I want to prove that a voltage signal I apply in the primary of an ideal transformer(or 2 coupled coils) will produce the same voltage signal in the secondary, e.g. a square wave(+- V) in the primary yielding a square wave in the secondary by using the fact that V=L*(di/dt).
So let´s suppose I have a square wave generator. At the moment the square wave reaches +V then V=L1* (di1/dt) and the current I1 in the primary ramps up linearly, causing then V/(L1)*t = i(t), and that means (di1/dt) = V/L1.
This increasing current will then create a magnetic field B= k1 * i(t), where k1 is a constant and the induced voltage in the secondary is going to be V2= -(d(phi)/dt) = - k1 * (di1/dt) = -k1 * (V/L1).
The outcome shows the secondary is going to have the same wave shape signal as the primary, which in this case is a square wave. So assuming I have a V(t) applied to this ideal transformer, the secondary signal is going to be the proportional to this V(t).
Finally, my question is: Can I prove the secondary signal will be alike the the primary by the way I did above? Which is by using the inductor formula V=L*(di/dt)?