Is Biot-Savart Law valid for time-varying currents unlike Ampere's law?

Question

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: $\mathrm dB= (\mu_0/4\pi)(I)(\mathrm dL~ X~\hat r)/r^2$

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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?

Answer 1

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.