The total flux ($\Phi$ ) through an solenoidal inductor of length $l$ and $N$ turns is proportional to the current through the inductor and the inductance $L$ of the inductor according to
$$\Phi =L \cdot I $$
$$\Rightarrow \Phi =\frac{\pi r^2\mu_0 N^2 }{l}\cdot I \tag{1}$$
Clearly, in this case, if we double the amount of turns (whilst holding the length and current constant) we increase the total flux by a factor of 4. This is because the inductance of a solenoid depends quadratically on $N$.
Now I have just learnt about the concept of reluctance (denoted $R$) and magnetomotive force (denoted $m.m.f$) where $m.m.f$ is defined to be $m.m.f \equiv N\cdot I$. These two quantities are related to each other by the total flux via the formula
$$m.m.f = R\cdot \Phi \tag{2}$$
Now applying the above formula (eq 2) to the case of the solenoidal inductor, if we hold the current constant but double the number of turns (N) of the inductor, the $m.m.f$ only doubles (since $m.m.f \equiv N\cdot I$). But according to equation (1), the total flux must quadruple since $\Phi \propto N^2$. Thus in order for equation (2) to remain valid, the reluctance must necessarily half. That is, in order for equations 1 and 2 to be simultaneously valid, the reluctance must be inversely proportional to the number of turns so that $R\propto \frac{1}{N}$. However, everywhere I look, I find that reluctance is always given by equations that are independent of $N$. For instance, wikipedia states that $R=\frac{l}{\mu A}$. How can this be? How can equations 1 and 2 be simultaneously true whilst reluctance is independent of $N$? Any help on this issue would be most appreciated!
Electromagnetism – Does Inductance Depend on the Number of Turns in a Solenoid and Reluctance?
electric-circuitselectric-currentelectromagnetic-inductionelectromagnetismmagnetic fields
Related Solutions
I agree that this seems mysterious on first meeting.
Multiplication by $n$ is shown to be correct by something called Stokes's Theorem, which you won't have met yet, and which lets us translate the basic equations of electromagnetism (called their Maxwell Equations) between their local and "spread out" forms.
But at an easier level, think of a very long, time varying flux tube, like the one I've drawn below.
Now imagine separate loops around the flux tube. Each of these loops has flux $\phi$ through them, and each separately will have an EMF $-\frac{{\rm d}}{{\rm d}\,t}\phi$ between their ends.
You link them together in series, and it is just like linking voltage sources, or batteries together in series. The total EMF across the series cells is $-n\,\frac{{\rm d}}{{\rm d}\,t}\phi$. Now imagine bringing the loops together in a neatly wound solenoid. Electrically nothing changes: it doesn't matter if you put long, zero resistance wires between batteries in series, their EMFs still add in exactly the same way.
Since the whole purpose of an inductor is react to a change in current with a back EMF, the physics doesn't change if you state the equation $V = -n\,\frac{{\rm d}}{{\rm d}\,t}\phi$ as $L I = -n\,\phi$; you simply differentiate this one to get the form I just derived.
The trouble arises, I believe, because you're considering the field to be due to a current in a wire of zero thickness, so the flux density approaches infinity as you approach the wire, and this makes the flux integral blow up. If you consider current spread over a finite cross-sectional area of wire this problem goes away. There are other mathematical difficulties, of course, but they can be handled by approximation methods, and you'll find formulae for flux due to a circular loop on the internet.
Best Answer
Reluctance is independent of the number of turns. Your confusion results from the fact that people use the word flux to refer to two closely related but different quantities. Recall that magnetic flux is the integral of $\vec B$ through a surface. To speak of a flux, we must first define what this surface is. In the analysis of magnetic circuits, this surface is usually the cross section of a relevant part of the magnetic circuit. In this case, this would be the cross section of the inductor winding with area $\pi r^2$. With this definition, the flux is $\Phi_1 = \pi r^2 B$ where $\vec B$ is the magnetic field within the coil (assumed uniform here for simplicity). From the reluctance of the magnetic circuit, we have $\mathcal F=\mathcal R\Phi_1 $, where $\mathcal F=NI$ is the mmf and $\mathcal R$ is the reluctance.
The other magnetic flux we may speak of is the one through the winding, more relevant for electrical analysis, e.g. through the definition of inductance. Consider the loop formed by the coil, including whatever external circuit it may be connected to. This loop is the boundary of a surface, and it is this surface through which we calculate the flux to define inductance, because this is the flux whose rate of change gives the electromotive force along the loop. It is not necessarily easy to visualize this, but each magnetic field line in the inductor core intersects the surface bounded by the loop $N$ times, so the flux is $\Phi_2=N\pi r^2B=N\Phi_1.$ Inductance is calculated as $L =\Phi_2/I$.
Putting everything together, $$L=\frac {\Phi_2} I = \frac {N\Phi_1} I = \frac {N^2} {\mathcal R}$$ which is proportional to $N^2$, as expected.