Timeline for Shouldn’t the magnetic field of a solenoid be affected by length?
Current License: CC BY-SA 4.0
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| May 7, 2023 at 20:42 | history | edited | Adam Herbst | CC BY-SA 4.0 |
making more self-contained by completing the analysis of the infinite solenoid
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| May 7, 2023 at 19:51 | comment | added | Adam Herbst | @EM_1 It's probably similar to proving the shell theorem for either gravitation or electrostatics: when you're inside a spherical shell of uniform charge/mass density, the net field is zero. Yes, you could do it using an integral of the Coulomb/Newton law for a point mass/charge over the distribution, but it's much easier to invoke Gauss' law, which says the flux of the field is proportional to the charge/mass enclosed in a surface. For a spherical surface somewhere inside the shell and concentric with it, symmetry thus implies that the field vanishes. | |
| May 7, 2023 at 19:42 | comment | added | Adam Herbst | @EM_1 If you pick a given field point and then increase the radius, the line elements of a given current loop are indeed further from the field point, but there is more length in the loop. Remember for example that the Biot-Savart law says that $d\mathbf{B} \propto \frac{Id\mathbf{\ell} \times \hat{\mathbf{r}}}{r^2}$. You are increasing both the number/length of $d\mathbf{\ell}$'s as well as $r$, and the two effects cancel. But proving this using Biot-Savart for an arbitrary field point might be impossible in closed form. | |
| May 7, 2023 at 14:39 | comment | added | EM_1 | Great answer. I have a relevant question. How can we argue or explain the independence of the field on the radius, in the case of an infinite solenoid? I thought about energy. If two solenoids of different radii carry the same current, they produce the same $B$. But if the radius is so large, the field point is far away from the current source, so $B$ should be less. But to maintain the same current in both solenoids, the battery would have to supply more energy to the larger radius solenoid. But that's an indirect claim about energy, not about the field itself. Any insight? | |
| Jan 26, 2021 at 7:47 | comment | added | ProfRob | +1 but maybe edit the phrase "Ampère's Law only applies to a solenoid of infinite length." Ampere's law always applies; it is whether it can be used simply to find the B-field that is in question. | |
| Jan 25, 2021 at 19:24 | history | edited | Urb | CC BY-SA 4.0 |
added 2 characters in body
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| Jan 25, 2021 at 18:46 | comment | added | Kahootsux | Thanks, that makes a lot of sense! | |
| Jan 25, 2021 at 18:36 | history | answered | Adam Herbst | CC BY-SA 4.0 |