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## Main Question or Discussion Point

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?

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?