Thought experiment and faraday cage

If the two plates have the same electric potential, there is no electric field inside. It is usually better to think of these problemes in terms of Laplace's equation $$Delta phi = 0$$

instead of Gauss's law. A function satisfying Laplace's equation is harmonic and has lots of nice properties, but in particular it is a well-posed problem, meaning any solution satisfying the boundary conditions is the only possible solution.

In the case of two parallel plates with the same potential $phi_0$, the solution $$phi(x,y,z) = phi_0 = text{ct}$$ satisfies the boundary condition. Thus, it is the only possible solution, and $$E = -nabla phi = 0$$ as we wanted to see.

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Note: Laplace's equation is valid when your fields are time-invariant, and it is easy to derive from Maxwell's equations. From Faraday's law $$nabla times E = -frac{partial B}{partial t} = 0$$ where the last equality is due to time-invariance. Thus, $E$ is conservative and there is an electric potential $phi$ such that $$E = -nabla phi$$ Taking the divergence and applying Gauss's law for free space, $nabla cdot E = 0$ we get $$nabla cdot E = -nabla cdot nabla phi = 0$$ whence $$Delta phi = 0$$

This is the way a Faraday cage works:

Source

Two metal plates isolated from each other will never cancel between them an external static electric field.