Series, parallel and mixed circuits

Explain the difference between connecting two resistors in series and in parallel using the path analogy: in series the current has a single path and flows through both; in parallel it has two paths and divides. Draw both circuit diagrams and indicate where the current can flow in each.

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

The figure shows two setups, each with a battery and two light bulbs, labelled setup a and setup b.

Indicate which setup is in series and which is in parallel, explaining your reasoning based on the paths available to the current, and explain what happens in each setup if one of the bulbs blows: does the other go out or stay lit?

In an old set of fairy lights connected in series, if one bulb blows, they all go out. Explain why this happens (what happens to the single path for the current) and why in modern fairy lights, connected in parallel, a blown bulb does not turn off the rest.

Home sockets are connected in parallel. Give two reasons: what voltage each appliance receives when connected in parallel (the same mains voltage) and what happens when you turn off the TV (other appliances keep working). Imagine a house wired in series and describe two absurd things that would happen.

In series, resistances add together: $R_T = R_1 + R_2$. Calculate the total resistance of a $30\ \Omega$ resistor and a $50\ \Omega$ resistor in series, and then the current allowed to pass by a $16\ \text{V}$ battery connected to the combination. Explain why in series the total resistance is always greater than either of the individual resistances.

Two identical resistors of $10\ \Omega$ are connected in parallel. Calculate the equivalent resistance knowing that with two identical resistors it is exactly half, and explain why in parallel the total resistance is less than either of the two (think about the paths: adding a new path makes it easier for current to flow).

The parallel formula for two resistors is $R_T = \frac{R_1 \cdot R_2}{R_1 + R_2}$. Apply it to calculate the equivalent resistance of a $10\ \Omega$ resistor and a $15\ \Omega$ resistor in parallel, and check that the result is less than the smaller of the two, as must always be the case in parallel.

In a series circuit, the battery voltage is shared between the resistors. Explain the idea with an example: a $9\ \text{V}$ battery with two identical light bulbs in series —what voltage does each one receive, approximately? What if there were three identical bulbs? Reason why they shine less brightly than a single bulb connected to the same battery.

In a parallel circuit, the total current is shared between the branches. Explain the idea with two different branches: a large light bulb and a small light bulb connected simultaneously to the same battery —which one do you think more current flows through and why? What do the two branches have in common?

The figure shows a $12\ \text{V}$ battery with two resistors in series, of $4\ \Omega$ and $8\ \Omega$, and the circuit current covered.

Calculate: (a) the total resistance of the circuit, (b) the covered current using Ohm's law, and (c) the voltage across the terminals of each resistor. Check that the two partial voltages add up to the $12\ \text{V}$ of the battery.

Two resistors in parallel are connected to a $12\ \text{V}$ battery: one of $6\ \Omega$ and another of $4\ \Omega$. Calculate the current flowing through each branch (each branch has the same $12\ \text{V}$) and the total current leaving the battery. Explain the rule: in parallel, the voltage is the same across all branches and the currents add together.

The figure shows a combination circuit: a resistor in series with a block of two resistors in parallel, powered by a $10\ \text{V}$ battery.

Solve the circuit step by step: (a) the equivalent resistance of the parallel combination of $R_2$ and $R_3$, (b) the total resistance by adding $R_1$ in series and (c) the covered total current. Explain which path that current takes and where it splits.

In the Technology workshop, you assemble a combination circuit connected to a $24\ \text{V}$ power source: a resistor $R_1 = 4\ \Omega$ in series with a parallel block made up of $R_2 = 12\ \Omega$ and $R_3 = 6\ \Omega$. A classmate claims that less current will flow through the $4\ \Omega$ resistor than through the $6\ \Omega$ one because "since it is at the beginning of the circuit, the $4\ \Omega$ one slows down and uses up almost all the current before it reaches the others."

1. a) Calculate the total equivalent resistance of the circuit and the total current supplied by the source.
2. b) Determine the voltage across each part of the circuit and the current flowing through each of the three resistors ($R_1$, $R_2$, and $R_3$).
3. c) Use the calculated values to refute your classmate's claim with physical arguments and explain which of the two parallel branches carries more current and why.

You have a $9\ \text{V}$ battery and want to light up a scale model using bulbs designed to work at $3\ \text{V}$. If they receive more than $3\ \text{V}$ they burn out immediately, and if they receive less than $2\ \text{V}$ their light is barely visible.

1. a) A classmate proposes connecting $4$ bulbs arranged in two parallel branches of $2$ bulbs in series each. Calculate the voltage each bulb would receive in this setup and predict whether the circuit will work or get damaged.
2. b) Analyse the alternatives of connecting the $4$ bulbs all in series or all in parallel: calculate the voltage each one would withstand in both cases and explain the insurmountable drawback of each option.
3. c) Design the optimal setup: how many $3\ \text{V}$ bulbs should you connect and how (in series or in parallel) so that they operate exactly at their nominal voltage of $3\ \text{V}$ using the $9\ \text{V}$ battery without burning any out? Justify it with your calculations.

Two identical light bulbs in series shine much less brightly than a single bulb with the same battery. Explain the complete chain of reasons: what happens to the total resistance when you put two in series, what happens to the current as a result, and why this translates to less brightness in each bulb.

You plug more and more appliances into a parallel power strip and the circuit breaker in your house ends up tripping. Explain why: what happens to the currents in each branch (they add up in the main line), what limit the circuit breaker has, and how this relates to not plugging multi-way adapters into one another.

You need to power a small electric motor that must operate at exactly $8\ \text{V}$ to avoid burning out, but in the workshop, you only have a $12\ \text{V}$ battery. To obtain that voltage, you decide to build a voltage divider by connecting two resistors in series. You have $2\ \Omega$ and $4\ \Omega$ resistors in your component box. Assume the motor draws a negligible current compared with that through the resistors, so it does not alter the voltage division.

1. a) Calculate the current that will flow through the series circuit and the voltage drop across each of the two resistors.
2. b) Justify in parallel with which of the two resistors you should connect the motor to obtain the required $8\ \text{V}$, explaining the voltage divider rule in series circuits.
3. c) A classmate suggests replacing the resistors with two identical $4\ \Omega$ resistors to "make it symmetrical." Demonstrate with calculations what voltage the motor would receive in that case and predict whether it would operate correctly at its rated value.

In a scale model, you connect two branches in parallel to a $12\ \text{V}$ power source: one with a $3\ \Omega$ heater and another with a $6\ \Omega$ exhaust fan.

1. a) Calculate the current flowing through each branch and the total current delivered by the source.
2. b) A classmate claims that "since they are connected to the same battery, both branches must draw exactly the same current." Refute their claim by demonstrating what mathematical relationship exists between the resistance values and the current distribution in parallel.
3. c) If the wire to the $6\ \Omega$ exhaust fan is accidentally disconnected due to a fault, predict what will happen to the $3\ \Omega$ heater (will it turn off, will its current change, or will it continue working exactly the same?). Justify your answer.

To illuminate a scale model, you connect a light bulb with a fixed resistance $R = 3\ \Omega$ to a $9\ \text{V}$ battery. Your classmate proposes adding an identical second bulb ($3\ \Omega$) in series and claims: "By placing two identical bulbs in series, the battery splits its power, and each one will shine with exactly half the power it had when there was only one."

1. a) Calculate the current and the power dissipated by the bulb when it is connected alone to the $9\ \text{V}$ battery.
2. b) Calculate the current, the voltage across each bulb, and the power consumed by each one when both are connected in series to the same $9\ \text{V}$ battery.
3. c) Compare the results: is your classmate's claim correct? Justify why the power of each bulb drops to one-fourth ($1/4$) and not to one-half.

In the laboratory, you connect two parallel resistors of $6\ \Omega$ and $3\ \Omega$ to a $12\ \text{V}$ power supply.

1. a) Calculate the equivalent resistance of the circuit and the total current supplied by the source.
2. b) A student adds a third resistor of $2\ \Omega$ in parallel, claiming that "since more resistors are added, the circuit will offer greater total opposition and the source current will drop". Calculate the new equivalent resistance and the new total current to check whether their prediction was correct.
3. c) Use an everyday analogy (such as the open lanes of a toll booth or the exit doors of a building) to justify why adding parallel branches always reduces the total resistance of the circuit, even if you connect a huge resistor of $1.000\ \Omega$ in parallel.

An old string of Christmas lights consists of $10$ identical incandescent light bulbs connected in series to a $230\ \text{V}$ domestic mains socket.

1. a) Calculate the voltage across each bulb under normal operating conditions.
2. b) If the filament of one of the bulbs burns out (opening the circuit), predict what happens to the others. If you place the test probes of a voltmeter between the two contacts of the broken bulb's socket (with the string plugged in), deduce with reasoning what it will read: $0\ \text{V}$, $23\ \text{V}$, or $230\ \text{V}$.
3. c) Some of these bulbs had a special conductive wire that created an internal bypass closing the circuit when the filament burned out. If $2$ bulbs burn out with this system, calculate the voltage that the $8$ remaining bulbs will now receive, and predict the risk involved if more bulbs continue to burn out.

A power strip connected to the $230\ \text{V}$ mains indicates a maximum permissible current of $10\ \text{A}$. The following devices are plugged into it at the same time: a TV ($150\ \text{W}$), a computer ($350\ \text{W}$), and a game console ($200\ \text{W}$).

1. a) Justify why electrical outlets in homes and power strips are connected in parallel and not in series, explaining what would happen to the voltage and the other appliances if they were connected in series and you turned one of them off.
2. b) Calculate the total power consumed and the current flowing through the main cable of the power strip with these three devices running. Is the installation operating within the safety margin?
3. c) If an $1.800\ \text{W}$ space heater and an $800\ \text{W}$ hairdryer are also plugged into the same power strip, predict through calculations whether the $10\ \text{A}$ limit of the power strip will be exceeded and explain what physical risk this overload poses due to the Joule effect.