Exercises: Electrical quantities and Ohm's law

Complete the table of the three electrical quantities in your own words: the name of each quantity, the letter representing it, the unit in which it is measured, and the name of that unit. Given data: voltage is measured in volts, current in amperes, and resistance in ohms. Add a sentence explaining what each one measures.

The pipe analogy helps us understand a circuit: voltage is like water pressure, current is like flow rate, and resistance is like a narrowing of the pipe. Explain the analogy in your own words and use it to deduce what happens to the "flow rate" if the pressure increases or if the pipe becomes narrower.

The figure shows a circuit with a battery and a resistor, with the quantities labelled, but the current value is covered.

Name the components of the circuit, explain what each labelled quantity represents (V, R, and I) and in which unit each is measured, and use Ohm's law to explain whether the covered current will be large or small with that resistance.

State Ohm's law in words, without formulas: what happens to the current when the voltage doubles with the same resistance, and what happens to it when the resistance doubles with the same voltage. Then write the formula $V = I \times R$ and check that your two sentences satisfy it.

Every electrical load offers a certain resistance to the flow of current. Explain what happens to this resistance in an old light bulb as its filament gets thinner with use, and explain, using Ohm's law, why many light bulbs blow just when switched on.

A light bulb is labelled "$60\ \text{W}$" and an LED bulb is labelled "$9\ \text{W}$". Explain what this number means (power: how much electrical energy it consumes every second) and deduce which of the two will use more electricity if they are on for the same number of hours.

On a home electricity bill, consumption is given in kWh (kilowatt-hours). Explain what this unit measures (energy, not power: it combines how much power and for how long) and mentally calculate how many kWh an appliance of $1\ \text{kW}$ consumes when left on for one hour.

Review the concept of conductors and insulators: classify copper, plastic, tap water, dry wood, pencil graphite, and rubber, and explain what this classification has to do with the resistance each material offers to the flow of current.

A circuit has a $9\ \text{V}$ battery and a $100\ \Omega$ resistor. Apply Ohm's law $V = I \times R$ by rearranging for current and calculate its value in amperes. Then, without doing any further calculations, explain whether the current would increase or decrease if the resistor were replaced with a $50\ \Omega$ one.

A current of $0.5\ \text{A}$ flows through a resistor of $12\ \Omega$. Calculate the voltage applied across its terminals and explain, using the pipe analogy, what this result means: how much "pressure" is needed to push that flow rate through that narrowing.

A heater connected to the $230\ \text{V}$ mains allows a current of $2\ \text{A}$ to pass. Calculate the resistance of the heater and reason what would happen to the current if its internal resistance were to double due to a fault, while keeping the same mains voltage.

The figure shows the Ohm's law triangle, a trick to remember the three formulas: covering the letter you are looking for leaves the calculation you need to carry out.

Explain how to use the triangle and write, from it, the three formulas of Ohm's law: for voltage, current, and resistance. Check each formula with this case: $V = 12\ \text{V}$, $I = 2\ \text{A}$, $R = 6\ \Omega$.

Electrical power is calculated using $P = V \times I$. An appliance connected to $230\ \text{V}$ allows $4\ \text{A}$ to flow: calculate its power in watts and explain whether it would exceed the limit of a socket that tolerates up to $1,000\ \text{W}$. What maximum current does that socket allow at $230\ \text{V}$?

A $1,000\ \text{W}$ heater is on for $2$ hours. Calculate the energy consumed in kWh (power in kW multiplied by time in hours) and repeat the calculation for a $200\ \text{W}$ games console turned on for $5$ hours. Compare the two consumption figures and explain why a small appliance running for a long time can use almost the same energy as a large appliance running for a short time.

The price of electricity is about $\$3$ per kWh. Calculate how much it costs to leave a $50\ \text{W}$ fan turned on all night ($8$ hours) and how much $8$ hours of a $2,000\ \text{W}$ heater cost. Express both results in pesos and compare which has a greater impact on the bill.

The figure shows a circuit for which we know the resistance and the current passing through it, but the battery voltage is covered.

Calculate the battery voltage using Ohm's law and explain, step by step, how you rearranged the formula $V = I \times R$ and what units you are using. Check that your result is coherent: is it a reasonable household battery?

A classmate claims: "If we want an electric heater to put out twice as much heat, we just need to replace its internal resistor with one that has twice the ohms, because the higher the resistance, the more heat it retains."

1. a) Quantitatively refute their statement: calculate the current and the power dissipated by a $20\ \Omega$ resistor connected to the $230\ \text{V}$ mains versus a $40\ \Omega$ resistor connected to the same voltage.
2. b) Justify, combining Ohm's law ($I = V/R$) and the power formula ($P = V \times I$), whether you need to increase or decrease the appliance's resistance to produce more heat.
3. c) If a heater's wire were to break and you decided to trim it to splice it together, leaving it with half of its original length (which reduces its resistance by half), predict whether it will heat more or less than before and what safety hazard this would entail.

In the Technology workshop, you have two incandescent light bulbs designed for $230\ \text{V}$: bulb A ($R_A = 1,150\ \Omega$) and bulb B ($R_B = 575\ \Omega$).

1. a) If they are connected in parallel to the $230\ \text{V}$ mains, calculate the current and the power dissipated by each one. Justify which one shines brighter.
2. b) A student predicts that if both are connected in series to that same $230\ \text{V}$ outlet, bulb B will still shine brighter than bulb A because "its lower resistance allows the current to flow better". Prove whether their prediction is correct by calculating the common current of the series circuit and the power that each bulb will dissipate in this configuration.
3. c) Explain why the result is completely reversed between the parallel circuit and the series circuit based on the relationship between current, voltage, and power.

Estimate the monthly cost ($30$ days) of two appliances: a $2,000\ \text{W}$ heater used $3$ hours a day and a $20\ \text{W}$ cell phone charger plugged in $10$ hours a day. With a price of $\$3$ per kWh, calculate both, compare them, and explain why saving energy on the charger is much less effective than controlling the heater.

A cell phone battery stores about $4,000\ \text{mAh}$ and the phone consumes on average $400\ \text{mA}$. Estimate how many hours the battery lasts with this average consumption, explain what the unit mAh means (how much current over how much time), and reason why with a bright screen and games the battery lasts much less.

The circuit breaker (MCB) protecting a home's kitchen trips if the total current exceeds $16\ \text{A}$ at a voltage of $230\ \text{V}$. To prepare dinner between 20:00 and 21:30, the following appliances must be used:

• Oven: $1,840\ \text{W}$ (needs $45$ continuous minutes of operation).

• Ceramic hob: $1,380\ \text{W}$ (needs $30$ continuous minutes to boil).

• Dishwasher: $2,070\ \text{W}$ (quick program lasting $45$ continuous minutes).

• Microwave: $920\ \text{W}$ (needs $15$ minutes to heat pre-cooked meals).

1. a) Numerically prove through current calculations why the power supply would cut off if the oven and the dishwasher were turned on simultaneously.
2. b) Design a detailed schedule of use between 20:00 and 21:30 so that all appliances complete their task on time without the circuit breaker tripping at any moment. Justify your proposal by calculating the maximum current reached during the most heavily loaded interval.

Analyze this simplified electricity bill: consumption of $180\ \text{kWh}$ at $\$3$ per kWh, plus a fixed charge of $\$200$ and $21\ \%$ VAT on the total. Calculate the final amount step by step and propose two specific measures that would lower the next bill, estimating how much each would save.

The management of a school wants to replace the $150$ $60\ \text{W}$ incandescent light bulbs in the hallways with $9\ \text{W}$ LED bulbs that provide the same amount of light. Each LED bulb costs $\$3.50$. The lights remain on for $6$ hours a day during the $180$ days of the school year. The price of electricity is $0.15\ \text{\$/kWh}$.

1. a) Calculate the total initial expense required to purchase the LED bulbs for the entire school.
2. b) Determine the energy saved in one school year (in kWh) and the annual economic savings achieved on the electricity bill.
3. c) Calculate the payback period: in how many months or school years will the initial investment be fully recovered thanks to the savings on the electricity bill?
4. d) Provide a reasoned conclusion on why the massive replacement of inefficient lighting is one of the first sustainability measures undertaken by institutions.

A small incandescent flashlight bulb has the nominal ratings "$4.5\ \text{V} - 0.3\ \text{A}$" engraved on it. In the workshop, you do not have a $4.5\ \text{V}$ battery, but a fixed $9\ \text{V}$ power supply.

1. a) Calculate the internal resistance of the filament under normal working conditions and the nominal power it dissipates.
2. b) If you were to connect the bulb directly to $9\ \text{V}$, calculate the new current and the power it would attempt to dissipate (assuming that the resistance does not change). Justify why the power is multiplied by $4$ when the voltage is doubled and predict what will physically happen to the filament due to the Joule effect.
3. c) Design a technical solution: calculate the value in ohms ($\Omega$) and the minimum power that an auxiliary resistor must be able to dissipate when connected in series with the bulb so that it operates at its nominal ratings connected to the $9\ \text{V}$ power supply without burning out.