When I was taking a trip down memory lane, I recalled a conversation with one of my friends which goes like this:
He: Why they have to teach the courses that I will never use in my life?
Me: Courses like?
He: Electromagnetics. (Electromagnetics is a course where we study about Maxwell’s Equations)
Me (Surprised): If you do not use electromagnetics what do you use?
He: Other simpler material like Kirchhoff’s Laws.
Well, this made me think, and I started to have a monologue.
Is electromagnetics the useless course that they teach in Electrical Engineering?
No, The knowledge of electromagnetic is very important for electrical engineers. The job of an engineer is to solve real-world problems. We solve such problems by renovating the problems into the simplest version possible. We do not use Maxwell’s Equation directly because they involve partial differential equations which are complicated to solve. And as an engineer, we diminish complications. Instead of solving Maxwell’s equations, we modify them into the simplest form and solve them.
How do we convert the law of electromagnetism into the simplest form?
If somebody asks you to determine the acceleration of the object and provide you the value of force on that object, what will you do? Well, you simply question him the value of mass, and you divide the force by mass. The enquired value of mass is concentrated at a point. We call this point a center of mass. In physics, we call this a point mass discretization. We have lumped the entire mass of the object in one point rather than distributing it throughout the object. We call such parameters a Lumped Parameters.
If somebody asks you to determine the current through a resistor and provide you the value of voltage, you divide the voltage by resistance to get current. In electrical circuits, we lumped the circuit parameters (In this case resistance). We call such parameters a Lumped Circuit Parameters.
How does discretization simplify Maxwell’s Equations?
Discretization does not simplify Maxwell’s equation. But discretization is the result of simplifying Maxwell’s equations. Consider the object, suppose a general element (Whose behavior is unknown ), as shown in the figure. If we define the current through the element and the voltage across the elements (As electrical engineers, we are always interested in voltage and current), we can define the behavior of that element. S_{a} is the surface through which current enters the element, and S_{b} is the surface through which current leaves the element. The behavior of that element is unknown. As a result, we have to apply Maxwell’s equation.
J is called current density.
I_{a}-I_{b} = \frac{\partial q}{\partial t}This equation implies that the current leaving the element subtracted from the current entering the element is equal to the charge build over time. We still can simply the above equation by considering an element where the charge buildup over time is ZERO.
I_{a}=I_{b}This equation implies that there is no charge buildup in the element. From this principle, we can lump filament heater into the resistor, light bulb into the resistor, and so on.
How do you define voltage across the lumped element?
Until now, we have learned to lump the resistor. Let us assume the circuit containing resistors, as shown in the figure below.
Let us apply the Maxwell’s equation in above circuit.
\oint E \,dl = -\frac{\partial \phi}{\partial t}If we simplify the problem by considering the term in RHS is Zero (i.e. rate of change of flux is zero).
\int_{CA}^ E.dl + \int_{AB}^ E.dl + \int_{BC}^ E.dl = 0\\ V_{CA}+ V_{AB}+V_{BC}=0This equation is the simplified version of Maxwell’s equation, which implies the sum of the voltage across the elements in a closed loop of a circuit is zero.
Why we study Maxwell’s equation when there is a simplified version?