Sunday, April 18, 2010

Flux through a surface and EMF around a loop

Faraday's law of induction makes use of the magnetic flux ΦB through a surface Σ, defined by an integral over a surface:

 \Phi_B = \iint\limits_{\Sigma(t)} \mathbf{B}(\mathbf{r}, t) \cdot d \mathbf{A}\ ,

where dA is an element of surface area of the moving surface Σ(t), B is the magnetic field, and B·dA is a vector dot product. The surface is considered to have a "mouth" outlined by a closed curve denoted ∂Σ(t). When the flux changes, Faraday's law of induction says that the work \mathcal{E} done (per unit charge) moving a test charge around the closed curve ∂Σ(t), called the electromotive force (EMF), is given by:

|\mathcal{E}| = \left|{{d\Phi_B} \over dt} \right| \ ,

where |\mathcal{E}| is the magnitude of the electromotive force (EMF) in volts and ΦB is the magnetic flux in webers. The direction of the electromotive force is given by Lenz's law.

For a tightly-wound coil of wire, composed of N identical loops, each with the same ΦB, Faraday's law of induction states that

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.
A vector field F(r, t) defined throughout space, and a surface Σ bounded by curve ∂Σ moving with velocity v over which the field is integrated.

Faraday's law of induction makes use of the magnetic flux ΦB through a surface Σ, defined by an integral over a surface:

where N is the number of turns of wire and ΦB is the magnetic flux in webers through a single loop.

In choosing a path ∂Σ(t) to find EMF, the path must satisfy the basic requirements that (i) it is a closed path, and (ii) the path must capture the relative motion of the parts of the circuit (the origin of the t-dependence in ∂Σ(t) ). It is not a requirement that the path follow a line of current flow, but of course the EMF that is found using the flux law will be the EMF around the chosen path. If a current path is not followed, the EMF might not be the EMF driving the current.

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