Most people would probably be bored with this converation by now. Lucky for you I like to hear myself talk, at least when it comes to certain subjects.
I can think of three ways to attack the problem, each spelled out below.
First and easiest – consult an expert. We have the expert words of Zahn and there is no doubt what he says on this exact question. Even if you’re not sure of the details of how to prove it to yourself, you must know that Zahn didn’t just make a mistake when he specifically said (paraphrasing) lots of people get confused about implications of switching, but there is no voltage induced when switching with stationary leads and static flux pattern.
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Second way – intuition based on the physics. I suggest maybe if you study the various incarnations of Faraday’s law you’ll see they all involve some motion or time variation and in fact are interrelated. This is something you should do on your own.
But I’ll throw out some thoughts for consideration which I freely admit are not rigorous proof but I hope may provide some intuition. Consider two cases of coil moving next to magnet and magnet moving next to coil. In the case of loop moving next to magnet, we explain the force using F = q V x B. But in the case of magnet moving next to loop, velocity is zero and we explain the force as induced voltage due to rate of change of flux linked by the loop. The two problems can be identical except for different choice of inertial reference frame viewed in and by necessity the predicted force is the same. This gives us some idea that motion and time rate of change are the basic common ingredients and they are related to each other. Do a third experiment where there is no motion, but time rate of change of the flux in the plane of the loop. Here we have no obvious motion. But we are looking only at one plane and that’s all the loop knows about. We could probably create that same time varying voltage in the place of the loop by setting up (with permanent magnets) a 3-d spatially-varying flux pattern moving which is static in its own reference frame but moving with respect to the loop. It’s somewhat loose and not exact but imo goes to extend the intuition of the unifying factors underlying all induced voltage: motion or time rate of change.
Third Way: Attempt at math application of Maxwell’s law of induction.
I would challenge you to find a calculus textbook that will tell you how to you find the derivative of a surface integral whose defining perimeter contour spontaneously morphs to something completely different. Go get your calculus books, I’ll wait....
Didn’t find anything? That’s because it’s not there. There is no way to solve that because it’s an invalid changing contour. I don’t know how to talk about a proof that doesn’t exist. Let’s instead talk about some similar but realistic time varying contours leading to a scenario that resembles your switching and see how they act.
See attachment figure.
We have in Case 1 a voltmeter in circuit with insulated wire which is roughly broken into two loops by a rubber band that keeps it physically together (but not electrically contacting) in the middle. The wires are located in a plane just above a permanent magnet which produces flux density B over area A in the plane of the wire, in a location that links loop 1 but not loop 2. (if you want a more realistic situation to explain why no flux in loop 2, then consider instead that the wires are in the plane of the airgap of a C-shaped permanent magnet / core so that there is flux in the airgap but no significant flux elsewhere in the plane).
The flux linked in the circuit for case 1 is 1*BA (since loop 2 contributes nothing).
Now see case 2. To get from case 1 to case 2 we grab the right side of loop 2 and pull it straight left. Now we have two turns with opposite polarity, the flux linked by the circuit is zero. Note that in getting from case 1 to case 2, loop 2 had to cross over the area of B (there would have been voltage induced in the process).
Now see case 3. To get from case 1 to case 2, we grab the right side of loop 2 and twist it and pull it to the left. The result is two turns of same polarity. The flux linked by the circuit is 2*A* B. Note that in getting from case 1 to case 3, loop 2 had to cross over the area of B (there would have been voltage induced in the process).
Let’s go from case 2 to case 3. We have to grab the left side of loop 2 and flip it. Two sides of loop 2 will cross through B and there will be voltage induced in the process.
What you are proposing is putting a switch on the right side of the figure and transitioning direclty from 2 to 3 by use of switch. No wires cross over the area of B and no voltage is induced. Is it conflict with Faraday? Nope, this is not an evolution of the contour, it involves opening and closing circuits to switch between two completely different contours in an unnatural way which is not a simple evolution of a single contour.
By the way, I should have used “turn” vs “loop” in this figure, but I’m not going back to change it.
About the last formula Vcoil(t), I agree and in fact, I have set to zero the resulting emf
That should be the answer. Zero times any finite number of turns will still be zero.
I think that is undeniable that F= B*A1 and if A=N*A1... F=B*N*A1. This F exist even if there are not electron around!!! The flux is line of force of Magnetic field concatenated with an area A (contour or surface delimited by...).
Well...for t=0 we have F'=x; for t1=1ms we have F"=x+y (where y can be positive or negative). Please consider that my contour is always the same (I don't move the measurement point). What I do is reversing the second turns direction (refer to explanation of my experiment in previous post).
Your contour is not always the same. You have disconnected and reconnected to new wires. It is a spontenaous change in circuit topology, it is not a continous evolution of a given topology.
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(2B)+(2B)' ?
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