# Electromagnetism— why do we only care about flux inside the loop? In other words, why is the area in 'φ=AB' the area bounded by the loop?

When we calculate the emf induced by placing a loop of wire inside a changing magnetic field, why do we use the area bounded by the loop to calculate total flux and then the rate at which it changes? Why does flux outside of the loop not affect the emf?

Because of Stokes’ Theorem in vector calculus, which relates a line integral around a closed curve to a surface integral over the interior of that curve (or, more generally, over any surface bounded by that curve).

$$\nabla\times\mathbf{E}=-\frac{\partial\mathbf{B}}{\partial t}.$$
Integrate it over a surface $$S$$ whose boundary is the closed loop $$C$$ of the wire, and then apply Stokes' Theorem to the left side. You get
$$\oint_C\mathbf{E}\cdot d\ell=-\frac{d}{dt}\iint_S\mathbf{B}\cdot d\mathbf{S}.$$