Given a loop of wire in a magnetic field, Faraday's law of induction states:
\mathcal{E} = -\frac{d\Phi_B}{dt}
where:
\Phi_B \ is the magnetic flux through the loop,
\mathcal{E} is the electromotive force (EMF) experienced,
t is time
The sign of the EMF is determined by Lenz's law.
Using the Lorentz force law, the EMF around a closed path ∂Σ is given by:[12][13]
\mathcal{E} =\oint_{\part \Sigma (t)} d \boldsymbol{\ell} \cdot \mathbf{F} / q = \oint_{\part \Sigma (t)} d \boldsymbol{\ell} \cdot \left( \mathbf {E} + \mathbf{ v \times B} \right) \ ,
where dℓ is an element of the curve ∂Σ(t), imagined to be moving in time. The flux ΦB in Faraday's law of induction can be expressed explicitly as:
\frac {d \Phi_B} {dt} = \frac {d} {dt} \iint_{\Sigma (t)} d \boldsymbol {A} \cdot \mathbf {B}(\mathbf{r},\ t) \ ,
where
Σ(t) is a surface bounded by the closed contour ∂Σ(t)
E is the electric field,
dℓ is an infinitesimal vector element of the contour ∂Σ,
v is the velocity of the infinitesimal contour element dℓ,
B is the magnetic field.
dA is an infinitesimal vector element of surface Σ , whose magnitude is the area of an infinitesimal patch of surface, and whose direction is orthogonal to that surface patch.
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.
The surface integral at the right-hand side of this equation is the explicit expression for the magnetic flux ΦB through Σ. Thus, incorporating the Lorentz law in Faraday's equation, we find:[14][15]
\oint_{\part \Sigma (t)} d \boldsymbol{\ell} \cdot \left( \mathbf {E}(\mathbf{r},\ t) + \mathbf{ v \times B}(\mathbf{r},\ t) \right) = -\frac {d} {dt} \iint_{\Sigma (t)} d \boldsymbol {A} \cdot \mathbf {B}(\mathbf{r},\ t) \ .
Notice that the ordinary time derivative appearing before the integral sign implies that time differentiation must include differentiation of the limits of integration, which vary with time whenever Σ(t) is a moving surface.
The above result can be compared with the version of Faraday's law of induction that appears in the modern Maxwell's equations, called here the Maxwell-Faraday equation:
\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \ .
The Maxwell-Faraday equation also can be written in an integral form using the Kelvin-Stokes theorem:[16]
\oint_{\partial \Sigma (t)}d \boldsymbol{\ell} \cdot \mathbf{E}(\mathbf{r},\ t) = - \ \iint_{\Sigma (t)} d \boldsymbol {A} \cdot {{ \partial \mathbf {B}(\mathbf{r},\ t)} \over \partial t }
Comparison of the Faraday flux law with the integral form of the Maxwell-Faraday relation suggests:
\frac {d} {dt} \iint_{\Sigma (t)} d \boldsymbol {A} \cdot \mathbf {B}(\mathbf{r},\ t)= \iint_{\Sigma (t)} d \boldsymbol {A} \cdot {{ \partial \mathbf {B}(\mathbf{r}, \ t)} \over \partial t } - \oint_{\part \Sigma (t)} d \boldsymbol{\ell} \cdot \left( \mathbf{ v \times B}(\mathbf{r},\ t) \right) \ .
which is a form of the Leibniz integral rule valid because div B = 0.[17] The term in v × B accounts for motional EMF, that is the movement of the surface Σ, at least in the case of a rigidly translating body. In contrast, the integral form of the Maxwell-Faraday equation includes only the effect of the E-field generated by ∂B/∂t.
Often the integral form of the Maxwell-Faraday equation is used alone, and is written with the partial derivative outside the integral sign as:
\oint_{\partial \Sigma}d \boldsymbol{\ell} \cdot \mathbf{E}(\mathbf{r},\ t) = - { \partial \over \partial t } \ \iint_{\Sigma} d \boldsymbol {A} \cdot { \mathbf {B}(\mathbf{r},\ t) } \ .
Notice that the limits ∂Σ and Σ have no time dependence. In the context of the Maxwell-Faraday equation, the usual interpretation of the partial time derivative is extended to imply a stationary boundary. On the other hand, Faraday's law of induction holds whether the loop of wire is rigid and stationary, or in motion or in process of deformation, and it holds whether the magnetic field is constant in time or changing. However, there are cases where Faraday's law is either inadequate or difficult to use, and application of the underlying Lorentz force law is necessary. See inapplicability of Faraday's law.
If the magnetic field is fixed in time and the conducting loop moves through the field, the flux magnetic flux ΦB linking the loop can change in several ways. For example, if the B-field varies with position, and the loop moves to a location with different B-field, ΦB will change. Alternatively, if the loop changes orientation with respect to the B-field, the B•dA differential element will change because of the different angle between B and dA, also changing ΦB. As a third example, if a portion of the circuit is swept through a uniform, time-independent B-field, and another portion of the circuit is held stationary, the flux linking the entire closed circuit can change due to the shift in relative position of the circuit's component parts with time (surface Σ(t) time-dependent). In all three cases, Faraday's law of induction then predicts the EMF generated by the change in ΦB.
In a contrasting circumstance, when the loop is stationary and the B-field varies with time, the Maxwell-Faraday equation shows a nonconservative[18] E-field is generated in the loop, which drives the carriers around the wire via the q E term in the Lorentz force. This situation also changes ΦB, producing an EMF predicted by Faraday's law of induction.
Naturally, in both cases, the precise value of current that flows in response to the Lorentz force depends on the conductivity of the loop.
