The second equation is often known as the local version of Faraday's Law. It would be helpful to put in the coordinates.
$$\nabla \times E(x,y,z)= -\frac{\partial B(x,y,z)}{\partial t} $$$$\nabla \times E(x,y,z,t)= -\frac{\partial B(x,y,z,t)}{\partial t} $$ When you have an expanding loop, the coordinates of a point of the loop are not fixed, but instead change over time. To apply the equation, you have a changing coordinate system, and so you have to apply a Lorentz transformation. The math is messy, but you can see the induced field appear when you substitute a Lorentz transformation into the equation.
