the field of a monopole, with non-zero divergence. 𝐁₃ circulates around the z-axis and grows linearly with radius — exactly the field inside a straight wire carrying a uniform current along z (Part 5). Simulation engineers use the same check: a numerical solution with significant ∇·𝐁 indicates a meshing or solver problem ("spurious modes").

Summary

Concept	Expression	Key point
Magnetic flux	Φ_B = ∫ 𝐁·d𝐀 (= BA cos θ)	Measured in webers (V·s)
Flux linkage	λ = NΦ_B	What a coil "sees"
Gauss's law for magnetism	∮ 𝐁·d𝐀 = 0	No net flux out of any closed surface
Differential form	∇·𝐁 = 0	𝐁 is divergence-free; no monopoles
Vector potential	𝐁 = ∇ × 𝐀	Route to antenna radiation calculations
Inductance	L = NΦ_B / I	Flux linkage per ampere
Transformer	V_rms = 4.44 f N Φ_peak	Core flux set by volts per hertz

Magnetic flux is the integral of the normal component of 𝐁 over a surface; with N turns it becomes flux linkage.
Magnetic field lines always close on themselves, so the net flux out of every closed surface is zero.
No magnetic monopole has ever been observed; magnetism arises from moving charges and electron spin.
Flux continuity makes the flux through a loop well-defined, makes magnetic circuits behave like electric circuits, and underpins inductors, transformers, ferrite components and loop antennas.

So far the fields have been static. In Part 4: Faraday's Law of Electromagnetic Induction, we let the magnetic flux change with time and discover that it creates an electric field — the principle behind generators, transformers and wireless power transfer.

Put It Into Practice

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Understanding Magnetic Flux and Gauss's Law for Magnetism