Gauss's law for magnetism: magnetic field lines never begin nor end but form loops or extend to infinity as shown here with the magnetic field due to a ring of current.
Faraday's experiment showing induction between coils of wire: The liquid battery (right) provides a current which flows through the small coil (A), creating a magnetic field. When the coils are stationary, no current is induced. But when the small coil is moved in or out of the large coil (B), the magnetic flux through the large coil changes, inducing a current which is detected by the galvanometer (G).
In a geomagnetic storm, a surge in the flux of charged particles temporarily alters Earth's magnetic field, which induces electric fields in Earth's atmosphere, thus causing surges in electrical power grids. (Not to scale.)
A diagram of Faraday's iron ring apparatus. The changing magnetic flux of the left coil induces a current in the right coil.
Magnetic-core memory (1954) is an application of Ampère's law. Each core stores one bit of data.
Faraday's disk, the first electric generator, a type of homopolar generator.
Left: A schematic view of how an assembly of microscopic dipoles produces opposite surface charges as shown at top and bottom. Right: How an assembly of microscopic current loops add together to produce a macroscopically circulating current loop. Inside the boundaries, the individual contributions tend to cancel, but at the boundaries no cancelation occurs.
Alternating electric current flows through the solenoid on the left, producing a changing magnetic field. This field causes, by electromagnetic induction, an electric current to flow in the wire loop on the right.
Faraday's homopolar generator. The disc rotates with angular rate {{mvar|ω}}, sweeping the conducting radius circularly in the static magnetic field {{math|B}} (which direction is along the disk surface normal). The magnetic Lorentz force {{math|v × B}} drives a current along the conducting radius to the conducting rim, and from there the circuit completes through the lower brush and the axle supporting the disc. This device generates an emf and a current, although the shape of the "circuit" is constant and thus the flux through the circuit does not change with time.
A wire (solid red lines) connects to two touching metal plates (silver) to form a circuit. The whole system sits in a uniform magnetic field, normal to the page. If the abstract path {{math|∂Σ}} follows the primary path of current flow (marked in red), then the magnetic flux through this path changes dramatically as the plates are rotated, yet the emf is almost zero. After Feynman Lectures on Physics {{Rp|ch17}}

The Maxwell–Faraday equation (listed as one of Maxwell's equations) describes the fact that a spatially varying (and also possibly time-varying, depending on how a magnetic field varies in time) electric field always accompanies a time-varying magnetic field, while Faraday's law states that there is emf (electromotive force, defined as electromagnetic work done on a unit charge when it has traveled one round of a conductive loop) on the conductive loop when the magnetic flux through the surface enclosed by the loop varies in time.

- Faraday's law of induction

The Maxwell–Faraday version of Faraday's law of induction describes how a time-varying magnetic field corresponds to curl of an electric field.

- Maxwell's equations
Gauss's law for magnetism: magnetic field lines never begin nor end but form loops or extend to infinity as shown here with the magnetic field due to a ring of current.

4 related topics with Alpha

Overall

The shape of the magnetic field produced by a horseshoe magnet is revealed by the orientation of iron filings sprinkled on a piece of paper above the magnet.

Magnetic field

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Vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials.

Vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials.

The shape of the magnetic field produced by a horseshoe magnet is revealed by the orientation of iron filings sprinkled on a piece of paper above the magnet.
Right hand grip rule: a current flowing in the direction of the white arrow produces a magnetic field shown by the red arrows.
A Solenoid with electric current running through it behaves like a magnet.
A sketch of Earth's magnetic field representing the source of the field as a magnet. The south pole of the magnetic field is near the geographic north pole of the Earth.
One of the first drawings of a magnetic field, by René Descartes, 1644, showing the Earth attracting lodestones. It illustrated his theory that magnetism was caused by the circulation of tiny helical particles, "threaded parts", through threaded pores in magnets.
Hans Christian Ørsted, Der Geist in der Natur, 1854

In a modified form that accounts for time varying electric fields, Ampère's law is one of four Maxwell's equations that describe electricity and magnetism.

are called the Ampère–Maxwell equation and Faraday's law respectively.

Lorentz force acting on fast-moving charged particles in a bubble chamber. Positive and negative charge trajectories curve in opposite directions.

Lorentz force

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Combination of electric and magnetic force on a point charge due to electromagnetic fields.

Combination of electric and magnetic force on a point charge due to electromagnetic fields.

Lorentz force acting on fast-moving charged particles in a bubble chamber. Positive and negative charge trajectories curve in opposite directions.
Lorentz' theory of electrons. Formulas for the Lorentz force (I, ponderomotive force) and the Maxwell equations for the divergence of the electrical field E (II) and the magnetic field B (III), La théorie electromagnétique de Maxwell et son application aux corps mouvants, 1892, p. 451. V is the velocity of light.
Charged particle drifts in a homogeneous magnetic field. (A) No disturbing force (B) With an electric field, E (C) With an independent force, F (e.g. gravity) (D) In an inhomogeneous magnetic field, grad H
Right-hand rule for a current-carrying wire in a magnetic field B
Lorentz force -image on a wall in Leiden
Lorentz force -image on a wall in Leiden

Variations on this basic formula describe the magnetic force on a current-carrying wire (sometimes called Laplace force), the electromotive force in a wire loop moving through a magnetic field (an aspect of Faraday's law of induction), and the force on a moving charged particle.

The electric field in question is created by the changing magnetic field, resulting in an induced EMF, as described by the Maxwell–Faraday equation (one of the four modern Maxwell's equations).

Alternating electric current flows through the solenoid on the left, producing a changing magnetic field. This field causes, by electromagnetic induction, an electric current to flow in the wire loop on the right.

Electromagnetic induction

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Production of an electromotive force across an electrical conductor in a changing magnetic field.

Production of an electromotive force across an electrical conductor in a changing magnetic field.

Alternating electric current flows through the solenoid on the left, producing a changing magnetic field. This field causes, by electromagnetic induction, an electric current to flow in the wire loop on the right.
Faraday's experiment showing induction between coils of wire: The liquid battery (right) provides a current that flows through the small coil (A), creating a magnetic field. When the coils are stationary, no current is induced. But when the small coil is moved in or out of the large coil (B), the magnetic flux through the large coil changes, inducing a current which is detected by the galvanometer (G).
A diagram of Faraday's iron ring apparatus. Change in the magnetic flux of the left coil induces a current in the right coil.
A solenoid
The longitudinal cross section of a solenoid with a constant electrical current running through it. The magnetic field lines are indicated, with their direction shown by arrows. The magnetic flux corresponds to the 'density of field lines'. The magnetic flux is thus densest in the middle of the solenoid, and weakest outside of it.
Rectangular wire loop rotating at angular velocity ω in radially outward pointing magnetic field B of fixed magnitude. The circuit is completed by brushes making sliding contact with top and bottom discs, which have conducting rims. This is a simplified version of the drum generator.
A current clamp

Michael Faraday is generally credited with the discovery of induction in 1831, and James Clerk Maxwell mathematically described it as Faraday's law of induction.

Faraday's law was later generalized to become the Maxwell–Faraday equation, one of the four Maxwell equations in his theory of electromagnetism.

Some examples of closed surfaces (left) and open surfaces (right). Left: Surface of a sphere, surface of a torus, surface of a cube. Right: Disk surface, square surface, surface of a hemisphere. (The surface is blue, the boundary is red.)

Magnetic flux

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Surface integral of the normal component of the magnetic field B over that surface.

Surface integral of the normal component of the magnetic field B over that surface.

Some examples of closed surfaces (left) and open surfaces (right). Left: Surface of a sphere, surface of a torus, surface of a cube. Right: Disk surface, square surface, surface of a hemisphere. (The surface is blue, the boundary is red.)
For an open surface Σ, the electromotive force along the surface boundary, ∂Σ, is a combination of the boundary's motion, with velocity v, through a magnetic field B (illustrated by the generic F field in the diagram) and the induced electric field caused by the changing magnetic field.
Area defined by an electric coil with three turns.

Gauss's law for magnetism, which is one of the four Maxwell's equations, states that the total magnetic flux through a closed surface is equal to zero.

The relationship is given by Faraday's law: