Elementary circuits

Explain in your own words the difference between connecting two light bulbs in series and connecting them in parallel, and give an everyday example of each type of connection.

Two resistors of $100\ \Omega$ and $200\ \Omega$ are connected in series to a $12\ \text{V}$ battery. Calculate the total resistance of the circuit and the current flowing through it.

Three identical resistors of $10\ \Omega$ are connected in series. What is the total resistance? What if there were ten resistors of $10\ \Omega$? Reason the rule you have used.

A circuit has a $9\ \text{V}$ battery and a single resistor of $100\ \Omega$. Applying Ohm's law, $V = I \cdot R$, calculate the current flowing through it.

Two resistors of $100\ \Omega$ and $300\ \Omega$ are connected in series to a $12\ \text{V}$ battery. Calculate the current flowing through the circuit and the voltage across the terminals of each resistor, verifying that the two voltages add up to the battery voltage.

A parallel circuit powered at $10\ \text{V}$ has two branches: one with a resistor of $50\ \Omega$ and another with one of $100\ \Omega$. Calculate the current in each branch and the total current, and reason why the branch with less resistance carries more current.

Consider two setups: the first consists of two resistors in series ($R_1 = 220\ \Omega$ and $R_2 = 330\ \Omega$) connected to a $9\text{ V}$ battery with an ammeter; the second consists of two light bulbs in parallel connected to a $4,5\text{ V}$ battery. Calculate the current shown on the ammeter in the series setup and explain what happens in the parallel setup if one of the light bulbs burns out.

Analyse the mixed circuit in the figure:

1. a) Calculate the total current $I$ supplied by the battery and the voltage across the parallel combination ($R_2$ and $R_3$).
2. b) Quantitatively predict what would happen to the current supplied by the battery and to the power dissipated by $R_1$ if an accidental short circuit occurs across the terminals of $R_3$ (a zero-resistance bridge between its terminals). Explain what danger this poses to $R_1$.
3. c) What resistance value should a new component $R_4$, placed in parallel with $R_2$ and $R_3$, have so that the total current $I$ of the circuit increases to exactly $0,75\ \text{A}$?

You want to protect an LED diode ($V_F = 2\ \text{V}$, nominal operating current $I = 20\ \text{mA}$, and maximum destructive current of $30\ \text{mA}$) connected to a $9\ \text{V}$ battery. In the workshop, you have $1/4\ \text{W}$ ($0,25\ \text{W}$) resistors from the standard commercial series: $220\ \Omega$, $330\ \Omega$, $390\ \Omega$, and $470\ \Omega$.

1. a) Calculate the exact theoretical value of the required resistor and justify which of the available commercial values you should choose to ensure that the LED operates properly without being destroyed.
2. b) With the chosen commercial value, calculate the actual current that will flow through the circuit and check whether the $0,25\ \text{W}$ resistor can handle the power it must dissipate without burning out.
3. c) Predict what would happen to the LED if the $220\ \Omega$ resistor were used by mistake.

Two incandescent light bulbs with resistances of $10\ \Omega$ and $40\ \Omega$ are connected to a $12\ \text{V}$ power source. A student claims: "The $10\ \Omega$ bulb will always shine brighter than the $40\ \Omega$ one, whether we connect them in series or in parallel, because it offers less resistance and lets more energy through."

1. a) Calculate the power dissipated by each bulb when connected in series and determine which one shines brighter.
2. b) Calculate the power dissipated by each bulb when connected in parallel and determine which one shines brighter.
3. c) Critically analyse the student's reasoning: explain why they are right in one setup but mistaken in the other, justifying which power formula ($P = I^2 \cdot R$ or $P = V^2 / R$) is convenient to use in each case to understand the result.

An experimental heater for a greenhouse consists of a $20\ \Omega$ resistor and requires dissipating a minimum thermal power of $2,5\ \text{W}$ to function properly. It is initially powered by a new $9\ \text{V}$ battery.

1. a) Calculate the current and the power dissipated with the new battery.
2. b) With use, the battery voltage drops to $7\ \text{V}$. Calculate the percentage reduction experienced by the dissipated power and determine with justification whether the heater will continue to achieve its objective.
3. c) A student suggests replacing the resistor with a $40\ \Omega$ one to "make better use of the little energy left". Refute their proposal quantitatively by calculating the resulting power at $7\ \text{V}$, and calculate what value the resistance should actually have to regain exactly $2,5\ \text{W}$ with the battery at $7\ \text{V}$.