Flux and flux linkage through a solenoid

Question

If there is a solenoid of length L , radius R, number of turns N immersed in uniform magnetic field B with axis of the solenoid being parallel to the field, then what will be the flux linkage and the flux through the solenoid ? I am a bit confused with the terms , I think the flux through the solenoid should be BN(pi)(R^2) . But then my book says that the magnetic flux through the solenoid is B(pi)(R^2) and the flux linkage is N times the flux through the solenoid .The book is saying that the flux through the solenoid is B(pi)(R^2) not BN(pi)(R^2) . Please could you help me clarify this what is the flux through the solenoid and what is the flux linkage ?

Answer 1

You might be confused because the flux through one solenoid coil is $BA$, where $B$ is the magnetic field and $A$ is the area, so if you stack a bunch of coils then it should be $N*BA$.

However, this is not the case. What magnetic flux concerns itself with is how much magnetic field passes through the coil. Think of it this way:

Consider a hose that is running. We will let water be the magnetic field in our example. Now, I put a ring with the same radius as the opening of the hose, and I ask what is the flux of water, that is how much water is passing through the ring? Let's say it's a certain amount W. Now, to the first ring I add a few more rings, like a solenoid, and I ask, how much flux of water is passing through it? It's going to be W, since adding more rings does not increase how much water is passing though, and analogously, this is what magnetic flux is quantifying -- how much magnetic field is passing through.

What magnetic flux linkage is quantifying is, in a sense, how much the total impact the magnetic field has on the coils, that is, if the magnetic field is passing through $N$ coils, the effect of the field is multiplied $N$ times, and that's why we quantify it as, in this example, $N*BA$.

I hope that you do revisit the definitions after reading the analogies. They tell you what is and what isn't. The analogies can only go so far as to describing the physical quantity in question.

Answer 2

Perhaps you are confused by the equation for EMF induced in the solenoid by extension of Faraday's law:$$\varepsilon = -N\frac{d\Phi_B}{dt} $$ The total EMF induced in the solenoid is the sum of the EMFs induced in each turn of the solenoid. This is why the change in magnetic flux over time for every turn,$\frac{d\Phi_B}{dt}$, which gives the EMF induced in that turn of conductor, is multiplied by N, the total number of turns.

But in the case of magnetic flux through a solenoid, the equation is $$d\Phi_B = B\cdot dA$$ This is because the magnetic flux is determined for a cross section of the solenoid, and the number of turns N, a parameter in the perpendicular space-direction to our plane of our reference (which is the cross section of the solenoid) does not have to be considered in this context.

Answer 3

I think the flux through the solenoid should be BN(pi)(R^2) .

No, the book is correct, the magnetic flux (the amount of magnetism) going through the solenoid is the magnetic field strength $B$ multiplied by the area of the solenoid $A=\pi R^2$. The magnetic flux linkage is the amount of magnetism multiplied by the number of turns of the coil. The point of the flux linkage is that a current induced in the solenoid by a changing magnetic field will be proportional to the flux linkage, not the flux.