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.