Ismat you are 100% correct that one refers to energy being stored in the electric and magnetic fields, but these too are real fields and capable of doing work once established. Of course externally applied em fields induce significantly larger electric and magnetic flux densities in materials dependent upon their relevant permittivity and permeability respectively.

Thus the relatively weak magnetic fields induced in and around a copper wire say may in turn induce magnetic flux densities in say an iron core some 10,000 times stronger. By careful design we may control these very strong magnetic flux densities and fields and produce electrical torque in the air-gap of an electrical machine which will drive mechanical and electrical loads of several MW or even GW.

At the heart of this are the tiny magnetic domains and their alignment of the electron orbital spin (very similar to polarisation in dielectrics) and yes whilst the actual ongoings and details at an atomic level are probably not yet fully understood and explained at a micro level, at a macro level and based on experience one may lump this into parameters such as inductance and capacitance. We also know that permittivity and permeability can be three dimensional leading to multi-dimensional Tensors which is not accounted for in many circuit simulators.

Thus whilst analogies are indeed really necessary, instrumental and very useful one must also understand and highlight limitations where applicable.

Both,Inductor and Capacitor can be treated as Time Varying Voltage Sources in A.C..

The Self Induced EMF induced in the Inductor due to the Rate of Change of Flux Linkages with the coil due to the Alternating current flowing through that Inductor Coil Opposes the Applied Voltage and Reduces the Current.Similarly,the Mutually Induced Emf induced in the Inductor Coil due to the alternating Current flowing through a neighbouring coil may be aiding or opposing the applied Voltage,depending on whether the Mutual Flux aids or opposes the Self Flux.These Induced Voltages can be treated as as Alternating Voltage Sources or as Reactance Drops(of Voltage) across the Inductor Coil.

Similarly, the Capacitor gets charged and Discharged alternately in the case of Alternating Current and the Voltage across the Capacitor can be treated as a Voltage Source or as a Voltage Drop across the Capacitive Reactance,and it affects the Current through the Capacitance.

In the case of D.C. there will be only Transient Induction or Charging/Discharging.

For D.C.Voltage the Inductance acts initially as an Open Circuit and Finally as a Short Circuit to the Current.Whereas for D.C. Voltage the Capacitor acts initially as a Short Circuit and Finally as an Open Circuit to the Current.These Properties are used in Pulse and Digital Circuits.

P.S.

Check this link.....

http://www.allaboutcircuits.com/vol_2/chpt_4/2.html

A nice classic explanation by Tony Kuphaldt... I mean something more here...

Dear Cyril Mechkov,

The linear capacitor model is an admittance: YC = IC/VC = sC or IC = sC x VC i.e. IC = C x sVC = C x d(VC)/dt i.e. the capacitor is a current source controlled by the time-derivative of the imposed voltage.

Similar an inductor may be modeled as a voltage source controlled by the time-derivative of the imposed current. VL = L x d(IL)/dt

In our SPICE programs the primary variables are the node voltages and the

impedance branch currents. All the two terminal branches are nonlinear controlled sources (controlled by imposed signal or controlled by time derivative of imposed signal).

Best regards

ERIK LINDBERG

Hi Erik! Nice to see you again! We now expect only Lutz to fill our "team":)

As usual, your thoughts about elements, circuits and electrical phenomena are original and surprising (maybe that's why we feel some hidden sense of sympathy for each other:) So I need some time to realize all this wisdom above - "the capacitor is a current source" and "the inductor is a voltage source"...

It seems the four possible combinations between the two elements (capacitor and inductor) and sources (voltage and current) are possible?

I agree with your viewpoint at reactive elements; they can be considered as functional time-dependent converters - integrators and differentiators. Here is a link from the class excercises with my students in 2004 dedicated to this topic:

http://www.circuit-fantasia.com/my-students/ske2004/classes/current-ske-4.htm

Then we created a library of elementary circuit building blocks (something like a "Mendeleev's periodic table" in analog electronics:)

http://www.circuit-fantasia.com/my-students/ske2004/intro/library.htm

IMO when we name some element with the word "source", we mean that it produces voltage (current) while in the case when "the capacitor is a current source controlled by the time-derivative of the imposed voltage" the capacitor is just an impedance element connected in series... and it acts just as the resistor in the simple current source...

Now, to increase the interest in the topic... and to show that it is important to answer this question... I want to ask you a riddle:)

What does the circuit in the attached picture represent - a capacitor, inductor or a voltage source?

Hi Cyril, lots of questions in one question. Thanks to contributions by colleagues above who answered some queries. I will only comment on your saying " inductor is a piece of wire" and your saying that it seems that something is "mystic". Let me give a simple analogy which might remove this mystic. Assume that you have 10 Km flat road of 5 meters wide. How much can the speed limit be. Let it be 80Km/h. Now assume the same road leading to the the top of a mountain Shatha (in Saudi-Arabia) where I have been once (See attached google 3D image, and believe it I was there). The 5 meters wide road is circulating the mounting in a helical way (like an inductor). How much the speed limit will be?. Why. Certainly you can't speed up on a 10 Km helical road like you speed up on a 10km flat road (of course, you have the option of crumbling down fast some 1000-2000 meters !!!). Similarly, the electrons can not speed up on a 1 meter of straight wire comparing to how speed they can attain if the 1 meter wire was wound up in a coil. In the first case, the outward forces on the car necessitates a slow down while in the wire case forces by the magnetic fields produced by different parts of the wound wire interact to lower down the rate of flow of electrons.

To explain, See also the second drawing. In the case of a straight wire (a), every part of the wire (assume of a length dx) will form around it a magnetic field that has no interaction with the nearby parts. Such magnetic fields tend to act in reverse; i.e. they are in a direction to induce a current opposing the current direction that originally produced them-basic physics).

However, in the case of a wound coil, the magnetic fields in the different parts will interact with each other. Hence in the case of 2 consecutive turns of a coil (c), the magnetic fields combine together outside the wires and cancel each other between them resulting in attraction force (i.e. all N turns attract each other and they would prefer that all turns become one turn with N times the current). Also, the in any single turn (b) opposing parts of the turn form magnetic fields that augment each other at the center of the turn while oppose each other outside it hence repelling each others (i.e. any single turn would like be straightened if given the opportunity). However, turns and parts of the wound coil are forced to stay in their destined places relative to each other (or the alternative is that turns of the coil are glued together or turns themselves are scattered away). In such situation, the magnetic fields combined act to oppose the current causing them more than in the case of the magnetic fields of the straight wire case. Hence electrons find it difficult to accelerate (decelerate) in a wound coil more than in a straight wire. Magnetic fields interaction in the coil are shown in (d). Thus a coil has higher inductance than a straight wire and a coil with a magnetic core has even higher inductance (cause the ferro-material of core inside coil provide easier path for magnetic field inside the coil producing more opposition to current changes). By the way, an inductor of high inductance is sometimes called a chock coil, it acts to chock the current. Thanks. AlDmour.

Sorry, it seems that you can't upload more than one picture/file. The picture of Magnetic Fields (belonging to previous note) is now attached. Thanks. @AlDmour

Impressive explanations, analogies and pictures, Ismat! You have deeply penetrated into electromagnetic phenomena. Thank you!

Ismat, is that you on top of the mountain :) :) :)

This is the coolest collection of inductor pictures.

The only thing I was desperately looking for was the magnetic field in a TOROID. My favorite toy :)

Also, your analogy of the mountain-climbing car is phenomenal, but, doesn't explain one thing, including all of your descriptions above. Let me try to add a few little things to your analogies, to complete them. Here is goes:

In all of your explanations, you explained the OPPOSING FORCE as a BAD thing. However, this is what stores ENERGY. So, we have to slightly modify your mountain-climbing car to reflect this ...

Why don't we add a massively big spring attached to the side of your car, and at the end of the spring, there is a 2000kg. piece of steel block. Your car also weighs 2000 kg.

CASE 1: According to your description above, on a FLAT ROAD, the faster you go, the more the 2000kg. steel block will slow you down ... So, this is almost like a RESISTOR. Velocity is CURRENT in this case ...

CASE 2: If you try to climb a mountain at 20 kph, 40kph, 80kph, the spring will open more, more, more, and will not only slow you down, but WILL STORE ENERGY. F=mV^2/R. Rotational energy is quadratically proportional to the force. This is pretty much identical to the energy the inductor stores, E=1/2LI^2. So, that force, F, will try to slow you down, but, this is not wasted energy. IT is completely stored in the TENSENESS OF THE SPRING. The faster you go, the longer the spring gets, and the more energy is stored ...

So, now, what happens if you try to slow down ? You can't !!! The steel block that is attached to the spring will try to PULL YOU UP AT THE SAME SPEED YOU WERE GOING. This is identical to what happens, if you try to remove the I from the inductor, after charging it up. It TRIES TO KEEP THE SAME CURRENT GOING.

So, even if you take your foot off the pedal, the spring will pull you up, and you will keep losing energy in the spring, and it will shrink, shrink, shrink, and you will eventually stop. But, while it is shrinking, you are climbing the mountain FOR FREE. So, the energy that was stored in the angular momentum is now being turned into increased kinetic energy in your car (i.e., moving up). This is just like what happens when the inductor turns its magnetic energy back to electrical energy .... and eventually, there is no energy left. and the car stops. But, you are at the top of the mountain.

Am I missing anything ?

Tolga, as far I understand matters an INDUCTOR is a circuit element representation of energy stored in a magnetic field and a CAPACITOR a circuit element representation of energy stored in an electric field. In either case the INDUCTANCE and CAPACITANCE is limited by the physical properties of the material subject to EM fields.

This is in line with what Erik has noted and there are various ways in which one may choose to represent such energy storage.

As Cyril has asked a CAPACITOR opposes due to the internal electric dipole moment induced by the applied electric field to the materials polar structure and which gives rise to the so called "displacement current".

In the same way an INDUCTOR stores energy in the magnetic field which too is induced by the "current" or "charge" flow.

Any such dynamic energy storage effected by an external source will be accompanied by a tendency to restore matters within the material leading to such stored energy being returned.

As such under transient conditions and when subject to externally applied fields, and even though they might be considered passive elements, they may be considered as "time varying sources" during such time as they return such stored energy as we commonly see in electrical power systems and as Erik has noted one use in the simulation of such circuit elements.

The answer to Cyril question lies in the concept of resonance where energy stored in an LC circuit is known to be time varying at a frequency dependant upon the material properties (L & C) of the circuit components. This is already a well known and widely exploited phenomenon and which supports Cyril's notion of them being able to be considered time-varying sources as Eric has noted even though they might be passive elements and materials.

Yes, Tolga, surely Google made that shot the same day I was there, who went there beside me!. It was a horrible "car riding a mountain" experience!.

I am happy that you liked my analogy. I am happy more cause you took the analogy and developed it in a fantastic way. I would call it now "AlDmour-Soyata analogy model" !. I respect what Gregory is saying :" there are various ways in which one may choose to represent such energy storage". In fact, making analogies of non-tangible worlds of currents and voltages with physical tangible worlds of mechanics is a useful tool for understanding. I think this is what Cyril is looking to explore out of his diligent questions like this. He touched on an important point of that it is not easy to conceptualize how inductance/inductors act. Generally, it is not an easy task to give an analogy explaining inductance act. For the capacitors, lots of easy to visualize, for example, the water tank model in which the height represent the voltage and the flow of water represents the current which stops when the tank is full. However, for inductance the easiest way to say is that energy is stored in magnetic field build up but this reveals no thing of the mystery. Thanks. @AlDmour.

Tolga, you're about to solve (or at least reduce) the energy problems of mankind utilizing energy instead of wasting it:) Your model represent an inductor (the inert mass) with some internal resistance (the spring). Now you have only to solve the differential equations describing the process:)

Brilliant thoughts, Gregory! I would add only one comment to the assertion that storage elements (capacitor and inductor) are sources. Maybe this concept would be more acceptable if we distinguish two separate phases in this process - CHARGING (the element acts as a load) and DISCHARGING (the element acts as a source)?

Ismat, thanks again for the fantastic photos from heaven and for the clever comments... and especially for "making analogies of non-tangible worlds of currents and voltages with physical tangible worlds of mechanics is a useful tool for understanding"... it sounds wonderful!

I would ask you only to add my name to the "analogy team":) since I have been using even simpler and human friendly analogy when explaining the odd inductor's behavior to my students. I play with an imaginary situation in which the car of a "nut" (their colleague) does not start and he asked them to push it (of course, for good payment... since his girlfriend is in the car and he does not want to discredit in her eyes:)

Regards, Cyril

Sure Cyril. Let it be "AlDmour-Soyata-Mechkov analogy model". You provided the incubator for such ideas to flourish. You deserve the largest credit. This is though that I didn't see the "nut"-car and inductor analogy point!. Thanks. AlDmour

Ismat, you are a great person, both scholar and teacher! Thanks for including me in the "triad", now I feel immortalized in the electronics history:)

In the "nut"'s analogy, the inert car is the inductor, the students are the voltage source... During the first phase, they are the "source" charging the "inductor"; then during the second phase, the moving by inertia car is the "source"... It is interesting if, after stepping the car, they continue to hold on to it:)

Why don't we make it, AlDmour-Soyata-Mechkov-Diana (ASMD) model :)

Analogies are important in this way:

Since humans store knowledge in a very HIERARCHICAL-DATABASE fashion, they learn a lot quicker if they build additional knowledge on top of what they already know. Since you have already driven in a mountain, and have seen a spring in your life, it is easy to describe another real foreign concept like the inductor by using a model based around these .. However, of course, there will be missing links in this MAPPING. Something will always be missing, or wrong.

For example, Ismat's original analogy explained the concept in an abstract form, my add-on tried to model the ENERGY STORAGE. Just like Gregory said, inductors store energy in magnetic field, and, capacitors store it in electric field. The analogy for that is, the "TENSE SPRING" as you drive real fast up the mountain is like the magnetic energy storage, and the HEIGHT of an object is another form of energy storage due to the gravity pulling you down(e.g., electric field).

The best thing these analogies can explain is this: Since I made EFFORT to tense-up that spring, I STORED ENERGY in the spring.

DISCLAIMER: I do not recommend anyone going 80 kph up a windy mountain, especially with a spring attached to the car containing a 2000kg. steel block. Please do not attempt ! :)

Tolga:)

Ismat you are 100% correct that one refers to energy being stored in the electric and magnetic fields, but these too are real fields and capable of doing work once established. Of course externally applied em fields induce significantly larger electric and magnetic flux densities in materials dependent upon their relevant permittivity and permeability respectively.

Thus the relatively weak magnetic fields induced in and around a copper wire say may in turn induce magnetic flux densities in say an iron core some 10,000 times stronger. By careful design we may control these very strong magnetic flux densities and fields and produce electrical torque in the air-gap of an electrical machine which will drive mechanical and electrical loads of several MW or even GW.

At the heart of this are the tiny magnetic domains and their alignment of the electron orbital spin (very similar to polarisation in dielectrics) and yes whilst the actual ongoings and details at an atomic level are probably not yet fully understood and explained at a micro level, at a macro level and based on experience one may lump this into parameters such as inductance and capacitance. We also know that permittivity and permeability can be three dimensional leading to multi-dimensional Tensors which is not accounted for in many circuit simulators.

Thus whilst analogies are indeed really necessary, instrumental and very useful one must also understand and highlight limitations where applicable.

Greg, here are my thoughts:

Analogies are great for INSTRUCTIVE purposes. When you are teaching someone a concept the first time, starting with analogies is very useful.

But, once you understand it conceptually, then, you can dive into formulas, theorems, etc ...

Exactly... Tolga! We, teachers, have to explain first qualitative things by qualitative means (analogies) and then - quantitative things by quantitative means (formulas)!

Fully agree.

Actually, let me throw this at you: I am claiming that, EVERYBODY learns better with analogies. Whether he is a regular dude like me, or GAUSS. I wasn't a dumb kid growing up :) but, I understood things in a very BOOLEAN-way :) It either TOTALLY didn't make sense to me, or, I got it and I could explain it to everyone.

I either totally didn't get it , and would sound like an idiot :)

or, I understood it so well that, I could explain what a QUADRATIC RESIDUE is, to my mom :) She has a high school degree !

In that sense, ANALOGIES are universal. I betcha, the same analogies can help EVERYONE, even a mathematician, electrical engineer, etc ... Not just the BEGINNERS.

Throughout a lot of my school life, primary, secondary, I would have my peers understand some concepts so quickly, and I would think that, I am not that bright :) There would be that TRIGGER that would make me understand a concept so well that, I would OWN IT for the following 100 years. It would fit perfectly on top of the neural net I built ... And, it would always be an analogy ... So, my brain was really reluctant to start a new ROOT of the tree, and, it resisted building anything on top of the existing knowledge tree unless it fit 100% perfectly.

Greg's comment about :

"ongoing and details at an atomic level are probably not yet fully understood" when it comes to magnetic storage ... is one example. I betcha, my brain is spending a lot of energy when I am sleeping to make sense out of how the inductor works :) At the atom, photon, spintronics levels ...

It was a relief for me to find that, there are a lot of students like me. I know how to handle them very well, and I know that, if they are taught the right way, they can learn things a lot faster, and learn it right. I wonder if I am the exception, or a very common learner ?.

Tolga, you are simply irresistible ... so much energy there is in you that I can feel it right here on the other side of the globe! To second you, I will say that I think only by analogies... and this allows me to see things that others do not see, can not see or do not want to see... which leads to the same result - knowing but lack of understanding of things...

I am tempted, however, to ask a question that has long excited me. All these questions about using analogies, teaching, learning, etc. belong to other humanitarian sections as Education, Science education, Teaching, Science teaching, etc. We, engineers teaching in technical schools and universities, are amateurs in this area, but we are enthusiasts... Why none of them (educators, pedagogues, psychologist...) participate in our discussions (I regularly include these topics in questions thus kindly inviting them)? Maybe they think that they have to deal only with some "pure pedagogy" - to teach pedagogues that then will teach other pedagogues... and so on and so forth... thus existing in a closed system?

Very creative discussions over a lap of couple of ours. Puzzled where to start. For naming, let it as you say Tolga." AlDmour-Soyata-Mechkov-Diana (ASMD)" ...or may I, make it, ASMADA where the extra A is for "Analogy". V. Good Tolga. Welcome to the club Gregory. It remains that Tolga go and do register the patent for us!...

For the analogies, yes, Georgy, magnetic and electric fields can be very real but mostly at the higher layer of their applications which you called probably micro /macro level. Hence still the use of analogy to fully tangible systems can be useful. We are seeking to understand how magnetic fields store/release energy while they are not seen or tangible. Cyril starts with analogies and then moves to formulas which is a good approach. Making good analogies can be useful and right to the point. I sometimes make use of them to avoid complex mathematical derivations which proves something that is either naive or can be argued with wording. Math, at the end, is one thing that tries to formulate our wordings and feelings. The complexity stems sometimes because we blinded ourselves trying to use mathematical treatment in its abstract format forgetting that we previously provided this abstract mathematical description to describe our feelings/notes. Speed is distance lapsed in unit time, but distance itself is given a meaning by us that describes what we feel when we say 1 meter for example. Similarly, unit time of 1 sec or 1 hour has some predefined feeling within us. Hence 10 m/s also describes something we feel as well. Going further, acceleration of 10 m/s^2 can further have meaning to us as the speed increase from 0 to 10m/sec in 1 sec. Assuming that a 1 Kg mass has a meaning for us, then the force of 10 Newton (which can describe how painful/harmful would be to be hit by a 1kg object accelerating in 1 m/s^2). With this development, believe me, at school time, the only feeling for what a force is was that it is a vector quantity that I need to calculate it (usually in Newtons) and include it in order to solve dynamic problems. The use of Newton's name as unit instead of kg.m/s2 probably hided lots of the meanings underpins the force concept.

If that was the case with relatively tangible worlds, then it will be easier to be caught in simply equations to calculate field strength, flux,...etc with discontinuities from lower layers of abstraction. E.g. we keep forgetting that the voltage is Energy per unit charge (with Volts is merely a name of a person same as Newton's name given to Kg.m/s2 ). We can name hundreds of the concepts based on Volts and Ampers (which is another scientist name meaning charges/unit time or Columbus/sec.) I can second to Tolga say "science is a hierarchy of concepts". I any hierarchy, communication between layers in the hierarchy should be maintained or otherwise the organization fails. Hence, the organization of science concepts can be understood if we kept these links maintained. This is the basic theme to understand concepts. Analogy, on the other hand, is some secondary tool that is useful when you/your students have built discontinuities in their conceptualization of their field for many reasons; i.e. e.g. they forgot that Volt is Energy spent to displace charges which lead to two options: either to review course on basic physics/ electromagnetics or to speed up things by making an analogy to hydraulics as Energy spent to displace water around is the pump pressure analogous to the battery Voltage. Thanks. @AlDmour

I like to please Tolga more and let him know that the behavior of his favorite Toy/Doughnut (I mean the Toroid) can be as well explained by the same concept he himeself developed. It is just we need to know that rotational kinetic energy is slightly different from the translational kinetic energy. In the first one, energy is stored in the translational movement in the pulling string spring during acceleration which acts to act against trials to decelerate during which the energy stored is released back. In the case of rotation accelerating around a hill means that you constantly keeping the direction perpendicular (tangent to circle) or else some one will crumble out. In doing so, you are storing kinetic energy due to two direction, one the perpendicular direction and one another is the inward direction (working against the centrifuge force/ remember what I said on that the loop likes to be straightened). Hence, in the case of rotation you are storing kinetic energy more than in the case of translation for the same speed. This is in analogy to the wound inductor (or Toroid) which stores more energy than a straight line, or in another terms: of higher inductance than a straight wire. Thanks. @AlDmour

One more thing to say, please, I still remember my biology teacher who used to use much analogies. He used to explain the "explained" !. This is a poor teaching style. Analogies have to be concise as possible and only when needed. Otherwise, the students either get bored or think that you are teaching them mechanics rather than electronics. Once students start asking questions on your example then this a bad sign, they probably forgot your original concept and you have gone too far in describing the analogue world. Thanks. @AlDmour.

You are right, Ismat! Maybe there is a whole science, or rather part of science (pedagogy, heuristics, psychology ...), which deals with the use of analogies...

Really, teachers should not overdo the use of analogies, although I have a tendency to do that; so I try to control myself. These are known and trivial truths (for "pure" pedagogues), but I would like to tell them from my practical perspective of a man who spent his life dealing with understanding the simple truths about electrical circuits (BTW, I have not said it .

The main (the only) purpose of analogies is to convey the wll-known ideas, notions, knowledge about familiar things from everyday life of the people to the the new and seemingly strange things in the subject (circuitry, in our case)... nothing else... So, the useful analogy should be something that is already well-known for the student; otherwise it is useless and irritating...

The analogy can be not only non-electrical; it can be electrical (a simpler circuit). So, electrical equivalent circuits can be considered as electrical analogies of electronic circuits. So, we can use analogical (usually, simpler) circuits from one electrical subject (e.g., Instrumentation) in the next electronics courses (e.g., Analog electronics, Digital circuits, Microprocessor devices...)

We can use electronic analogies of electronic circuits as well. For example, emitter, source, cathode, op-amp followers are based on the same idea but use different active devices...

(I have not said it before, but my doctoral dissertation was devoted precisely to this - developing a set of heuristic tools for processing electronic circuits)

There is nothing "mystic" about the action of the capacitor or the inductor, but they may seem that way until the way their internal physics is clearly explained.

The capacitor and inductor merely take time to react to applied voltages and their behaviour is a natural result of the reaction time. They are still passive components.

The parallel plates of the capacitor take time for the charge to build up when a voltage is applied, and it takes time for that charge to drain away. That is why they appear to resist the applied voltage. They aren't actually "resisting" it, they just take time to catch up.

Likewise the inductor takes time to build up its magnetic field using the electrical energy applied to it. When that voltage is removed, the magnetic field starts to collapse and turns back into electrical energy, which is why the inductor "appears" to try and keep current flowing through itself.

No magic involved, just physics. Making analogies that hint that the components actively oppose or resist the applied voltages is a misleading anthropomorphic way to explain the very simple and unconscious thing that these passive components do.

Physics and the defining equations should suffice for those who know basic mathematics (including differential / integral calculus).

Yes, Nicholas, both inductors and capacitors are passive elements but this does not mean that their stored powers can't be re-used acting as power supplies. The "mystic" of both capacitors and inductors can be explained by basic mathematics as Fernando is saying, however, mathematics usually comes at later stages in the development of concepts and sciences. I doubt that the Italian physicist Alessandro Volta in the eighteenth century cared much about equations when experimenting with his basic cells at the time. I doubt also that Orsted made calculations on magnetic field before noticing the sudden movement of compass needle when he moved away a current carrying conductor. Hence understanding how things work starts with notices and descriptions and making analogies is one way of describing/simplifying a phenomena. Analogies are necessary for students to grasp ideas that may look primitive to intellectuals like you. If you are in teaching you would surely know that sole use of equations is something that can't help much. It is either that the students will consider your topic as either tough (can't be described in words but equations).or impractical (as they can't see the themes/applications behind the whole lots of equations and mathematical derivations). Thanks. @AlDmour.

Ismat I agree. Physics and math came into being based on observation and enquiry. The eloquent Exponential function was derived based on evidence from measurements made (on first order systems) which appeared to have a constant rate of change. This led to the next profound development namely Euler's identity.

Maths and Physics themselves are constructed analogies to help us better understand matters. They are not absolute truth but accurate representations.

The analogy is a "bridge" between the well-known and new knowledge...

They oppose or 'resist' CHANGES in voltage (capacitor) or current. As the defining equations state. Simple for those who know the difference between v(t) and dv(t)/dt and between i(t) and di(t)/dt.

Finally, I have asked the extremely interesting question about how to create virtual electrical elements in electronics:

https://www.researchgate.net/post/How_do_we_create_virtual_electrical_elements_in_electronics_Are_they_really_elements_or_circuits