Averagesupernova said:
@tedward all this talk of electric fields seems to cloud what's really at hand. You might question how the subject can be discussed by ignoring this. I'll tell you how. The same way we do basic circuit analysis without ever worrying about the lowly electron. It simply isn't needed.
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The turns ratio of primary to secondary in a transformer does not fail, ever, as long as copper losses and loading is accounted for. So, yes, there is X number of volts dropped across each turn and to delve farther into that, X volts per unit of length within a turn. And yes, it's measurable. There is nothing to say there needs to be more than one turn total. I've gotten mixed signals from you concerning this. It's been my perception that your opinion is that turn per turn will never add up in a coil to the total voltage.
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In the Lewin and
@mabilde setup, all we really need to worry about is that we have a secondary coil with two discrete resistances wired across it. We need not worry about fields until we start trying to measure voltages on portions of the coil so as to not introduce errors into the measurements. How this is done I've already covered.
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One last exercise for our brains:
Suppose we have a transformer that has two secondary coils wound in such a way so they supply the same voltage. Many transformers are configured like this. Hook them in series so they are additive. For sake of discussion let's say each secondary is putting out 10 volts. We don't know how many turns there are and we don't care. Put in series they would together put out 20 volts. Now take and put a resistor across the 20 volts output of the two secondaries in series. Next break the connection made between the two secondaries and insert another resistor. We now have the same setup as the Lewin experiment only we have more turns. Somehow all the readings in this setup are accepted and I assume it is because it is easy for people to visualize that there is a transformer. The way of thinking changes. Also people can be sloppy with lead placement with no consequences.
(warning: Long response - even for me). A lot of the talk about electric fields was to explain how there can be no field in a conducting wire, and hence no voltage between two points along a wire. This comes up especially when trying to talk about the 'scalar potential' argument, a voltage convention that DOES sum to zero around any loop, and is cited as a justification of Kirchoff's law always holding. But I don't think you're interested in this convention. We both agree (I hope) by what we mean by voltage. V = IR across a resistor, it's measured by a voltmeter, it's the work on a charge between two points, etc. But to understand how a transformer really works, you have to discuss electric field just a bit. It is, after all, central to Faraday's law, as is the magnetic field, and shouldn't be ignored. But I'll try to only reference the field when I absolutely have to.
Apologies if I sound patronizing, but just to make sure we're on the same page, electric field is the force per unit charge on a small test charge. It's a vector that points from a higher potential to a lower potential, i.e always in the direction of DECREASING potential. This is just like the gravitational field here on Earth (as in g = 9.81 N/kg), which points toward the ground, in the direction of decreasing potential energy. The connection between voltage / potential and electric field is that a potential difference is the integral, or sum, of an electric field over the length between two points. To keep things simple, if electric field is constant, then potential difference or voltage is just the electric field times the length of the path, so V = Ed. So if one point has a potential of 8V, and another at 5V, then there must be a path between them whose electric field points towards the 5v point, and whose sum over the length is 3V. The average field here is 3V divided by whatever the length is.
So for a DC battery - measure the terminals with a voltmeter, and you are measuring the sum of the electric field between those points along the path of the voltmeter leads. Move the wires around with the ends fixed, and nothing happens, as the path changes but the voltage doesn't. It's very cool when you think about it - the field could vary over space but the sum is always the same along any path.
Enter the transformer - say it has many turns, don't care how many. When you measure the terminals with a voltmeter, you are measuring the sum of the electric field between an imaginary line between those terminals, running OUTSIDE the coil, along the path of the voltmeter leads. This is what we refer to as the voltage of the transformer, or the output. You can define the voltage of an inductor the same way, but we don't care about self-inductance for now. What causes this voltage? The changing magnetic flux inside the coil volume, and Faraday's law. An induced electric field is created in a circular pattern, which pushes charge through the coils. The more coils, the more wire exposed to the induced electric field, and greater emf, or voltage measured at the terminals.
But here's the thing - if the transformer isn't hooked up to anything, the charge has nowhere to go and piles up at the open terminals (almost instantly), one positive and one negative. These accumulated charges generate their own field, pointing from positive to negative, in the OPPOSITE direction through the coils. The charge builds up at the ends exactly enough so that the induced field is canceled out in the conducting wire itself. It has to, otherwise more charge would flow until it did balance. This happens even if the transformer is hooked up to a load - the charge just builds up at the load resistor but current still flows, the same way a crowd forms when many people are trying to enter through the same narrow doorway. Either way, the electric field is zero IN the conducting coils. So if voltage is measured THROUGH this wire path, the sum should be zero. So we've got a conflict in measurements.
Now the way you measure this non-voltage very important. If you put the voltmeter on adjacent turns, you will measure one turn worth of emf. If you measure 5 turns apart, measuring on a path outside, you'll get 5 turns worth of emf, like you described. But to get an accurate measurement along this path, you would have to measure a small angle on the same turn - say a quarter turn, with the voltmeter outside. Your voltmeter should say zero. Measure 4 of these quarter turns in succesion, moving the leads as you go, and you'll still get zero. But as we said, measuring from point to the same position on the next turn, and you WILL get an emf. Why? Path dependence. You can't avoid it.
Think of a wire loop with a break in it. If you put your leads on opposite ends of the break, your measurement circuit, even with twisted leads, has a circular area for flux to pass through and you'll get an emf. But if you put the leads almost next to eachother on the same side of the break, no flux passes through your loop (remember the voltmeter and leads are outside the loop), so no emf. Adding up many of these small measurements around the circle, and you get zero. This happens on the whole coil as well. Measure top to bottom, you get the full emf, as the flux passes through every turn. Measure adjacent points on the same turn one at a time, in such a way that no flux gets through, and the sum of all the readings will be zero. It's an awkward and cumbersome way to measure the voltage (which is why you will rarely see anyone try it) but that's what we mean when we say the voltage of a path THROUGH the coils.
It's a lot easier to perform / visualize with only one loop, that doesn't cross itself, lying in a plane. It's basically the Omega shape - a circle with two connected feet that connect to the rest of the circuit. If there's a changing flux through the circular part, and you hook up a voltmeter underneath, connecting the feet, completing the circle, you will measure the emf. This is because your measurement path includes a full circle for flux to pass through. But now hold your voltmeter above the circular part, in the same plane, connected to the same two points. Now your measurement loop is a crescent shape, and it lies OUTSIDE the circle. No flux passes through this crescent, so you will measure no emf. That's path dependence. The difference in measured voltage is due to the fact that the Induced e-field is non-conservative: loops don't have to sum to zero, and different paths can give different measurements.
Lewin's circuit is basically the same idea, but now the entire circuit is the single loop transformer. Now we have the resistors as a load, whose total drop is equivalent to the induced emf. If you measure the wire path, making sure your voltmeter loops never allow any flux to pass through, You'll measure only the resistor's drop which is non-zero, and equal to the emf passing through the entire circuit. Now, draw a weird path that goes through the resistors around the loop, but doubles back around the flux in the middle , returns to the wire and completes this C-shaped loop, the voltages will add to zero, as this new section of the path subtracts out the emf from the flux. Essentially measuring the voltages across the left and right sides does the same thing - different paths, different voltages.
For your proposed circuit model with the two transformers, it works out to the same thing. Say one's on top and one's on the bottom. Let's replace them with single loop transformers, like the omega shape, and make the emf whatever you like. The top one is an omega, the bottom one is an upside down omega, and they each have changing magnetic flux passing through them. Any loop around the whole circuit that connects the feet of the omegas will measure the transformer voltages, and everything sums up to zero. This is because this loop does not include any of the flux. But measure a loop that follows the actual wire path, and you will measure the total emf. This is because the entire loop has flux through it, and therefore has a net emf. Again this is better to see in a diagram, I will try to post some when I get a chance.
