Exercises: Electrical quantities and Ohm's law

Define in your own words the three basic electrical quantities (voltage, current, and resistance), state the SI unit for each, and the instrument used to measure it.

Explain what a resistor is used for in a circuit. Use as an example the typical case of an LED diode connected to a battery: what would happen if the LED were connected directly, without a resistor?

The figure shows a basic direct current circuit consisting of a battery, a resistor, and an ammeter, with the current indicated by an arrow, as shown in the figure.

Name the three components of the circuit, explain the function of each one, and indicate which quantities are shown in the diagram with their values and units.

Explain what a conductor, an insulator, and a semiconductor are, with an example of a material for each and a use within an electrical circuit or appliance.

Explain what a short circuit is, why the current increases so much in that case (the path offers almost zero resistance), and two possible consequences for the installation.

What is a fuse and how does it protect an installation? Explain what happens inside it when the current exceeds its rating, and what must be checked before replacing it with a new one.

The hydraulic analogy compares a circuit with water pipes. Explain which electrical quantity corresponds to water pressure, flow rate, and a narrowing of the pipe, and how this analogy helps you understand Ohm's law.

A multimeter measures voltage, current, and resistance. Explain two checks you must carry out before measuring (selecting the appropriate quantity and range, connecting the test leads to the correct terminals) and why you should not measure the resistance of a component with the circuit switched on.

A circuit has a battery of $V = 12\ \text{V}$ and a resistor of $R = 60\ \Omega$. Applying Ohm's law $V = I \cdot R$, calculate the current flowing through it, expressing the result in the corresponding SI unit.

An electric heater has a resistance of $R = 88\ \Omega$ and is connected to a $V = 220\ \text{V}$ mains supply. (a) Calculate the current flowing through it applying Ohm's law. (b) Calculate its power using the expression $P = V \cdot I$, stating the SI unit of power. (c) Explain what this power value means in terms of energy consumption.

In the circuit in the figure, the battery is $24\ \text{V}$ and the ammeter reads $0{,}5\ \text{A}$, but the value of the resistor is unknown, as shown in the figure.

Rearrange Ohm's law for resistance and calculate its value in ohms ($\Omega$). Check your result by verifying that $V = I \cdot R$ holds true.

The resistance of a cable depends on its geometry: it is proportional to its length and inversely proportional to its cross-sectional area. (a) A cable of $5\ \text{m}$ has a resistance of $2\ \Omega$; calculate the resistance of an identical one of $15\ \text{m}$. (b) Reason what would happen to the resistance if the cross-sectional area of the original cable were doubled. (c) Using the water pipe analogy, explain why a long, thin cable "strangles" current more than a short, thick one.

A red LED requires a current of $0{,}02\ \text{A}$ and has a voltage drop of $2\ \text{V}$. It is connected in series with a resistor to a battery of $9\ \text{V}$, so that the resistor must absorb the remaining $7\ \text{V}$. Calculate the value of the series resistor applying Ohm's law, and explain what would happen to the LED if the chosen resistor were much smaller.

A torch operates on a $6\ \text{V}$ battery and draws $0{,}5\ \text{A}$. (a) Calculate its power. (b) Calculate the resistance of its bulb. (c) If the battery degrades and supplies only $4\ \text{V}$, calculate the new current assuming the same resistance, and explain why the torch shines less brightly.

Reason using Ohm's law: (a) what happens to the current if you double the resistance while keeping the voltage constant; (b) what happens if you double the voltage while keeping the resistance constant; (c) what combination of changes would leave the current exactly the same. Support each reasoning with a numerical example.

A laptop charger has a power rating of $15\ \text{W}$ and is used for $2\ \text{h}$. Calculate the energy consumed in $\text{kW} \cdot \text{h}$ and its cost if the price is $0{,}20\ \text{€}$ per $\text{kW} \cdot \text{h}$. Comment on why the cost is so small even though the power seems high (short usage time).

A classmate claims that if you connect two traditional incandescent light bulbs designed for $230\ \text{V}$, one rated at $60\ \text{W}$ and the other at $100\ \text{W}$, in series to a $230\ \text{V}$ outlet, the $100\ \text{W}$ bulb will still shine brighter because "it has a higher rated power." (a) Model both bulbs as pure resistors and calculate the rated ohmic value of each one. (b) Calculate the current that will flow through the series combination, the voltage that each bulb will actually experience, and the power that each will dissipate. (c) Based on the results, predict which one will shine brighter in this circuit and reasonably refute your classmate's claim, explaining why the behavior changes compared to the usual parallel connection.

In the Technology department, replacing $20$ desktop computers with more efficient equipment is being evaluated. Two options are compared for an annual use of $180$ school days and $6\ \text{h}$ per day:

• Option A: Keep the current towers, upgrading their power supplies to achieve an average consumption of $150\ \text{W}$ per computer, with a total intervention cost of $800\ \text{€}$.

• Option B: Purchase mini-PCs that consume an average of $40\ \text{W}$ per computer, with a total initial outlay of $4100\ \text{€}$.

(a) With an energy price of $0{,}18\ \text{€}/(\text{kW} \cdot \text{h})$, calculate the energy consumed per year in $\text{kW} \cdot \text{h}$ and the economic cost of electricity for each alternative.

(b) Calculate the payback period (in school years of operation) required to offset the higher initial investment of option B through electricity savings compared to option A.

(c) If the expected useful life before replacing the equipment is $5$ years, technically and economically justify what decision you would advise the school management to make.

A two-wire copper line of $15\ \text{km}$ in length ($

ho = 0{,}0175\ \Omega \cdot \text{mm}^2/\text{m}$) with conductors of $70\ \text{mm}^2$ cross-section connects a small hydroelectric plant generating $1{,}2\ \text{MW}$ to an industrial park.

(a) Calculate the total resistance of the line (there and back).

(b) Determine the current that would flow and the power that would be lost due to the Joule effect in the cables if transmission were carried out directly at low voltage at $400\ \text{V}$. Interpret the result and demonstrate the physical infeasibility of this option.

(c) If the voltage is stepped up to $25\ \text{kV}$ for transmission, calculate the new line current and the percentage that the Joule losses represent relative to the total generated power.

(d) In view of the results, justify why the voltage is not stepped up to even higher values (for example, to $400\ \text{kV}$) in this specific installation, arguing at least two technical, economic, or safety factors that limit the voltage.

On a socket outlet line protected by a $16\ \text{A}$ circuit breaker and a $2{,}5\ \text{mm}^2$ copper cable ($

ho = 0{,}0175\ \Omega \cdot \text{mm}^2/\text{m}$) with a length of $20\ \text{m}$ from the main distribution board, the following appliances are connected simultaneously to a power strip at $230\ \text{V}$: an $1800\ \text{W}$ heat gun, a $1200\ \text{W}$ space heater, and a $700\ \text{W}$ vacuum cleaner.

(a) Calculate the current demanded by each load and the total current on the line if all three operate at full power.

(b) Predict with reasoning what will happen to the circuit breaker and assess the thermal risk to the installation.

(c) If the protection does not trip immediately due to its thermal delay curve, calculate the percentage voltage drop that would occur on the line (round trip) and check whether it exceeds the regulatory limit of $3\ \%$ established by the REBT for residential indoor circuits.

(d) A technician suggests replacing the $16\ \text{A}$ circuit breaker with a $25\ \text{A}$ one to prevent nuisance tripping. Technically refute this dangerous advice and propose the correct load redistribution solution.

A $25\ \text{m}$ cable reel has copper conductors with a cross-section of $1{,}5\ \text{mm}^2$ ($

ho = 0{,}0175\ \Omega \cdot \text{mm}^2/\text{m}$). The manufacturer states on its nameplate that at $230\ \text{V}$ it supports a maximum power of $1000\ \text{W}$ fully coiled and $3000\ \text{W}$ fully uncoiled.

(a) Calculate the total electrical resistance of the line (round trip).

(b) If a $2200\ \text{W}$ heater is connected while keeping the cable completely coiled, calculate the current flowing, the thermal power dissipated by the Joule effect inside the drum, and the voltage drop at the load.

(c) Refute the erroneous claim that "the cable's resistance increases when it is coiled": explain the real thermodynamic cause why the permissible power drops drastically when the drum is not uncoiled.

(d) Predict what physical consequences the cable's PVC insulation will suffer if that $2200\ \text{W}$ load is maintained for an extended period without uncoiling.

An autonomous data logger is powered by a lithium cell with a nominal voltage of $3{,}7\ \text{V}$ and a capacity of $3000\ \text{mA} \cdot \text{h}$. The system repeats indefinitely a duty cycle of $5\ \text{minutes}$:

• For $15\ \text{seconds}$, it measures magnitudes and transmits data via radio frequency, consuming a current of $120\ \text{mA}$.

• For the remaining $4\ \text{minutes}$ and $45\ \text{seconds}$, it stays in deep sleep mode, with a residual consumption of just $0{,}06\ \text{mA}$.

(a) Interpret the physical meaning of the unit $\text{mA} \cdot \text{h}$ and calculate the total energy stored in the battery expressed in $\text{W} \cdot \text{h}$ and in joules ($\text{J}$).

(b) Calculate the charge consumed per cycle (in $\text{mA} \cdot \text{h}$) and the equivalent average current absorbed continuously by the equipment.

(c) Estimate the battery life of the data logger (in days). If, due to self-discharge and the low-voltage cutoff margin, only $80\ \%$ of the nominal capacity is usable, recalculate the expected real battery life.

(d) A technician suggests keeping the transmitter continuously on at $120\ \text{mA}$ to avoid the complexity of programming sleep states. Calculate the new battery life and justify why power management using a duty cycle is essential in embedded systems.

In the laboratory, you have a direct current power supply set to $12\ \text{V}$, a signaling lamp with specifications $12\ \text{V}$ and $6\ \text{W}$, and a digital multimeter. The multimeter has an internal resistance of $0{,}2\ \Omega$ on its current range (protected with a $1\ \text{A}$ fast-acting fuse) and $10\ \text{M}\Omega$ on its voltage range.

(a) A student mistakenly connects the ammeter probes directly between the positive and negative terminals of the $12\ \text{V}$ power supply. Model the equivalent circuit of this setup, calculate the current that would attempt to flow through the multimeter, and predict which component will act to protect the equipment.

(b) Describe in detail the correct wiring to simultaneously measure the voltage across the lamp and the current flowing through it using two multimeters, specifying how each one must be inserted into the circuit.

(c) Determine the systematic insertion error introduced by the ammeter when measuring the real current of the lamp, and explain why an ideal ammeter should have strictly zero internal resistance while that of an ideal voltmeter should be infinite.

Design the lighting for a scale model using four identical LEDs ($2\ \text{V}$, $0{,}02\ \text{A}$ each) and a $9\ \text{V}$ battery: propose a connection diagram (series, parallel, or series-parallel groups), calculate the necessary resistors using Ohm's law, and justify your scheme compared to the alternatives (what voltage each group withstands, what happens if an LED fails).