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For this above question, how is even a back emf induced in the circuit because of the coil. Doesn't Faraday's Law say the a change in flux threading an external coil will induce an emf. So then how can the coil induce a back emf in its own circuit?

Below is the solution. Which looks right if a back-emf could be produced in the first place.

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  • $\begingroup$ The problem is not well stated. You are given a graph of $V$ over time, but the problem does not say where $V$ is measured. The circuit has four nodes; between the battery's negative terminal and the resistor, between the resistor and the switch, between the switch and the coil, and between the coil and the battery's positive terminal. Voltage is always measured between two nodes, but the problem does not specify which two. $\endgroup$ Sep 23 '17 at 15:38
  • $\begingroup$ Furthermore, the $V$ vs. time graph is unrealistic no matter which two nodes you pick. The problem asks you to show how the "back EMF" influences the current, but no such thing is shown on the graph. Let's suppose that the graph is supposed to show the voltage between the two ends of the coil. If that's the case, then when the switch opens, the voltage will not just go to zero, it will go way past zero,... negative... That's what "back EMF" looks like. $\endgroup$ Sep 23 '17 at 15:47
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The circuit has an EMF $\mathcal E_0$ in the form of the battery of $12~\mathrm V\,.$

The current $I$ through the circuit is not constant right from the beginning.

It was zero when the circuit was open.

After a sufficient amount of time-interval, $I$ would attain a steady value $I_0\,.$

Prior to that $\dot I \ne 0\,.$

It can't go from $0$ to $I_0$ at an instant.

So, as the current $I$ changes at the rate $\dot I(t),$ there then arises the induced electromotive force which would tend to run the current in such a direction so as to oppose the flux change.

So, applying the law of conservation of energy, we get $$\mathcal E_0 + \underbrace{\left(-~ \mathrm L~\dot I(t)\right)}_\textrm{Back EMF} = RI(t);\tag I $$ assuming the direction of the current driven by the battery as positive.

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  • $\begingroup$ But how can a current in a coil induce an emf in the same coil? Doesn't Faraday's Law specifically say the magnetic field has to be external to the coil? $\endgroup$
    – Nanoputian
    Oct 27 '16 at 9:51
  • $\begingroup$ Faraday-Maxwell Law only says this: $\textrm{curl}~\mathbf E = -\partial_t ~\mathbf B\,.$ Is it that how electromotive force acts on the same coil which produces it bothering you? $\endgroup$
    – user36790
    Oct 27 '16 at 9:54
  • $\begingroup$ Sorry, I haven't learnt that notation. It is just that I have learnt that if you have a square coil for example, the produced magnetic field of one side of the coil can't effect the opposite side of the coil because they are part of the same circuit. Isn't that what is happening in this question kind of? $\endgroup$
    – Nanoputian
    Oct 27 '16 at 12:26
  • $\begingroup$ Why do you think a current element can't get affected by the fields created by the other current elements in the same coil? It's not some mysterious creature which has consciousness. They will indeed get affected by the field created by the current and charge density at other parts not at that moment, but at retarded time. $\endgroup$
    – user36790
    Oct 27 '16 at 14:46
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    $\begingroup$ @Nanoputian, Sure, a part of the current element will get affected by the fields due to the other current elements; but you have to consider other non-electromagnetic forces also; after-all no charges are allowed to go out from the, say, wire. Also, you have to consider the whole coil; does there exist a net force from such interactions? Well, think about it. $\endgroup$
    – user36790
    Oct 28 '16 at 3:11
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Hy there! I am searching for an answer in those questions myself.

Without any bad intention i would like to point out, that to the question :"So then how can the coil induce a back emf in its own circuit? " the right answer would be: It doesn't!

Even if by some miracle it does violate the physic laws how electrostatic forces work, no reply here did answer the question.

There is an observed and measured behavior of electricity in coils, then many wrong conclusion were made and absurd theory emerged. With the wrong words, even the relative 'simple' and logical science, physic will become incomprehensible. (For example, Counter electromotive force in generators and backemf in a DC PULSED coil are not the same.) The same goes to the word 'induction', we define a certain way to make an electron move - induction.

Induction are between a coil changing magnetic field, or a static magnetic field moving and any electrical conductor nearby, where electrons are forced to move. In a coil connected to an electric EMF source the moving electrons are the cause and the magnetic field is the effect - period.

What they call a coil's induction or selfinduction (L)(which is a misnomer i belive) is none other than an effect of the physical and geometrical propertys of the coil which affects the movement of the electrons (and consecvently the generated magnetic field). Everything in the universe is particle movement and interaction, be it smaller or bigger. Electrons included. They are complex particles and as such have a physical form, shape, and of course inertia.

'...The circuit has an EMF E0 in the form of the battery of 12 V. The current I through the circuit is not constant right from the beginning. It was zero when the circuit was open. After a sufficient amount of time-interval, I would attain a steady value I0. Prior to that I˙≠0. It can't go from 0 to I0 at an instant. So, as the current I changes at the rate I˙(t), ...'

so far so good but then it happens:

'...there then arises the induced electromotive force which would tend to run the current in such a direction so as to oppose the flux change...'

'ARISES'..?? What the hell, its not the Lich King from WOTLK! Where the hell is this current? Even this simple diagram show a fine decaying current strenght, or should i say electron movement after disconnection (and we miss the rising voltage across the coil with reverse polarity, as the current stop, that will be the peak of the voltage).

In this case what he calls this phenomenal electromotive force is none other than the electrons in the coil gathering at one end of the coil. The electrons in the coil were forced to a circular motion with great centrifugal force making them move on and above the surface of the wire. At the moment of cut off, nearly all the electron will stop almost immediately in the straight wire, but in the coil, because of their given inertia they can move with a much more free mean path on the surface, and stack for a moment at one end of the coil causing an electron surge there and a deficit/absence at the other end (or generaly in the whole coil), what you can measure. Of course if it is not directed elsewhere and sufficiently large enough, it will tear your break(switch..etc) in the circuit along with the electric components. If not, it will just settle in a while in the coil (after all its just a shortcut with negative charge at one end and 'positive' at the other in the moment the electrons movement cease).

It is an amazingly twisted way to say for a bunch of still moving electron in the coil after disconnection, that gather at one end leaving a void at the other, that a mysterious electromotive force a.r.i.s.e.s. and run a current (so there are more than one current in the coil?) to oppose the flux change, meaning strengthening the decaying current, maintaining the magnetic field, what is the product by the way of the moving electrons, which are stopping now.

From this you can see that any sentence stating any current, that want to act against the flux change is a badly interpreted misconception. The collapse of the magnetic field is the stopping of the electrons, it doesnt have any or little influence on the current itself (nor in the 'generating' voltage spike), you can't hack the cause-effect relation, no matter what game you play with words. And just as bad the popular supression of this so called 'backemf'. ANYTHING that is not radiated or lost in some other form can be harnessed back from the coil cutting your operating loss by anything between 50-90%, with a diode AND!! a storage element (capacitor, battery..etc). Like a car which have a 'regenerative braking'.

Needless to say, pulsed DC have numerous advantage over AC for inductive loads, which mostly act against itself for the above mentioned reasons (electron movement/inertia) the higher the frequency (called impedance). It is the worst design/idea ever to work coils with AC. Like throwing out more money, the harder you work. There would be no need for power factor corrections and your punny houshold electrics wouldnt suck 2000W from the grid, while doing 800W or less useful work... (Have to say here AC, specially high frequency can have its uses, for example the free resonance of a (R)LC circuit)

Conclusion: - Coils never induce anything in themselves, and if you dont belive me grab a scope or any meter and measure what you belive an increasing current in a coil generate ahead or after itself working against the change at any part of the coil, and post it please.

Tip: A correctly constructed and operated coil is a current amplification device. Even the 'sample answer' diagramm showing the current with red line in the question above show extra current, modifying the measured current in the circuit. (I wont tell where, its your homework :)

With correct physical-geometric form and engenieering you can have a vast variety of effects for the same electron movement (current). This in itself should be a very exciting major in scientific study.

If you are open minded and looking for answers what you can't get from those forums, or from scientific teachings, which taught the people answering here ,(curios about what are the magnetic fields, electrostatic forces and how they work in general) i recommend you a book titled: The Ultimate Reality by Joseph H. Cater. Just read the part you are interested, and if more question pop up than read that part too.

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