Qualitative statement
The most widespread version of Faraday's law states:
The electromotive force around a closed path is equal to the negative of the time rate of change of the magnetic flux enclosed by the path.[13][14]
The closed path here is, in fact, conductive.
Quantitative
The definition of surface integral relies on splitting the surface Σ into small surface elements. Each element is associated with a vector dA of magnitude equal to the area of the element and with direction normal to the element and pointing "outward" (with respect to the orientation of the surface).
Faraday's law of induction makes use of the magnetic flux ΦB through a hypothetical surface Σ whose boundary is a wire loop. Since the wire loop may be moving, we write Σ(t) for the surface. The magnetic flux is defined by a surface integral:
Φ B = ∬ Σ ( t ) B ( r , t ) ⋅ d A , {\displaystyle \Phi _{B}=\iint \limits _{\Sigma (t)}\mathbf {B} (\mathbf {r} ,t)\cdot \mathrm {d} \mathbf {A} \,,} {\displaystyle \Phi _{B}=\iint \limits _{\Sigma (t)}\mathbf {B} (\mathbf {r} ,t)\cdot \mathrm {d} \mathbf {A} \,,}
where dA is an element of surface area of the moving surface Σ(t), B is the magnetic field (also called "magnetic flux density"), and B·dA is a vector dot product (the infinitesimal amount of magnetic flux through the infinitesimal area element dA). In more visual terms, the magnetic flux through the wire loop is proportional to the number of magnetic flux lines that pass through the loop.
When the flux changes—because B changes, or because the wire loop is moved or deformed, or both—Faraday's law of induction says that the wire loop acquires an EMF, E, defined as the energy available from a unit charge that has travelled once around the wire loop.[15][16][17] (Note that different textbooks may give different definitions. The set of equations used throughout the text was chosen to be compatible with the special relativity theory.) Equivalently, it is the voltage that would be measured by cutting the wire to create an open circuit, and attaching a voltmeter to the leads.
Faraday's law states that the EMF is also given by the rate of change of the magnetic flux:
E = − d Φ B d t , {\displaystyle {\mathcal {E}}=-{\frac {\mathrm {d} \Phi _{B}}{\mathrm {d} t}},} {\displaystyle {\mathcal {E}}=-{\frac {\mathrm {d} \Phi _{B}}{\mathrm {d} t}},}
where E is the electromotive force (EMF) and ΦB is the magnetic flux.
The direction of the electromotive force is given by Lenz's law.
The laws of induction of electric currents in mathematical form was established by Franz Ernst Neumann in 1845.[18]
Faraday's law contains the information about the relationships between both the magnitudes and the directions of its variables. However, the relationships between the directions are not explicit; they are hidden in the mathematical formula.
A Left Hand Rule for Faraday's Law. The sign of ΔΦB, the change in flux, is found based on the relationship between the magnetic field B, the area of the loop A, and the normal n to that area, as represented by the fingers of the left hand. If ΔΦB is positive, the direction of the EMF is the same as that of the curved fingers (yellow arrowheads). If ΔΦB is negative, the direction of the EMF is against the arrowheads.[19]
It is possible to find out the direction of the electromotive force (EMF) directly from Faraday’s law, without invoking Lenz's law. A left hand rule helps doing that, as follows:[19][20]
Align the curved fingers of the left hand with the loop (yellow line).
Stretch your thumb. The stretched thumb indicates the direction of n (brown), the normal to the area enclosed by the loop.
Find the sign of ΔΦB, the change in flux. Determine the initial and final fluxes (whose difference is ΔΦB) with respect to the normal n, as indicated by the stretched thumb.
If the change in flux, ΔΦB, is positive, the curved fingers show the direction of the electromotive force (yellow arrowheads).
If ΔΦB is negative, the direction of the electromotive force is opposite to the direction of the curved fingers (opposite to the yellow arrowheads).
For a tightly wound coil of wire, composed of N identical turns, each with the same ΦB, Faraday's law of induction states that[21][22]
E = − N d Φ B d t {\displaystyle {\mathcal {E}}=-N{\frac {\mathrm {d} \Phi _{B}}{\mathrm {d} t}}} {\displaystyle {\mathcal {E}}=-N{\frac {\mathrm {d} \Phi _{B}}{\mathrm {d} t}}}
where N is the number of turns of wire and ΦB is the magnetic flux through a single loop.
Maxwell–Faraday equation
An illustration of the Kelvin–Stokes theorem with surface Σ, its boundary ∂Σ, and orientation n set by the right-hand rule.
The Maxwell–Faraday equation states that a time-varying magnetic field always accompanies a spatially varying (also possibly time-varying), non-conservative electric field, and vice versa. The Maxwell–Faraday equation is
∇ × E = − ∂ B ∂ t {\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}} \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}
(in SI units) where ∇ × is the curl operator and again E(r, t) is the electric field and B(r, t) is the magnetic field. These fields can generally be functions of position r and time t.
The Maxwell–Faraday equation is one of the four Maxwell's equations, and therefore plays a fundamental role in the theory of classical electromagnetism. It can also be written in an integral form by the Kelvin–Stokes theorem:[23]
∮ ∂ Σ E ⋅ d l = − ∫ Σ ∂ B ∂ t ⋅ d A {\displaystyle \oint _{\partial \Sigma }\mathbf {E} \cdot \mathrm {d} \mathbf {l} =-\int _{\Sigma }{\frac {\partial \mathbf {B} }{\partial t}}\cdot \mathrm {d} \mathbf {A} } {\displaystyle \oint _{\partial \Sigma }\mathbf {E} \cdot \mathrm {d} \mathbf {l} =-\int _{\Sigma }{\frac {\partial \mathbf {B} }{\partial t}}\cdot \mathrm {d} \mathbf {A} }
where, as indicated in the figure:
Σ is a surface bounded by the closed contour ∂Σ,
E is the electric field, B is the magnetic field.
dl is an infinitesimal vector element of the contour ∂Σ,
dA is an infinitesimal vector element of surface Σ. If its direction is orthogonal to that surface patch, the magnitude is the area of an infinitesimal patch of surface.
Both dl and dA have a sign ambiguity; to get the correct sign, the right-hand rule is used, as explained in the article Kelvin–Stokes theorem. For a planar surface Σ, a positive path element dl of curve ∂Σ is defined by the right-hand rule as one that points with the fingers of the right hand when the thumb points in the direction of the normal n to the surface Σ.
The integral around ∂Σ is called a path integral or line integral.
Notice that a nonzero path integral for E is different from the behavior of the electric field generated by charges. A charge-generated E-field can be expressed as the gradient of a scalar field that is a solution to Poisson's equation, and has a zero path integral. See gradient theorem.
The integral equation is true for any path ∂Σ through space, and any surface Σ for which that path is a boundary.
If the surface Σ is not changing in time, the equation can be rewritten:
∮ ∂ Σ E ⋅ d l = − d d t ∫ Σ B ⋅ d A . {\displaystyle \oint _{\partial \Sigma }\mathbf {E} \cdot \mathrm {d} \mathbf {l} =-{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{\Sigma }\mathbf {B} \cdot \mathrm {d} \mathbf {A} .} {\displaystyle \oint _{\partial \Sigma }\mathbf {E} \cdot \mathrm {d} \mathbf {l} =-{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{\Sigma }\mathbf {B} \cdot \mathrm {d} \mathbf {A} .}
The surface integral at the right-hand side is the explicit expression for the magnetic flux ΦB through Σ.
