Equivalent Resistor \((R_e)\):

One can understand the equivalent resistor of an association as an imaginary resistor when subjected to the same \(\mathbb{V}\) of the association, a current of the same intensity that would flow in the association will also flow in the equivalent resistor.

Series Association:

Two or more resistors in series constitute an association when they are connected so that the same current sequentially passes through each of them. We can say that the intensity of the current is the same for all resistors and the total \(\mathbb{V}\) of the circuit is the sum of \(\mathbb{V}_i\)'s between the terminals of each \(i\) resistor, namely, the equivalent resistance of an series association is the sum of each resistance. $$R_e = R_1 + R_2 + \dots + R_n$$

Parallel Association:

A combination of resistors in parallel is characterized by the same \(\mathbb{V}\) across the terminals of each resistor, and the same \(\mathbb{V}\) across the terminals of the whole association. We can say that the intensity of the current flowing through the equivalent resistor is equal to the sum of the intensities of the currents flowing in each of the associated resistors. Consequently, the inverse of the equivalent resistance of the resistor is the sum of the inverse of the resistance of the associated resistors. $$ \frac{1}{R_e} = \frac{1}{R_1} + \frac{1}{R_2} + \dots + \frac{1}{R_N}.$$

Example 1

Short Circuit:

A circuit is placed in short-circuit when we connect their terminals via a negligible resistance wire, that is, almost no resistance. Using Ohm's law, moving the \(i\) in evidence, we have the following expression: $$ i = \frac{\mathbb{V}}{R},$$ where it is clear that a resistance value very close to zero generates an extremely high value for the current. This generates a great amount of heat in the circuit, capable of even burning it. The short circuit is dangerous and is a common cause of fires.

Steps for determining the equivalent resistance:

1. Put up letters on all nodes of the association (Reminder: node is the meeting point of three or more resistors)
2. Calculate the equivalent resistence for those resistors which are associated in series or in parallel, those resistors between two consecutive nodes (or between a node and a terminal). Redraw the scheme, with the equivalent resistor replacing the association.
3. Repeat the previous operation, as many times as necessary, always designing a new scheme. The final equivalent resistor is the one that is among the association's terminal nodes.