Example of information travelling faster than the speed of light?

Question

In the following situation, information seems to travel faster than the speed of light. Is there a mistake in my reasoning?

Suppose we have an electric circuit. For times $t<0$, there is no current in the circuit. Starting at $t=0$, the current is turned on. This will take a short time $\delta$. We say that $I(t)=f(t)$ for $t\in[0,\delta]$, where $f$ is a continuous function with $f(0)=0$ and $f(\delta)=1$. For $t\geq\delta$, we keep the current constant at $I(t)=1$. We take $\delta$ smaller than $1$ millisecond.

In the time intervals $t\in(-\infty,0)$ and $t\in(\delta,\infty)$, the current is constant. Therefore, we can use the law of Biot-Savart: $$\mathbf{B}(\mathbf{r})=\frac{\mu_0}{4\pi}\int_C\frac{Id\mathcal{l}\times\mathbf{r}'}{|\mathbf{r}'|^3}.$$ For $t<0$, this gives us $\mathbf{B}=0$ everywhere.

For $t>\delta$, however, this will induce a nonzero magnetic field in (almost) all of space.

Now consider an observer which is 1000 kilometers away from the circuit. As we saw before, there is no magnetic field here for $t<0$, but at $t>\delta$ there is a nonzero magnetic field. It doesn't matter what happens for $t\in[0,\delta]$. The point is that at time $\delta$, the observer will know that 1000 kilometers away from him, something has happened. So in $\delta$ time, information has been transferred over a distance of 1000 kilometers. But since we chose $\delta$ smaller than a millisecond, this means that the information has travelled with a speed larger than $10^9ms^{-1}$. This is faster than the speed of light.

Answer 1

Therefore, we can use the law of Biot-Savart

No, you cannot use the Biot Savart law in this situation. The Biot Savart law is derived from the magnetostatic assumption. That is, the assumption that $dJ/dt=0$ which is not a valid assumption in your case.

In the time intervals $t\in(-\infty,0)$ and $t\in(\delta,\infty)$, the current is constant.

Here you seem to be aware of this limitation, but have the mistaken impression that $dJ/dt=0$ means just over some finite time. The magnetostatic approximation is that $dJ/dt=0$ over all of space and all of time. Only in that case can the Biot Savart law be derived from Maxwell’s equations.

Now, obviously this condition never holds exactly. This is why the magnetostatic approximation is an approximation. It is important to know when it is a good approximation and when it is a bad approximation. It will be a good approximation only when the time that the current has been steady is much longer than the spatial distances divided by c.

So in this problem, at $1 \mathrm{\ ms}$ after the end of the current change the Biot Savart law will only be a good approximation for distances much smaller than $300 \mathrm{\ km}$. Obviously $1000 \mathrm{\ km}$ is not much smaller than $300 \mathrm{\ km}$ so the magnetostatic assumption is not valid and therefore Biot Savart is not applicable.