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In David J Griffiths textbook "Introduction to Electrodynamics", he states that the strength of the magnetic field a distance $s$ away from a steady current along a long wire is $$ B = μ_0(I)/((2\pi)s)$$ using Ampere's law. But when using the Bio-Savart law, he produces a different value that only gives the above-mentioned value when the wire is of infinite length. But Ampere's law is said to work with non-infinite currents, so what is going on here?

(Example 5.7 vs. 5.5 of Griffiths)

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  • $\begingroup$ Amperes Law is valid exactly only for infinit wires, but a very good approximation in the middle part of a long wire $\endgroup$
    – trula
    Commented Jul 31, 2023 at 13:26
  • $\begingroup$ Above formula is valid only for infinitely long current carrying wires. $\endgroup$
    – Spencer
    Commented Jul 31, 2023 at 13:27
  • $\begingroup$ Because a non-infinite wire would result in a non-steady current (since current piles at the end or at least dJ/dt and dp/dt are not equal to 0) ? $\endgroup$
    – user364438
    Commented Jul 31, 2023 at 16:24
  • $\begingroup$ @EmilSriram yes. Ampere's law is valid also for finite wires, but the system has to be stationary, thus no change of electric charge density can happen. This is for example, when the wire is shaped into a closed path and current is constant in time. Then Ampere's law works, but it is not that useful, because such system does not have the symmetry to express the circulation of magnetic field in a useful way as $2\pi s B$. $\endgroup$
    – Ján Lalinský
    Commented Aug 2, 2023 at 13:16

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the strength of the magnetic field a distance $s$ away from a steady current along a long wire is $$ B = μ_0(I)/((2\pi)s)$$ using Ampere's law. … But Ampere's law is said to work with non-infinite currents, so what is going on here?

Ampere’s law (with Maxwell’s correction) is a general law that always holds in classical electromagnetism. It is: $$\nabla \times \vec B=\mu_0 \vec J+\mu_0 \epsilon_0 \frac{\partial}{\partial t}\vec E$$ The expression you wrote is not Ampere’s law, but rather it is a specific solution of Ampere’s law for an infinite straight wire. So it is unsurprising that the specific solution is only valid for infinite wires.

Biot Savart can be used for finite wires, however it is important to make sure, when using Biot Savart, that there are no current endpoints. All current lines must form loops or go off to infinity. Endpoints have charge accumulation which produces a non-zero $\frac{\partial}{\partial t}\vec E$ that is not accounted for with Biot Savart.

Thus the two solutions are only expected to coincide for an infinite straight wire

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  • $\begingroup$ You're wrong here. "Ampere's law" has an established meaning, and does not contain displacement current term. Maxwell introduced it, and the fixed equation is called the Maxwell-Ampere law. $\endgroup$
    – Ján Lalinský
    Commented Aug 2, 2023 at 13:11
  • $\begingroup$ I have often seen it called simply Ampere’s law, often with a parenthetical comment about Maxwell’s correction. I have clarified with the parenthetical comment. “Wrong” is a little overblown, at worst “unclear” $\endgroup$
    – Dale
    Commented Aug 2, 2023 at 13:21
  • $\begingroup$ OK the Ampere references are good now. The Biot-Savart law can be used also for systems where charge accumulates, provided electric field everywhere is conservative field. This is e.g. when charging a capacitor. $\endgroup$
    – Ján Lalinský
    Commented Aug 2, 2023 at 13:27

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