This is basically a continuation of the post here.

Consider electrostatics in $1$-dimension (say, the $x$-axis). Now consider a positive charge $+q$ located at $x=0$, and two equal negative charges $-q$ are held fixed at $x=+a$ and $x=-a$. In this configuration, the total force on $+q$ at $x=0$ is zero i.e., the charge at $x=0$ is in equilibrium. Moreover, it is also a stable equilibrium i.e., if we slightly displace $q$ towards left or right, the it would oscillate about $x=0$. This means that it is possible to keep the charge $q$ in stable equilibrium by electrostatic forces alone.

But this again goes against Earnshaw's theorem. Again I must be missing something. Is it that when I say the charges at $x=\pm a$ are held fixed, I am outside the purview of Earnshaw's theorem?

This is basically a continuation of the post here.

Consider electrostatics in $1$-dimension (say, the $x$-axis). Now consider a positive charge $+q$ located at $x=0$, and two equal negative charges $-q$ are held fixed at $x=+a$ and $x=-a$. In this configuration, the total force on $+q$ at $x=0$ is zero i.e., the charge at $x=0$ is in equilibrium. Moreover, it is also a stable equilibrium i.e., if we slightly displace $q$ towards left or right, then it would oscillate about $x=0$. This means that it is possible to keep the charge $+q$ in stable equilibrium by electrostatic forces alone.

But this again goes against Earnshaw's theorem. Again I must be missing something. Is it that when I say the charges at $x=\pm a$ are held fixed, I am using mechanical forces and thus move outside the purview of Earnshaw's theorem?