# Inductance - a better analogy than the fact that it is similar to inertia

This question is to get a feel and good definition for what inductance actually is.

I have read about the analogy where inductance can be compared to inertia-that the flux through an inductor resists change.

But, while this analogy is fine for understanding induction as a standalone concept, I find it difficult to actually use this reasoning in problem solving. What I mean when I say that it is fine for understanding as a standalone concept is that, if someone asks me, "What is induction?" then I can answer, "It can be said to be like inertia-except that it resists change in flux."

Now , one problem where I can show how this analogy is not helpful is :

If I have been given two loops with some current going through them in the same direction, then what is the net magnetic energy of the system?

My approach: The net magnetic energy will have the energy due to the self induction of each loop. So, I have : $$\frac{1}{2}(L_1i_1^2+L_2i_2^2)$$. Now, mutual induction. How does this thing work? If using the inertia analogy, then it must resist change in flux. How exactly does it resist this change?

This is but one of the many places where I get stuck. To simplify, the main doubts are :

1. How does mutual induction affect energy?

2. Is there a better way to get the 'feel' of induction rather than the inertia example?

3. Mutual induction-I understand its formula but what exactly is it? This may seem a bit vague but, like we can get an idea for charge, mass, velocity by understanding the definition once, what is a good definition for inductance in general. Not the formula where it is the flux divided by the current but a good, solid definition like velocity is how fast an object covers a certain distance. The definition should be such that it can be applied for problem solving. As in, if I am stuck, then I should be able to start from the definition of inductance and work my way from there. Unlike now, where I am stuck wondering what part of induction is like inertia.

• What about the definition of inductance can not be applied to "problem solving"? If you want to get "a feel" for what inductance does, then you need to work with electrical circuits that contain inductances on the lab bench. Analogies will only get you so far and if you make one step further, then they will misguide you. One should avoid them and learn to use the actual definitions. Commented Oct 15, 2022 at 9:47
• See.. I am fine with one that does not rely on an analogy. A feel for the subject is appreciated but not necessary. What I would really like is one definition that completely defines induction as some 'solid' concept - the kind where it sticks in your head and you know when you are asked that, "Ah, this is the PERFECT definition. Nothing more, nothing less." You get me? Like the definition of, let's say, flux. The number of field lines passing through a surface. There are no symbols in this definition, but instinctively I can understand that it has something to do with flow through a surface. Commented Oct 15, 2022 at 12:48
• And pure feel doesn't work either. I still struggle with the definition of a field. Sure, I got that feel for it. It just comes from the word. But definition? I got no clue. Maybe if you ask me what's an electric field - force /charge, that's my answer. That is just a relation between quantities, not the stuff I am looking for here in this question. Commented Oct 15, 2022 at 12:56
• I agree that the literal definition of inductance is complex to apply to problems like high frequency circuit design. My preferred electrical engineer simplification is to think of inductance as the additional energy that must be supplied to magnetically "charge" a region of space such that it can sustain a certain current. Mutual inductance represents initial conditions, the "charging" of space by other currents that can increase or decrease the amount of energy that must be supplied by a new current to reach steady state. Commented Oct 15, 2022 at 13:26
• In addition to multiple questions, this is all fairly subjective. e.g. for point 3, why is $v=x/t$ "solid" but $L=\Phi/I$ not? And in assuming you have the same issues with capacitance? Commented Oct 15, 2022 at 18:49

This question is to get a feel and good definition for what inductance actually is.

I have read about the analogy where inductance can be compared to inertia-that the flux through an inductor resists change.

The thing about analogies is that a good analogy has three key aspects:

1. the mapping between the analogy and the original is clear and easy to remember

2. the analogy quantitatively reproduces the most important behavior of the original

3. the analogy is simpler than the original

The problem with almost all circuit theory analogies is that they fail on 3), inevitably the original circuit theory principle is simpler than the mechanical analogy. In addition, the inertia analogy fails on 1), but maybe it is just me that has a hard time remembering how the analogy goes.

So usually you are better off just learning the circuit theory concept directly, without analogy. In circuit theory the primary quantities of interest are voltage and current. So the clearest definition of inductance is: $$V = L \frac{d I}{dt}$$ I have yet to see any analogy that is simpler than that. And since $$V$$ and $$I$$ are easy to measure, it is not difficult to build up practical intuition directly.

Once we have the inductance formula we can simply integrate power to get energy: $$E= \int P \ dt = \int V I \ dt = \frac{1}{2} L I^2$$ So directly from standard circuit theory we know that an inductor stores some sort of energy. The energy can be put into the inductor at one time and then pulled out later.

A little bit of physics lets us know that this energy is stored in the magnetic field produced by the current. Now, with that small extra clue from outside of circuit theory, mutual inductance is not too difficult to understand. Magnetic fields can overlap. When the magnetic fields of two inductors overlap then it is possible for energy to be put into one inductor, go through the overlapping field to the other and be pulled out there.

For two mutual inductors the energy is $$E= \frac{1}{2} L_1 I_1^2 + \frac{1}{2} L_2 I_2^2 + M \ I_1 I_2$$ The mutual inductance parameter is $$M=\sqrt{k L_1 L_2}$$ where $$0 is a parameter that describes how much the magnetic fields overlap.

The inertia example is not very good for self inductance and I cannot see how it can possibly be used for mutual inductance. Just learn these concepts directly, without analogies. To get an intuitive feel you can solve lots of homework problems or build lots of actual circuits. My preference is for building circuits as the best way to gain intuition.