Series, parallel and mixed circuits

Two resistors of $10\ \Omega$ and $15\ \Omega$ are connected in series. Knowing that in series the resistances add up ($R_\text{eq} = R_1 + R_2$), calculate the equivalent resistance and explain what "equivalent resistance" means.

Three identical resistors of $9\ \Omega$ are connected in series. Calculate the equivalent resistance and explain the rule you used. What would the equivalent resistance be with four identical resistors of $9\ \Omega$ in series?

A series circuit with a $12\ \text{V}$ battery has two resistors: $R_1 = 10\ \Omega$ and $R_2 = 14\ \Omega$. (a) Calculate the equivalent resistance and the current flowing through the circuit. (b) Calculate the voltage drop across each resistor ($V_1 = I \cdot R_1$ and $V_2 = I \cdot R_2$). (c) Verify that $V_1 + V_2$ equals the battery voltage.

Two resistors of $12\ \Omega$ and $6\ \Omega$ are connected in parallel. Knowing that $\frac{1}{R_\text{eq}} = \frac{1}{R_1} + \frac{1}{R_2}$, calculate the equivalent resistance and verify that the result is smaller than the smaller of the two resistors, explaining why this always happens in parallel.

Three resistors of $12\ \Omega$, $6\ \Omega$ and $4\ \Omega$ are connected in parallel. Calculate the equivalent resistance using the expression $\frac{1}{R_\text{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$ and explain why the result is smaller than the smallest of the three.

A test bench supplies $20\ \text{V}$ to three parallel loads of $R_1 = 5\ \Omega$, $R_2 = 10\ \Omega$, and $R_3 = 20\ \Omega$. The main line is protected by a main fuse of $8\ \text{A}$, and the conductors of each branch withstand a maximum admissible current of $5\ \text{A}$. (a) Calculate the current in each branch and the total current. Justify whether the installation operates within safety margins. (b) Due to a mechanical overload, resistor $R_2$ degrades and abruptly drops to $2,5\ \Omega$. Predict which element will fail first: will the main fuse blow or will the line of the faulty branch burn out? Justify it with calculations. (c) Design a modification replacing the main fuse with individual protections in each branch, selecting standard commercial ratings ($1\ \text{A}$, $2\ \text{A}$, $3\ \text{A}$, $5\ \text{A}$, $6\ \text{A}$) so as to ensure selectivity (so that a fault in one branch does not cut off power to the other two).

Complete and explain these rules with a simple numerical example for each: (a) in a series circuit, the current is the same through all components and the voltages...; (b) in a parallel circuit, the voltage is the same across all branches and the currents...

Two resistors of $10\ \Omega$ and $15\ \Omega$ are connected in parallel to a $12\ \text{V}$ battery. (a) Calculate the equivalent resistance. (b) Calculate the current flowing through each branch. (c) Calculate the total current and verify that it equals the sum of the branch currents.

A student claims: "in a parallel circuit, the current is the same through all branches, just like in series." Correct the error by stating the correct rules for series and parallel circuits (what is shared and what is divided in each) and support your explanation with a numerical example using two different branches.

The circuit in the diagram is a mixed circuit: resistor $R_1 = 4\ \Omega$ is in series with the parallel combination formed by $R_2 = 12\ \Omega$ and $R_3 = 6\ \Omega$, all powered by a $12\ \text{V}$ battery, as shown in the diagram.

(a) Calculate the equivalent resistance of the parallel section. (b) Calculate the equivalent resistance of the entire circuit. (c) Calculate the total current supplied by the battery, stating its unit.

The circuit in the diagram has a $12\ \text{V}$ battery and two resistors in series, $R_1 = 4\ \Omega$ and $R_2 = 8\ \Omega$, as shown in the diagram.

(a) Calculate the total resistance of the circuit. (b) Calculate the current flowing through the circuit. (c) Calculate the voltage drop across $R_2$, which is the unknown marked in the diagram, stating its unit.

In a circuit powered by a $20\ \text{V}$ DC source, a resistor $R_1 = 4\ \Omega$ is connected in series with a parallel block made up of two resistors, $R_2 = 10\ \Omega$ and $R_3 = 15\ \Omega$. (a) During a control test, an operator measures with a voltmeter a voltage drop across $R_1$ of $20\ \text{V}$ instead of the theoretical design value ($8\ \text{V}$). Reasonably deduce what fault has occurred in the parallel block (has a branch short-circuited or opened?). (b) If under normal conditions resistor $R_3$ were to suffer an internal break and remain an open circuit, predict how the total current supplied by the source and the power dissipated in $R_1$ would vary, calculating their new numerical values. (c) Justify what auxiliary resistor should be placed in parallel with the original block ($R_2 \parallel R_3$) so that the voltage drop across $R_1$ increases to $12\ \text{V}$.