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.

What is Electromotive Force (EMF)?

Electromotive force can be defined as the induced voltage, produced by a changing magnetic flux, whose polarity is derived by knowing the direction of the induced opposing magnetic flux .

Consider the following example. Let there be a increasing (so, changing) uniform magnetic field with time in space. Let there also exist a circular wire containing a series resistor in this field, whose area vector is parallel to the direction of the described field. By Faraday's law, an electromotive force is induced in the loop, and this voltage is developed across the series resistor. The polarity of this voltage is of special interest.

Faraday's Law (along with Lenz's law) implies that nature tends to oppose changes in equilibrium (to maintain a 'balance'). So, because there is an increasing magnetic field through the loop, Faraday's Law states that an induced magetic field is produced by the loop whose increasing direction opposes that of the original field. The direction of the induced field defines the current through the loop, and thus the polarity of the voltage (EMF) across the series resistor.

The image below (provided by Addison Wesley Longman, Inc) illustrates this. Note that the B-field in blue is increasing with time, and that the resistor is not shown here but a positive voltage will be observed across point a to point b.

What is Electric Field?

Electric field is defined as force per unit charge.

This means that the magnitude of electric field is a measure of the force that one unit charge exerts on another of like or opposite charge. We often visualize electric field as imaginary lines sourcing from (or sinking to) a point charge like so:

In this image, the dots represent a point charge (+/- q) and the arrows represent electric field. From this image alone it is not clear how a force plays a role in the definition of electric field. This will become more apparent as we begin studying how two (or more) point charges (and the electric field associated with them) interact with each other. The explanation for this becomes more complicated and will introduce some mathematics but is necessary to fully understand electric field.

The units most commonly associated with electric field in engineering applications is volts per meter (V/m). This, however, does not give us a clear picture of the connection between electromagnetism (EM) and Newtonian mechanics. Meaning, defining electric field as force per unit charge can help us relate EM to our every day experiences.

So, then, let us play with this and see where this "force" is really coming from.

First, let us consider only one point charge (say, a positive charge). For our purposes, the picture above does not well describe the radial sourcing of electric field. Let us consider an imaginary sphere surrounding the charge, and imaginary lines of electric field being radiated in all directions:

Now that we have a better view of the charge and electric field lines, let us consider the electric field at a point on the sphere, which is a distance 'r' away from the point charge:

The magnitude of the radius vector I have defined here is equal to the radius of the imaginary sphere. We need to make sure that we are not confusing this vector as an electric field vector - they are not the same. The electric field lines above are imaginary, they just give us an idea of whether the charge is positive or negative (for now). So, to find the electric field strength precisely at the tip of the vector above (at that point on the surface of our sphere) we need to:

1. Find the amount of charge contained in our imaginary sphere
2. Divide the charge enclosed by the total surface area of our sphere
3. Divide by the electric permittivity (this will give us units of electric field)

The third step is what relates the electric field to the materials in space. If we consider the point charge in free space, the electric permittivity is a set, known value (constant). If there are other materials present, the electric permittivity constant is multiplied by another constant - the relative permittivity.

Addressing the steps above, we find the following:

1. The amount of charge enclosed in the imaginary sphere we have defined to be +q (measured in Coulombs [C]).
2. Dividing by the surface area of the sphere gives us
   .
3. The electric permittivity constant is represented by the Greek symbol epsilon (ε₀) and is equal to 8.854x10^-12 [F/m]. We now have the electric field at our point of interest:
   .
   

Doing a units analysis gives the following:

Or, from a different perspective,

So, it is shown above that given information about the charge enclosed, the electric field everywhere can be derived. Now, let us consider two charges (say, of like charge) that are placed a distance 'r' apart from each other. In this case the electric fields are opposing each other, so we imagine them to be similar to the following:

Due to the opposing electric field, a force is observed that acts upon both charges. To reason this phenomenon, consider the electric field due to the first charge (on left) at precisely the point where the second charge (on right) is located. Meaning, we will determine the electric field in the same manner as we did previously, and the 'r' in this case is the distance between the two charges.

So, the electric field due to q1 at the exact position of q2 is:

The subscript on the 'E' is interpreted as "The electric field at the position of charge 2, due to charge 1." To get the force on q2 due to q1, we multiply q2 by the electric field just calculated. So,

A units analysis confirms the result is in units of force:

 Similarly, consider the electrical field due to q2 at the position of q1.

The electric field in this case is

,

and the force on q1 due to q2 is thus

So, it can be concluded that the force felt by each of two charges is

.

This result is commonly known as Coulomb's Law. The fact that the result is positive indicates that the force felt by the two charges is repulsive. If the result were negative (meaning, one of the charges is opposite the other), the force felt by the two charges would be attractive.

This concludes the brief discussion of electric field and forces associated with it.