Electric Flux Density and Faraday's Law

Electric flux, a scalar field, and its density D, a vector field, are useful quantities in solving certain problems, as will be seen in this. Unlike E, these fields are not directly measurable; their existence was inferred from nineteenth- century experiments in electrostatics.

Example: Refring tothe figure below, a charge +Q is first fixed in place and a spherical, concentric, conducting shell is then closed around it. Initially the shell has no net charge on its surface. Now, if a conducting path to ground is momentarily completed by closing a switch, a charge -Q might be accounted for by a transient flow of negative charge from the ground, through the switch, and onto the shell. But what could provoke such a flow? The early experiments suggested that a flux from the +Q to the conductor surface induced, or displaced, the charge -Q onto the surface. Consequently, it has also been called displacement flux, and the use of the symbol D is a reminder of this early concept.
by definition, electric flux originates on positive charge and terminates on negative charge. in absence of negative charge, the flux terminates at infinity. Also by definition, one coulomb of electric charge gives rise to one coulomb of electric flux. Hence

In figure 3.2-a the flux lines leave +Q and terminate on -Q. This assumes that the two charges are of equal magnitude. The case of positive charge with no negative charge in the region is illustrated in figure b. here the flux lines are equally spaced throughout the solid angle, and reach out toward infinity.

If in the neighborhood of point P the lines of flux have the direction of the unit vector a and if an amount of flux crosses the differential area dS, which is a normal to a, then the electric flux density at P is

A volume charge distribution of density is shown enclosed by surface S. Since each coulomb of charge Q has, by definition, one coulomb of flux, it follows that the net flux crossing the closed surface S is an exact measure of the net charge enclosed. However, the density D may vary in magnitude and direction from point to point of S; in general, D will not be along the normal to S. If, at the surface element dS, D makes an angle with the normal, then the differential flux crossing ds is given by

where dS is the vector surface element, of magnitude dS and direction a. The unit vector an is always taken to point out of S, so that dw is the amount of flux passing from the interior of S to the exterior of S through dS.

Faraday's law (electromagnetism)

In physics, in particular in the theory of electromagnetism, Faraday's law of induction states that a change in magnetic flux generates an electromotive force (EMF, voltage difference). The law is named after the English scientist Michael Faraday, who discovered in 1831 on basis of observations that a change in a magnetic field induces an electric current. This is the phenomenon of electromagnetic induction.

Some thirty years after Faraday's discovery, between 1861 and 1864, the Scottish mathematical physicist James Clerk Maxwell formulated the mathematical expression relating the change in magnetic flux to the induced EMF. This relationship, known as Faraday's law of induction (to distinguish it from Faraday's laws of electrolysis), states that the EMF induced in a circuit is proportional to the rate of decrease of the magnetic flux that cuts across the circuit. By Ohm's law an EMF induces an electric current in a conductor.

When one rotates a circuit in a static homogeneous magnetic field, the magnetic flux cutting across the circuit is changed. Hence this rotation generates an electric current in the circuit, which means that the work done by rotating the circuit inside a magnetic field is converted into an electric current. Thus, Faraday's law is the theoretical basis of the dynamo and the electric generator.

Faraday's law of electromagnetic induction relates the electromotive force (EMF) to the time derivative of the magnetic flux Φ. The law reads

where k = 1 for SI units and one over c (the speed of light) for Gaussian units. The EMF is defined as

where the electric field E is integrated around a closed path C. The magnetic flux Φ through a surface S that has C as boundary is defined as the surface integral,

where dS is a vector normal (perpendicular) to the infinitesimal surface element dS and dS is of length dS. The dot stands for the inner product between the magnetic induction B and dS.

In vacuum the magnetic induction B is proportional to the magnetic field H. (In SI units: B = μ₀ H with μ₀ the magnetic constant of the vacuum; in Gaussian units: B = H.)

If, in the definition of the EMF, C is a conducting loop, then under influence of the EMF a current i_ind will run through it. The minus sign in Faraday's law has the consequence that the magnetic field generated by i_ind opposes the change in Φ; this phenomenon is known as Lenz' law.

If the surface S is constant in size and direction, a change in Φ is solely due to a change in B.

The magnetic flux can be varied by changing the angle between the field and surface. For the simple case of a planar loop bordering a constant area A, and a homogeneous field B of strength B, the flux is given by

where is a unit vector normal (perpendicular) to the plane S bordered by the loop. The initial condition α(0) = 0° is applied. Clearly, the flux is maximum, Φ(0), if α = 0°, i.e., if B is perpendicular to S. In the perpendicular case all of B passes through the surface. When B is parallel to surface, α = 90°, then there is no flux through the surface, Φ = 0, because B only grazes the surface.

When α(t) becomes a function of time by rotation of the loop, Faraday's law shows that an EMF is induced of the form,

When the rotation is uniform, i.e., α(t) linear in time t, it follows that the EMF has a sine dependence on time and an alternating current will be induced.

The flux through two surfaces that together form a closed surface is equal because of Gauss' law. Indeed, in the figure on the right the surfaces S₁ and S₂, which have the boundary C in common, form together a closed surface. Hence Gauss' law states that

where the minus sign of the first term is due to the fact that the flux is into the volume enveloped by the two surfaces. It follows that

and that the magnetic flux Φ can be computed with respect to any surface that has C as boundary.