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

Question

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.

Answer 1

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 $0<k<1$ 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.