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I have just finished learning the basics of magnetism, and it should be noted that I am not very familiar with Maxwell's equations.

Note:

  1. In the question, when I say "Ampere's Law", I am referring to the equation without Maxwell's correction.

  2. Also, when I say "Biot Savart Law", I am referring to the equation: $\mathrm dB= (\mu_0/4\pi)(I)(\mathrm dL~ X~\hat r)/r^2$


Consider an infinitely long straight wire, carrying a time varying current I(t) such that dI(t)/dt is non-zero. Also consider a point P which is at a distance r from the wire. Using Biot Savart Law, we find out that the magnetic field is $\mu_0\cdot I(t)/2\pi \cdot r$, at any instant t.

Now, I have read that Ampere's Circuital law is NOT valid for cases in which the currents are time varying. However, if we consider an Amperian loop along a circle of radius r and centre at the perpendicular from P to the wire, using symmetry arguments, we obtain the same value of field: $\mu_0 I(t)/2\pi\cdot r$. Since Ampere's law is invalid for such a current, the expression mentioned for the magnetic field must be incorrect.

So, can Biot Savart Law also NOT be used for time varying currents? Also, just out of curiosity, what would be the actual value of the magnetic field at time t?

My book (Halliday and Resnick) derives the equation for the magnetic field created due to a moving point charge. However, after the derivation, it states that the result obtained is not really valid, since "a point charge cannot be assumed as a steady current by any stretch of imagination". This makes me believe that even Biot-Savart Law is only true for non time varying currents. Am I right or wrong?

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  • $\begingroup$ I do not agree that this is a duplicate. I am merely questioning the validity of the equation, not asking to solve for the answer. $\endgroup$
    – Newton
    Jun 27, 2016 at 15:37
  • $\begingroup$ Ampere's law can't be used for time varying fields because the continuity equation contradicts it. We see $\nabla \cdot J= \frac{\partial \rho}{\partial t} \neq 0 $ $\endgroup$
    – Weezy
    Jun 27, 2016 at 15:47
  • $\begingroup$ What about Biot Savart Law? $\endgroup$
    – Newton
    Jun 27, 2016 at 15:48
  • $\begingroup$ Biot Savart law gives you the value of B at any location. When current changes with time so does the magnetic field and B becomes time dependent. $\endgroup$
    – Weezy
    Jun 27, 2016 at 15:50
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    $\begingroup$ This is just semantics, but I'm pretty sure most here are using "Ampere's law" to mean the form without the Maxwell term. $\endgroup$
    – octonion
    Jun 28, 2016 at 14:19

1 Answer 1

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The Biot Savart law is equivalent to Ampere's law without the Maxwell term under the assumption that the charge density has no time dependence. So if we have the usual situation where there are currents producing a magnetic field but no net charge density then the two formulas are actually equivalent. (In the case where there is changing charge density but induction or radiation effects are weak, Biot-Savart might still work. See this answer.)

The demonstration that they are equivalent is as follows (I only wrote down the key steps): $$B=\frac{\mu}{4\pi}\int d^3r' \frac{J(r')\times(r-r')}{|r-r'|^3}$$ $$=\frac{\mu}{4\pi}\int d^3r' \nabla\times\left(\frac{J(r')}{|r-r'|}\right)$$ Then use the formula for the curl of a curl (noting one term vanishes since the divergence of the current is zero under the assumption there is no time dependent charge density) $$\nabla\times B=-\frac{\mu}{4\pi}\int d^3r' J(r')\nabla^2\left(\frac{1}{|r-r'|}\right)$$ The laplacian of $1/|r|$ is $-4\pi\delta(r)$, so we get Ampere's law $$\nabla\times B=\mu J.$$

Also intuitively you can understand why Biot-Savart law can not hold exactly for time varying currents. When you have time varying currents the magnetic field needs to act as a wave. Imagine you had a current that suddenly vanished. The magnetic field should not instantly go to zero everywhere as it would if you applied the Biot-Savart law (or else you could instantly send a signal faster than light).

The correct modification of Biot-Savart for time varying currents is known as Jefimenko's equations. If you go to that wiki page you can see it looks like the Biot-Savart law but there is an additional term that depends on the derivative of the current, and also the 'retarded time' appearing in the equation is consistent with causality.

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  • $\begingroup$ The Biot-Savart formula is actually applicable to currents varying in time and is thus more general than the Ampere integral formula. Check my answer here: physics.stackexchange.com/questions/268023/… $\endgroup$ Mar 16, 2017 at 0:13
  • $\begingroup$ @Ján Lalinský, I derive Ampere's formula from the Biot-Savart law in this answer. They are equivalent. If you believe there is something subtle that is not correct here please let me know. $\endgroup$
    – octonion
    Mar 16, 2017 at 5:58
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    $\begingroup$ The first sentences are incorrect - the Biot-Savart law is in general not equivalent to Ampere's law. You assume that divergence of the current density vanishes, but this is not warranted - the question stated the current is varying in time, which allows for non-zero divergence of j. For example, if we supply the current from a capacitor, the divergence of j at its plates is nonzero. The Biot-Savart formula, in contrast to Ampere's law, is applicable even in situations where electric field is changing in time, provided it is accurately given by the Coulomb formula (no em induction). $\endgroup$ Mar 16, 2017 at 18:35
  • $\begingroup$ @Ján Lalinský, Okay fair enough. I edited the first paragraph and put a link to your answer. Note however that the OP is concerned about time varying currents not time varying charge density. $\endgroup$
    – octonion
    Mar 16, 2017 at 19:19
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    $\begingroup$ @Ján Lalinský, The current in wires varying does not necessarily imply there is charge building up, which is what you need for you to say Biot-Savart is not equivalent to Ampere's formula. In the set up of an infinitely long wire with the current uniformly changing in OP's question there is no divergence so Ampere's formula is derived from Biot-Savart. Even though there is no charge building up there are induction effects which are described by Jefimenko's equations I linked to. Still thank you for pointing out that the two formulas are not equivalent in the case of changing charge density. $\endgroup$
    – octonion
    Mar 17, 2017 at 0:02

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