Exercises: Kirchhoff's laws and loops

State Kirchhoff's current law (or Kirchhoff's first law) in your own words and explain it using a hydraulic analogy: what happens to the flow rate of water in a pipe that splits into two.

State Kirchhoff's voltage law (or Kirchhoff's second law) in your own words: when travelling completely around any closed path of a circuit, the sum of voltage gains and drops is equal to $\sum V = 0$. Explain it using the analogy of a circular walk on a mountain (you go up and down, but return to the same elevation).

Define these three elements of circuit topology in your own words: (a) node, (b) branch, and (c) loop. Give an example of each by describing a simple circuit with two batteries and three resistors.

$5\ \text{A}$ arrive at a node through a cable and $2\ \text{A}$ leave through another branch. Applying Kirchhoff's current law ($I_\text{entran} = I_\text{salen}$), calculate what current leaves through the third branch of the node and explain your reasoning.

A classmate states: "Kirchhoff's current law means that electric charge cannot accumulate at a point in the circuit: everything that enters must leave." State whether the assertion is true or false and justify it physically.

Match each concept to its definition: (1) node, (2) branch, (3) loop, with: (a) section of a circuit between two consecutive nodes, through which the same current flows; (b) closed path starting and ending at the same point without passing twice through the same branch; (c) point in the circuit where three or more conductors join.

$3\ \text{A}$ arrive at a node through one cable and $1\ \text{A}$ leave through one of the two remaining branches. Calculate the current leaving through the last branch by applying $I_1 = I_2 + I_3$ and reason why more current cannot leave than enters.

The rules for series and parallel circuits (current is the same in series, voltage is the same in parallel) are actually special cases of Kirchhoff's laws. Explain which law each corresponds to and why Kirchhoff's laws are useful for circuits that are neither series nor parallel.

Two cables with $4\ \text{A}$ and $2\ \text{A}$ arrive at a node, and $5\ \text{A}$ leave through a third one. (a) Calculate the magnitude and direction (entering or leaving) of the current through the fourth cable. (b) Check the result by writing $\sum I = 0$ with signs (positive for entering, negative for leaving).

In a loop with a $12\ \text{V}$ battery and two resistors, voltage drops of $5\ \text{V}$ and $7\ \text{V}$ are measured. (a) Verify that Kirchhoff's voltage law is satisfied by writing the signed sum. (b) Reason what would happen to the loop current if the measured drops added up to less than the battery voltage.

In a single loop, a $12\ \text{V}$ battery, a $4\ \text{V}$ battery connected in opposition (positive terminals facing each other), and a total resistance of $8\ \Omega$ are connected one after another. (a) Calculate the net voltage driving the current and its value $I$. (b) Calculate the voltage drop across the resistor. (c) Write the equation for the complete loop, $\sum V = 0$, verifying that it holds true.

The circuit in the diagram has two batteries and three resistors distributed over two loops, with the branch currents labelled, as shown in the diagram.

(a) Write the upper node equation relating the three branch currents ($I_1$, $I_2$ and $I_3$). (b) Write the equation for the left loop (the one containing the battery of $12\ \text{V}$, $R_1$ and $R_3$) by applying $\sum V = 0$. (c) Write the equation for the right loop. It is not necessary to solve the system: only set it up.

A two-loop circuit has a battery of $6\ \text{V}$ with $R_1 = 1\ \Omega$ in the left branch, a battery of $4\ \text{V}$ with $R_2 = 2\ \Omega$ in the right branch, and $R_3 = 2\ \Omega$ in the common branch. Calling $I_1$ and $I_2$ the currents in the side branches and $I_3 = I_1 + I_2$ the current in the common branch, set up (without solving) the three Kirchhoff equations for the circuit: the node equation and the two loop equations.

A current of $2\ \text{A}$ flows through a branch formed by a battery of $12\ \text{V}$ (traversed from its negative to its positive terminal) and a resistor of $3\ \Omega$. Calculate the potential difference between the ends of the branch by traversing it in the direction of the current (rises $12\ \text{V}$ in the battery and drops $I \cdot R$ in the resistor) and explain each term of your calculation.

Between two nodes there are two parallel branches: through the first, of resistance $2\ \Omega$, $6\ \text{A}$ flow. (a) Applying Kirchhoff's current law, calculate the current in the second branch if $9\ \text{A}$ arrive through the common cable. (b) Calculate the resistance of that second branch knowing that both withstand the same voltage. (c) Explain using Kirchhoff's current law and Ohm's law why the branch with smaller resistance carries more current.

A classmate claims to have solved a two-loop circuit (battery of $12\ \text{V}$ with $R_1 = 3\ \Omega$, battery of $8\ \text{V}$ with $R_2 = 2\ \Omega$, and $R_3 = 2\ \Omega$ in the common branch) and says that the currents are $I_1 = 2\ \text{A}$, $I_2 = 1\ \text{A}$ and $I_3 = 3\ \text{A}$. Check whether their solution is correct by verifying, with the given numbers: (a) Kirchhoff's current law at the node and (b) Kirchhoff's voltage law in the two loops.

A two-loop circuit has a battery of $14\ \text{V}$ in series with $R_1 = 2\ \Omega$ in the left branch, a battery of $9\ \text{V}$ in series with $R_2 = 1\ \Omega$ in the right branch, and $R_3 = 2\ \Omega$ in the common branch. (a) Set up the system of Kirchhoff equations with the branch currents $I_1$ and $I_2$ (the current in the common branch is $I_1 + I_2$). (b) Solve it and give the three currents with their units. (c) Check the solution by verifying the node and one of the loops.

The circuit in the diagram has two batteries in the side branches and a resistor in the common branch, with the current of that branch marked with a question mark, as shown in the diagram.

(a) Set up the system of Kirchhoff equations for the circuit. (b) Solve it and calculate the three branch currents, in particular that of the common branch marked in the diagram. (c) Check the result using Kirchhoff's current law at the node.

To charge a $6\ \text{V}$ battery, it is connected in opposition to a $12\ \text{V}$ charger through a $3\ \Omega$ current-limiting resistor, forming a single loop.

1. a) Calculate the charging current $I$ flowing through the loop (apply Ohm's law to the net voltage driving the current and to the resistance).

1. b) Calculate the power delivered by the charger, the power absorbed by the battery and the power dissipated by the resistor ($P = V \cdot I$ for each element).

1. c) Check that the power delivered is the sum of the powers absorbed and explain what this balance means.

In a two-loop circuit, the left branch has a battery of $20\ \text{V}$ with $R_1 = 4\ \Omega$, the right branch has a battery of $18\ \text{V}$ with $R_2 = 6\ \Omega$, and the common branch has $R_3 = 4\ \Omega$. (a) Set up and solve the system of Kirchhoff equations. (b) Using the currents obtained, calculate the power delivered by each battery and that dissipated by each resistor. (c) Verify that the total power delivered equals the total power dissipated and reason why this must always be the case.

A real battery of $12\ \text{V}$ has an internal resistance of $0{,}5\ \Omega$, which can be drawn in series with it. (a) When connecting a load of $5{,}5\ \Omega$ to it, calculate the circuit current and the voltage that would be measured across the terminals of the battery (it is no longer $12\ \text{V}$). (b) Explain why the terminal voltage drops as current increases. (c) Reason what would happen to that voltage if the load were a short circuit and why this is dangerous.

A circuit has only two nodes, $A$ (top) and $B$ (bottom), joined by three branches in parallel with each other:

• Branch 1: a $12\ \text{V}$ battery in series with a $2\ \Omega$ resistor, with the positive terminal of the battery towards node $A$.

• Branch 2: a $6\ \Omega$ resistor.

• Branch 3: a $4\ \text{V}$ battery in series with a $1\ \Omega$ resistor, with the positive terminal of the battery also towards node $A$.

1. a) Explain why this circuit cannot be solved by combining resistors in series and in parallel.

1. b) Set up the Kirchhoff equations needed to solve it, first defining the branch currents with their direction.

1. c) State how many independent equations are needed and why.

In the circuit of the figure, with two batteries and three identical $1\ \Omega$ resistors, the currents in the side branches have been measured, as shown in the figure.

(a) Calculate the current in the common branch by applying only the junction rule, without solving any system of equations. (b) With the three currents known, calculate the voltage drop across each resistor. (c) Check that the measurements are consistent by verifying the loop rule in both loops of the circuit.

Two batteries of different voltage, $6\ \text{V}$ and $4\ \text{V}$, each with an internal resistance of $0{,}5\ \Omega$, are connected in parallel (positive to positive) without any external load. (a) Draw the equivalent circuit and calculate, by applying Kirchhoff's voltage law to the loop formed by the two batteries, the current flowing between them even without a load. (b) Calculate the power dissipated by that current in the internal resistances. (c) Use the result to explain why cells or batteries of different voltage should never be connected in parallel.

You can do it on paper and send a photo with your answer.

Topic Challenge. How would you ensure emergency lighting for an isolated medical post? Design and validate a mixed DC circuit with two independent sources ($12,0\text{ V}$ and $9,0\text{ V}$) to power three critical loads.

1. Real data: Look up the specifications of three indicator LED lamps in online technical catalogs to deduce their commercial equivalent resistances.

2. Analytical solution: Draw the circuit schematic forming two active loops. Apply Ohm's law and Kirchhoff's laws to calculate analytically the branch currents and the voltage across each component.

3. Virtual simulation: Build the circuit in Tinkercad Circuits. Connect virtual voltmeters and ammeters at key points to record empirical measurements.

4. Reasoned proposal: Prepare a brief report using LibreOffice containing the schematic, the step-by-step algebraic development, and a comparison table between theoretical and simulated values, explaining any discrepancies.

Assessment criteria:

• Rigorous application of Kirchhoff's laws and mathematical solution of the system of equations.

• Functional and tidy assembly in Tinkercad with correct connection of the measuring instruments.

• Clarity and critical thinking when comparing analytical results with simulated readings in the table.