Sandbox testing.

$$ h(t)\nabla \times \tilde {\mathbf H} = \mathbf J +\epsilon_0 \dot {e}(t) \tilde {\mathbf E} $$

$$ h(t)\nabla \times \tilde {\mathbf H} = \mathbf J +\epsilon_0 \dot {e}(t) \tilde {\mathbf E} \tag{1}$$

$$ h(t)\nabla \times \tilde {\mathbf H} = \mathbf J +\epsilon_0 \dot {e}(t) \tilde {\mathbf E} \tag{1}\label{1}$$

This seems to be caused by the use of \label{}. Here is your first equation, with and without that command:

$$ h(t)\nabla \times \tilde {\mathbf H} = \mathbf J +\epsilon_0 \dot {e}(t) \tilde {\mathbf E} \tag{1}\label{1}$$

$$ h(t)\nabla \times \tilde {\mathbf H} = \mathbf J +\epsilon_0 \dot {e}(t) \tilde {\mathbf E} \tag{1}$$

Both look identical during editing, but the first is sometimes (but inconsistently) mangled in the output.

I'm unaware of how \label{} works in MathJax, and whether it is truly supported. But do you really need it? Why not use just \tag{}s, and hard-code any references in the text? (i.e. use $(1)$ instead of $\eqref{1}$.)