Magnetic Flux

Magnetic flux is a signed scalar quantity measuring the magnetic field passing through a specified, oriented surface. Usually denoted by $\Phi_B$ or $\Phi$, it is defined by integrating the field’s component perpendicular to that surface. It is a central concept in electromagnetism, especially in electromagnetic induction, where changes in flux are associated with an induced voltage around a circuit. (ocw.mit.edu)

Mathematical definition

For an oriented surface $S$, magnetic flux is

where $\mathbf B$ is the magnetic flux density, $\hat{\mathbf n}$ is the chosen unit normal to the surface, and $d\mathbf A=\hat{\mathbf n}\,dA$ is the vector area element. This surface integral adds the local contributions of the field over the entire surface. The surface may be curved, and the field may vary in magnitude and direction. (ocw.mit.edu)

The dot product selects the normal component of the field: a field tangent to a surface makes no local contribution. Reversing the surface orientation reverses the sign of the flux. Positive and negative contributions can cancel, so zero net flux does not necessarily mean that the magnetic field vanishes. These properties follow directly from the integral definition. (ocw.mit.edu)

For a uniform field through a flat surface of area $A$,

where $\theta$ is the angle between the field and the surface normal, not the surface itself. The flux is $BA$ when the normal points along the field, zero when the field lies in the surface, and $-BA$ when the normal points opposite to the field. This is the constant-field specialization of the surface integral. (ocw.mit.edu)

Magnetic flux can be defined through an imaginary surface; no wire or material membrane is required. For a wire loop, the relevant surface is one whose boundary follows the loop. (web.mit.edu)

Units and distinction from magnetic flux density

In the International System of Units (SI), magnetic flux is measured in webers, symbol Wb. The related unit of magnetic flux density is the tesla, symbol T:

Thus, a uniform field of one tesla passing normally through an area of one square metre produces one weber of flux. (bipm.org)

The distinction is between a local field quantity and a surface-integrated quantity. Magnetic flux density $\mathbf B$ describes the field at a point; magnetic flux describes its net normal contribution over a chosen surface. Consequently, the flux depends on the surface’s area and orientation as well as on the field. (bipm.org)

In magnetic materials, flux is calculated from $\mathbf B$, not directly from the magnetic field intensity $\mathbf H$. Their SI relationship is

where $\mathbf M$ is magnetization. The simpler relation $\mathbf B=\mu\mathbf H$, using magnetic permeability $\mu$, applies when an appropriate linear constitutive relationship is available. (web.mit.edu)

Closed surfaces and Gauss’s law for magnetism

One of Maxwell’s equations, Gauss’s law for magnetism, states that the net magnetic flux through any closed surface is zero:

Its differential form is

With the surface normal directed outward, outward and inward flux contributions balance. The law does not require the magnetic field to be zero on the surface or within the enclosed region. (web.mit.edu)

An important consequence is that the flux linked by a given loop is independent of the spanning surface selected. Two surfaces sharing the same boundary can be joined, with opposite orientations, to form a closed surface. Because the total flux through that closed surface is zero, their consistently oriented fluxes are equal. This permits a convenient curved surface to replace a more difficult flat one in a calculation. (web.mit.edu)

Faraday’s law and changing flux

Faraday’s law of induction relates magnetic flux to electromotive force (emf):

For a stationary contour $C$ bounding a fixed surface $S$, it takes the form

where $\mathbf E$ is the electric field. The contour direction and surface normal are related by the right-hand rule. A time-varying magnetic field can therefore produce a circulating electric field. (ocw.mit.edu)

Flux can change because the magnetic field changes, because a loop moves through a nonuniform field, or because its area or orientation changes. For a moving loop, the flux rule must include the magnetic force on charges associated with the loop’s motion, rather than only the electric-field circulation measured in the laboratory frame. (ocw.mit.edu)

The minus sign expresses Lenz’s law: the induced response opposes the change in magnetic flux. It does not imply that the induced magnetic field always opposes the existing field. For example, when the original flux decreases, an induced current can produce a field in the original direction. (ocw.mit.edu)

An induced emf does not by itself specify an electric current. The resulting current depends on the circuit’s electrical properties and whether a conducting path is available. (web.mit.edu)

Flux linkage and inductance

For a winding with multiple turns, the relevant quantity is flux linkage, usually denoted by $\lambda$ or $\Psi$:

If all $N$ turns link the same flux,

Flux linkage therefore accounts for the cumulative induction in the winding. It is distinct from the flux through a single turn, although both are expressed in webers in SI; “weber-turn” is sometimes used to emphasize the winding factor. (web.mit.edu)

For a linear coil of fixed geometry, self-inductance $L$ relates its current $I$ to the linkage produced by that current:

If $L$ is constant, the induced emf is $-L\,dI/dt$. Mutual inductance describes the corresponding linkage in one circuit produced by current in another. These relationships connect the spatial magnetic field with circuit-level descriptions. (ocw.mit.edu)

Measurement and applications

A pickup coil measures changes in flux linkage through its induced voltage. Integrating that voltage gives

Moving, rotating, or withdrawing a calibrated coil provides ways to convert a static magnetic field into a measurable changing flux. Measurement requires knowledge of the coil geometry, orientation, and voltage integration. The unit relationship $1\ \mathrm{Wb}=1\ \mathrm{V\,s}$ underlies this method. (indico.cern.ch)

Flux and flux linkage provide the common description of induction in electric generators, electrical transformers, and inductors: motion or changing currents alter the linked flux, producing an emf. For example, applying the flux rule to a coil rotating at constant angular speed $\omega$ in a uniform field gives

for a suitable choice of initial orientation. (web.mit.edu)

Flux quantization in superconductors

In superconductivity, magnetic flux is connected to the phase coherence of the superconducting state. The characteristic magnetic flux quantum is

where $h$ is the Planck constant and $e$ is the elementary charge. The factor $2e$ corresponds to the charge magnitude of a conventional Cooper pair. (physics.nist.gov)

Under appropriate conditions, flux trapped in a superconducting ring takes integer multiples of $\Phi_0$. More precisely, the generally quantized quantity is the fluxoid, which includes both magnetic flux and a contribution from the circulating supercurrent; flux alone coincides with it when that current contribution is negligible along the chosen integration path. Flux quantization was experimentally observed in superconducting rings in 1961. (ocw.mit.edu)

A superconducting quantum interference device (SQUID) exploits superconducting interference to detect extremely small magnetic signals. This quantum behavior does not mean that flux through every arbitrary surface is restricted to integer multiples of $\Phi_0$; the quantization concerns particular superconducting configurations and their phase constraints. (nist.gov)

References

1. Summary of Class 20: Faraday’s Lawocw.mit.edu
2. Electromagnetic Field Theory: A Problem-Solving Approach, Part 1ocw.mit.edu
3. Practical Realization of Electrical Units, CCEM/09-05bipm.org
4. Electromagnetic Fields and Energy, Section 9.2web.mit.edu
5. Gauss’ Integral Law of Magnetic Fluxweb.mit.edu
6. Electromagnetic Fields and Energy, Chapter 2web.mit.edu
7. Summary of Class 21: Faraday’s Lawocw.mit.edu
8. TEAL E&M: Faraday’s Lawweb.mit.edu
9. Electromagnetic Fields and Energy, Section 8.4web.mit.edu
10. Overview of Magnetic Measurementsindico.cern.ch
11. Problem Solving 10: Faraday’s Lawweb.mit.edu
12. CODATA Value: Magnetic Flux Quantumphysics.nist.gov