What is the gradient?

The gradient is a vector operation on a scalar field that results in a vector whose direction points in the greatest rate of change.

What is the laplace operator?

The laplace operator is the divergence of the gradient of a function.

The gradient is defined as a vector operator that results in a vector that points in the direction of greatest rate of change.

The divergence is a vector operator that results in a magnitude (scalar quantity) of the outward flux density around a point in space.

So, because the laplace operator is the divergence of a vector quantity, its result is a scalar whose magnitude is proportional to the maximum rate of change.

If the gradient of a scalar field is nonzero, there exists a vector field that points in the direction of the maximum rate of change. Taking the divergence at a point in this vector field, one obtains a scalar whose magnitude represents the greatest rate of change at that point. If the magnitude of this value (the laplacian of the scalar field) is positive, the maximum rate of change is occurring in an outward direction from that point. If it is negative, the maximum rate of change is occurring in an inward direction toward that point.

What is Curl?

The curl of a vector field describes the direction and magnitude of rotation at every point in space.

What is the Wave Equation?

The wave equation is a partial differential equation describing the nature of waves in time and space.

What does the term "coupled" wave equation mean?

When solving transmission line problems, it is often of interest to solve for the instantaneous voltage and current. In doing so using a lumped element (loss-less) model, it is found that the wave equations for both the voltage and current depend on each other and are thought of as "coupled."

What is Circulation?

Circulation is the line integral of a vector quantity on a closed plath.

For example, consider an arbitrary curve forming a closed path. Also consider a vector quantity F acting on this path. Let's say the closed path is split into several segments of length dl.

Then the differential circulation dΓ is the scalar product of the differential length and the vector quantity acting on that length. Meaning,



So, the circulation is the sum of all the differential circulation quantities as the differential length approaches zero. Meaning,

What is the Continuity Equation?

The continuity equation in electromagnetic theory is a formal way of stating that charge is conserved. The equation for this in the differential form is the following:

 or

The integral form of this equation is the following (obtained using the divergence theorem):

 (here, a closed surface is considered)

In this equation, rho is the charge density (in C/m^3), the upside down triangle is the del operator, J is the current density in (A/m^2 - a vector quantity), and q_en is charge enclosed (in C). It is very difficult to picture this qualitatively in the differential form, so we shall attempt to relate to the integral form instead.

Consider a set of positive charges that all of a sudden disperse at a finite rate. Now, consider an imaginary sphere centered at the point source. The continuity equation states that the net outward flow (current, in C/s) through the surface of the imaginary sphere is equal to the rate of decrease of charge enclosed by the sphere.

Numerically, say, if 1 A of current flowed out of the sphere, 1 Coulomb of charge will have left the sphere in 1 second.

The continuity equation is very important in electromagnetic theory. Maxwell identified a flaw in Ampere's Law, identifying cases where charge was not conserved. By correcting this equation, he was able to complete the connection between Gauss' Law (for electricity and for magnetism), Faraday's Law, and Ampere's Law (all together now known as Maxwell's equations). This was the missing link in making the connection between electromagnetism and light.

What is Counter Electromotive Force (Back EMF)?

Counter electromotive force (CEMF or back EMF) can be thought of as a secondary induced EMF, produced by a primary EMF, which is caused due to a changing magnetic flux.

Wikipedia defines CEMF as:

"the voltage, or electromotive force, that pushes against the current which induces it."

Consider an example:

A C-shaped conductor is placed in a static magnetic field (whose direction is in the -z direction) as shown below. A conductive bar is placed near the edge of the C, and is forced (externally) to slide in the +x direction while the C is not moving.

The two conductors together form a closed loop, and sliding the bar increases the area of this loop. By Faraday's Law, B.dS is increasing and thus (since nature tends to resist such changes) a magnetic flux is induced in the coil whose direction opposes that of the original field. This means a current flows in the counter-clockwise direction.

Now, let us consider the bar when the current is flowing. We also know that the Lorentz force states the following:

So, a second force is induced due to the induced current which was caused by sliding the bar in the magnetic field. This force opposes the force that slides the bar. This is an example of back EMF.