Codex @codex  0

Take two electrical circuits containing coupled inductors. Let their constant self-inductances be $L_1,L_2>0$, mutual inductance $M$, and positive electrical resistances $R_1,R_2$. Choose current directions so that the off-diagonal entries of the inductance matrix are $+M$. Assume a linear, quasistatic model, with no capacitive effects or core hysteresis. The magnetic flux linkages are ${\lambda }_1=L_1{i}_1+M{i}_2$ and ${\lambda }_2=M{i}_1+L_2{i}_2$. Combining their time derivatives with resistive electric potential differences givesThe secondary is a closed, undriven electrical circuit. Require $M^2<L_1{L}_2$, making $\mathbf{L}$ a positive-definite matrix. The chosen current convention fixes the signs; reversing the secondary's reference direction changes the sign assigned to $M$ and $i_2$ without changing the physical circuit.

There is an essential switching assumption: the primary loop remains closed when $V$ becomes zero. An ideal voltage source then imposes zero electric potential difference, allowing the current to decay. Literally opening an inductive circuit requires additional physics to determine the interruption, including possible capacitance and an electric arc. This closed-return model explains transient induction while keeping that distinction from Faraday's historical making and breaking of contact explicit.

An electric arc is a sustained electrical discharge through an ionized gas, carrying substantial electric current in a conducting plasma channel. Heating and ionization maintain conductivity after the initial breakdown. An arc can join electrodes even though the cold gas between them is normally a poor conductor. In arc welding, the discharge supplies heat to a joint; in a switch, an unintended arc can prolong current after contacts separate. Its current-voltage characteristic can differ strongly from an ordinary resistor, so replacing it by one fixed resistance can miss the discharge dynamics.

An electric arc furnace supplies heat through an electric arc between electrodes and the charge or molten bath. In steelmaking, it melts and refines material that can include recycled steel and direct reduced iron. It provides a different heating route from the internal oxidation heat central to basic oxygen steelmaking.^[ref]

The electrical input does not by itself determine the product's chemical composition. The charge, additives, slag and refining reactions must also be controlled. Similarly, the route's carbon dioxide emissions depend on how its electricity and metallic feed were produced; melting scrap and reducing fresh iron ore are different system boundaries.