When you are talking about an elctromagnetic wave that reflects on a surface (for example here), why do you say that the reflected magnetic field inverts and not the electric field?
\begin{align}
E_+ + E_- &= E_t\\
H_+ – H_- &= H_t
\end{align}
Is there a law that forces magnetic field to invert and electric field to stay the same? because Poynting vector can point backwards if you invert the electric field and leave the magnetic field as it is. Couldn't it be:
\begin{align}
E_+ – E_- &= E_t\\
H_+ + H_- &= H_t
\end{align}
And then you will get the exact opposite:
\begin{align}
\frac{E_-}{E_+} = \frac{n_2 – n_1}{n_2+n_1}
\end{align}
which is possitive when $n_2 > n_1$ and there the incident and reflected electric fields have the same direction.
Best Answer
But if you adopt the different sign convention you have just got the same result!
The E-field is a vector; by making the incident and reflected waves have opposite signs on the LHS in your revised formula you have started off by assuming that the reflected E-field is in the opposite direction to the incident E-field. Then when you find that the reflection coefficient is positive, you have merely confirmed your assumption.
Whichever way you do it, if $n_2>n_1$, then the incident and reflected fields are in opposite directions. It really does help if you write these equations down as vector expressions.