Conservation of Charge and Faraday's Law in Electromagnetism

Continuity of Current

The principle of conservation of charge states that charges can neither be created nor destroyed, although equal amounts of positive and negative charge may be simultaneously created, obtained by separation, destroyed, or lost by recombination.

Equation 5 indicates that $\mathbf{J}$, the current or charge per second, diverging from a small volume per unit volume is equal to the time rate of decrease of charge per unit volume at every point. The velocity is given by:

Faraday's Law of Induction

In terms of fields, we now say that a time-varying magnetic field produces an electromotive force (EMF) which may establish a current in a suitable closed circuit. An electromotive force is merely a voltage that arises from a conductor moving in a magnetic field or from changing magnetic fields, and we shall define it below. Faraday's law is customarily stated as:

Equation (1) implies a closed path, although not necessarily a closed conducting path; the closed path, for example, might include a capacitor, or it might be a purely imaginary line in space. The magnetic flux ($\Phi$) is that flux which passes through any and every surface whose perimeter is the closed path, and $d\Phi/dt$ is the time rate of change of this flux.

A nonzero value of $d\Phi/dt$ may result from any of the following situations:

1. A time-changing flux linking a stationary closed path.
2. Relative motion between a steady flux and a closed path.
3. A combination of the two.

The minus sign is an indication that the EMF is in such a direction as to produce a current whose flux, if added to the original flux, would reduce the magnitude of the EMF. This statement that the induced voltage acts to produce an opposing flux is known as Lenz's Law.

If the closed path is that taken by an $N$-turn filamentary conductor, it is often sufficiently accurate to consider the turns as coincident and let:

Where $\Phi$ is now interpreted as the flux passing through any one of the coincident paths. We need to define EMF as used in (1) or (2), that is:

and note that it is the voltage about a specific closed path. Replacing $\Phi$ in (1) by the surface integral of $\mathbf{B}$, we have:

Investigating EMF Components

Where the fingers of our right hand indicate the direction of the closed path, and our thumb indicates the direction of $d\mathbf{S}$. Let us divide our investigation into two parts by first finding the contribution to the total EMF made by a changing field within a stationary path (transformer EMF), and then we will consider a moving path within a constant field (motional, or generator, EMF). We first consider a stationary path. The magnetic flux is the only time-varying quantity on the right side of (4), and a partial derivative may be taken under the integral sign:

Applying Stokes' theorem to the closed line integral, we have:

Differential Relationships

The surfaces are perfectly general and may be chosen as differentials:

And: