Lorentz Force - Lorentz Force and Faraday's Law of Induction

Lorentz Force and Faraday's Law of Induction

Given a loop of wire in a magnetic field, Faraday's law of induction states the induced electromotive force (EMF) in the wire is:

where

is the magnetic flux through the loop, B is the magnetic field, Σ(t) is a surface bounded by the closed contour ∂Σ(t), at all at time t, dA is an infinitesimal vector area element of Σ(t) (magnitude is the area of an infinitesimal patch of surface, direction is orthogonal to that surface patch).

The sign of the EMF is determined by Lenz's law. Note that this is valid for not only a stationary wire — but also for a moving wire.

From Faraday's law of induction (that is valid for a moving wire, for instance in a motor) and the Maxwell Equations, the Lorentz Force can be deduced. The reverse is also true, the Lorentz force and the Maxwell Equations can be used to derive the Faraday Law.

Let Σ(t) be the moving wire, moving together without rotation and with constant velocity v and Σ(t) be the internal surface of the wire. The EMF around the closed path ∂Σ(t) is given by:

where

is the electric field and dℓ is an infinitesimal vector element of the contour ∂Σ(t).

NB: 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 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:

The Maxwell-Faraday equation also can be written in an integral form using the Kelvin-Stokes theorem:.

So we have, the Maxwell Faraday equation:

and the Faraday Law,

The two are equivalent if the wire is not moving. Using the Leibniz integral rule and that div B = 0, results in,

\oint_{\partial \Sigma(t)} d \boldsymbol{\ell} \cdot \mathbf{F}/q(\mathbf{r}, t) = - \iint_{\Sigma(t)} d \bold{A} \cdot \frac{d}{dt} \mathbf{B}(\mathbf{r}, t) + \oint_{\partial \Sigma(t)} \mathbf{v} \times \mathbf{B} d \boldsymbol{\ell}

and using the Maxwell Faraday equation,

\oint_{\partial \Sigma(t)} d \boldsymbol{\ell} \cdot \mathbf{F}/q(\mathbf{r},\ t) = \oint_{\partial \Sigma(t)} d \boldsymbol{\ell} \cdot \mathbf{E}(\mathbf{r},\ t) + \oint_{\partial \Sigma(t)} \mathbf{v} \times \mathbf{B}(\mathbf{r},\ t) d \boldsymbol{\ell}

since this is valid for any wire position it implies that,

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 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.

Note that the Maxwell Faraday's equation implies that the Electric Field E is non conservative when the Magnetic Field B varies in time, and is not expressible as the gradient of a scalar field, and not subject to the gradient theorem since its rotational is not zero. See also.

Read more about this topic:  Lorentz Force

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