Why does change in magnetic flux decrease as turns increase?

Question

Let us consider the primary coil of a transformer. As we apply a constant voltage (say 240V) over the primary coil, and vary the number of coil turns we should expect to see the below relation (rearranged form of Faraday's Law of Induction)

$$\frac{\varepsilon}{N}=\frac{\Delta\phi}{\Delta t}$$

So it must be that as we increase the number of turns on the primary, the change in flux would decrease. This is kind of an unintuitive result. With a constant voltage and more turns in the solenoid I would expect it to produce more flux?

Answer 1

This is a good example of how physics is not pure math. You need to understand what the equation describes.

If you have a coil of $N$ turns with the flux through the coil changing at a constant rate, then the induced EMF caused by this is given by

$$\epsilon=\frac{\Delta \phi}{\Delta t}N$$ where $\Delta\phi$ represents the change in flux through just one turn. Therefore, we see that the more turns we have, the larger the induced EMF is. This is important. A changing magnetic flux induces an EMF.

Now, if we want to use your version of the above equation, we should take it to mean the following: If I increase the number of turns in my coil, then in order to get the same EMF I would need a smaller change in flux through one turn per unit time. Notice that I did not say your equation means that if we hold the EMF constant and then add more turns the change in flux decreases. This is because this is not a good physical picture to have, even if it true mathematically.

The problem is that you are thinking the opposite way: that the induced EMF per turn determines the change in flux. This is not the case. There is no induced magnetic flux from the EMF. You have to be careful with how you interpret your equations when you start rearranging variables.

A contrived but similar example would be to say that since $v=\frac xt$, this must mean that as time goes on, velocity always decreases. Of course this is not true. At a constant velocity the displacement also changes with time, but it just goes to show how you need to know what is physically happening. You can't just take your equation and use something that is mathematically true to determine your physical interpretation.

Answer 2

The coil of wire is an inductor, so it obeys the equation $ \varepsilon = L \frac{di}{dt} $ in which $L$ is directly proportional to the number of loops and the change magnetic flux is directly proportional to the change in current. Since the emf is held constant, increasing the number of loops must decrease the derivative of magnetic flux. To understand why this is the case, you need to understand that inductors oppose changes in current (and thus magnetic flux) analogous to how resistors oppose current, so increasing $N$ will create more "resistance" to changes in magnetic flux.

Answer 3

As The number of turns increases the flux density decreases due to the fact that the formula only takes into account that you’re only winding it one way and back. If you (according to my experimental data) were to wind a solenoid over a ferromagnetic core in only one spot then the flux density would be greater which according to my experimental data does makes a greater magnetic motive force or MMF.