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

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.

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.

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.

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 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.

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.