= Solution
Take the <divergence> of the proposed <Ampère's law>. Since the <divergence of a curl> vanishes identically, it implies $\nabla\cdot\mathbf J=0$. However, local <charge conservation> requires the <continuity equation>
$$
\nabla\cdot\mathbf J=-\frac{\partial\rho}{\partial t}.
$$
For example, <electric current> entering a charging <capacitor> accumulates <electric charge> on its plates, so the <charge density> is time dependent. The proposed equation cannot describe that process.
From <Gauss's law>, $\nabla\cdot\mathbf E=\rho/\epsilon_0$. Thus $\nabla\cdot(\epsilon_0\partial_t\mathbf E)=\partial_t\rho$, which supplies precisely the missing <divergence>. Adding the <displacement current density> gives the <Ampère-Maxwell equation>:
$$
\boxed{\mathbf J_D=\epsilon_0\frac{\partial\mathbf E}{\partial t},\qquad \nabla\times\mathbf B=\mu_0\left(\mathbf J+\epsilon_0\frac{\partial\mathbf E}{\partial t}\right).}
$$
Its <divergence> now yields the correct <continuity equation>. Through a fixed <surface>, the associated <displacement current> is $I_D=\epsilon_0\,d\Phi_E/dt$, where $\Phi_E$ is the <electric flux>. In a vacuum <capacitor> gap, this equals the conduction <electric current> charging the plates, even though no charges cross the gap. <Charge conservation> fixes the required <divergence> of the correction; by itself it does not exclude adding a further <divergence-free vector field>.
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