Welcome to the World of Combined Circuits!

In our previous lessons, we looked at single resistors. But in the real world—like inside your smartphone or a toaster—resistors are rarely alone. They work together in teams! Today, we are going to learn how to calculate the "total" resistance when we group these components together. Don't worry if this seems a bit abstract at first; by the end of these notes, you'll be calculating complex circuits like a pro.

Quick Refresh: Remember that Resistance (\(R\)) is how much a component opposes the flow of Current (\(I\)). We calculate it using Ohm's Law: \(V = IR\), where \(V\) is the Potential Difference (p.d.).


1. Resistors in Series: The "Single Lane" Path

When resistors are connected in series, they are placed end-to-end in a single line. There is only one path for the current to flow.

How it Works:

  • Current (\(I\)): Because there is only one path, the current is the same through every resistor. Think of it like a single-lane road: every car that passes through the first toll booth must also pass through the second.
  • Potential Difference (\(V\)): The total voltage from the battery is shared across the resistors. \(V_{total} = V_1 + V_2 + ...\)
  • Total Resistance (\(R_{total}\)): Since the current has to fight through every single resistor one after another, the total resistance is simply the sum of all individual resistances.

The Formula:

\(R_{total} = R_1 + R_2 + R_3 + ...\)

Example: If you have a \(5 \Omega\) resistor and a \(10 \Omega\) resistor in series, the total resistance is \(5 + 10 = 15 \Omega\).

Analogy: Imagine walking through a series of narrow doorways. Each doorway slows you down. The more doorways you add in a row, the harder it is to get to the end!

Quick Review Box: In series, \(R\) increases as you add more resistors. The total \(R\) is always larger than the largest individual resistor.

Key Takeaway: For series circuits, just add them up!


2. Resistors in Parallel: The "Multiple Lane" Choice

In a parallel circuit, resistors are connected across the same two points. The current reaches a junction and has to split up into different branches.

How it Works:

  • Potential Difference (\(V\)): This is the "magic" of parallel circuits! The p.d. across each branch is exactly the same. If you have a 12V battery, every branch gets the full 12V.
  • Current (\(I\)): The total current from the source is the sum of the currents in the individual branches. \(I_{total} = I_1 + I_2 + ...\)
  • Total Resistance (\(R_{total}\)): Adding more branches actually decreases the total resistance! This is because you are providing more paths for the electricity to flow through.

The Formula:

\(\frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + ...\)

Did you know? Your house is wired in parallel. This is why you can turn off the light in the kitchen without the TV in the living room turning off too!

Common Mistake to Avoid: When using the parallel formula, students often forget to do the final "flip." If you calculate \(\frac{1}{R_{total}} = 0.5\), your answer isn't 0.5! You must calculate \(1 \div 0.5\) to get \(R_{total} = 2 \Omega\).

Analogy: Think of a supermarket checkout. If only one lane is open, there is high resistance. If the manager opens three more lanes (parallel paths), even though there are more "obstacles" (cashiers), the total resistance to the flow of customers decreases.

Key Takeaway: In parallel, \(R_{total}\) is always smaller than the smallest resistor in the group.


3. The Potential Divider: Sharing Voltage

A potential divider is a simple circuit that uses two or more resistors in series to "divide" the input voltage (\(V_{in}\)) into a specific output voltage (\(V_{out}\)).

The Potential Divider Formula:

If you want to find the voltage across Resistor 2 (\(R_2\)), use this:

\(V_{out} = \frac{R_2}{R_1 + R_2} \times V_{in}\)

Step-by-Step Logic:
1. Calculate the total resistance (\(R_1 + R_2\)).
2. Find the ratio of the resistor you're interested in compared to the total.
3. Multiply that ratio by the total voltage supply.

Using Sensors (LDRs and Thermistors):

In the H1 syllabus, you often see potential dividers used with input transducers:

  • LDR (Light Dependent Resistor): Resistance decreases when light intensity increases. (Mnemonic: LURD - Light Up, Resistance Down).
  • NTC Thermistor: Resistance decreases when temperature increases. (Mnemonic: TURD - Temperature Up, Resistance Down).

Practical Example: In a night-light circuit, as it gets dark, the LDR's resistance increases. This causes it to "grab" a larger share of the voltage, which can then be used to switch on a bulb!

Key Takeaway: The "bigger" resistor always gets a "bigger" share of the voltage in a series circuit.


4. Summary Checklist for Success

Before you tackle practice problems, keep these Top Tips in mind:

  • Check the Units: Are the resistors in \(\Omega\) or \(k\Omega\)? Always convert to \(\Omega\) (\(\times 10^3\)) before calculating if they are mixed!
  • Series = Current Same: If you find the current through one series resistor, you know the current through all of them.
  • Parallel = Voltage Same: If you know the voltage across one parallel branch, you know the voltage across all of them.
  • Simplify First: If you have a "messy" circuit with both series and parallel parts, look for the parallel groups first. Calculate their total resistance to turn them into one "equivalent" resistor, then solve the rest as a series circuit.

Don't worry if this feels like a lot of math at first. Physics is a skill—the more you practice "collapsing" these circuits on paper, the more natural it will become!