Properties of ideal solenoid: postulates or derivations?

[DavideGenoa]

My text of physics, Gettys's, proves that the magnetic field on the axis of a solenoid, in whose loops, of linear density $n$ (i.e. there are $n$ loops per length unit), a current of intensity $I$ flows, has the same direction as the loops' moment of magnetic dipole and magnitude $\mu_0 nI$ by using the fact that the magnetic field on the axis of a loop, whose radius is $R$ and whose moment of magnetic dipole is $\mathbf{m}$ (with $\|\mathbf{m}\|=I\pi R^2$), is$$\frac{\mu_0\mathbf{m}}{2\pi (d^2+R^2)^{3/2}}$$where $d$ is the distance from the centre of the loop, as we can calculate by starting from the Biot-Savart law. By considering the continuous layer of an infinite amount of loops composing the solenoid, with every loop flown though by an "infinesimal current" $nI ds$, the magnitude of the magnetic field on the axis of the solenoid result is given by the integral $$\int_{-\infty}^{+\infty} \frac{\mu_0nIR^2}{2((x-s)^2+R^2)^{3/2}}ds.$$

Gettys's Physics says that an ideal solenoid has a constant magnetic field at any point inside and a null magnetic field outside.
In order to show (without mathematically proving it with rigour) that the field is constant inside the solenoid, the book uses intuitive arguments to say that its direction is the direction of the moment of magnetic dipole of the loops everywhere and uses Ampère's circuital law to measure its magnitude as $\mu_0 nI$, but intuitive arguments are not mathematical proofs, of course, and, moreover, I have never found a mathematical proof of Ampère's law from the Biot-Savart law for a superficial distribution of current.

Are the constance of the field inside and its nullity outside mathematically derivable by using the infinite loops model or are they part of the definition of an ideal solenoid?

 

[Charles Link]

My text of physics, Gettys's, proves that the magnetic field on the axis of a solenoid, in whose loops, of linear density $n$ (i.e. there are $n$ loops per length unit), a current of intensity $I$ flows, has the same direction as the loops' moment of magnetic dipole and magnitude $\mu_0 nI$ by using the fact that the magnetic field on the axis of a loop, whose radius is $R$ and whose moment of magnetic dipole is $\mathbf{m}$ (with $\|\mathbf{m}\|=I\pi R^2$), is$$\frac{\mu_0\mathbf{m}}{2\pi (d^2+R^2)^{3/2}}$$where $d$ is the distance from the centre of the loop, as we can calculate by starting from the Biot-Savart law. By considering the continuous layer of an infinite amount of loops composing the solenoid, with every loop flown though by an "infinesimal current" $nI ds$, the magnitude of the magnetic field on the axis of the solenoid result is given by the integral $$\int_{-\infty}^{+\infty} \frac{\mu_0nIR^2}{2((x-s)^2+R^2)^{3/2}}ds.$$

Gettys's Physics says that an ideal solenoid has a constant magnetic field at any point inside and a null magnetic field outside.
In order to show (without mathematically proving it with rigour) that the field is constant inside the solenoid, the book uses intuitive arguments to say that its direction is the direction of the moment of magnetic dipole of the loops everywhere and uses Ampère's circuital law to measure its magnitude as $\mu_0 nI$, but intuitive arguments are not mathematical proofs, of course, and, moreover, I have never found a mathematical proof of Ampère's law from the Biot-Savart law for a superficial distribution of current.

Are the constance of the field inside and its nullity outside mathematically derivable by using the infinite loops model or are they part of the definition of an ideal solenoid?

I actually have proved by the Biot-Savart law that the field inside an infinite solenoid is uniform everywhere (and points in z-direction). It was a somewhat lengthy process setting up the integrals and it required complex variable analysis to evaluate the result. I would be happy to supply you with the details if you have further interest. The integrals also gave a null result outside the solenoid. Incidentally, Biot-Savart is an integral solution to ampere's law in differential form. Ampere's law in integral form is an alternative integral result using Stoke's theorem.

 

[DavideGenoa]

@Charles Link I heartily thank you! Of course I am interested! I study by myself and don't attend university courses yet, so I planned my studies in an order according to which I have studied some elements of complex analysis before elementary physics, and therefore I hope I will be able to follow your proof. Gettys's calculates the magnetic field generated by a loop only on its axis, while I think that we need the general expression for the magnetic field generated by a loop, which we should then integrate...