11.6. RESISTANCE OF A SYSTEM OF RESISTORS

What happens to the value of current when a number of resistors are connected in series in a circuit? What would be their equivalent resistance? Let us try to understand these with the help of the following activities.

Join three resistors of different values in series. Connect them with a battery, an ammeter and a plug key. You may use the resistors of values like 1 Ω, 2 Ω, 3 Ω etc., and a battery of 6 V for performing this Activity.

When resistors are connected in series, the same current flows through each resistor. By measuring the current with an ammeter at different positions in the circuit, you will find that the current remains constant regardless of where you measure it. The total voltage across the combination of resistors is equal to the sum of the individual voltage drops across each resistor. Mathematically, if you denote the total voltage as V, and the individual resistances as R1, R2, and R3, then the total potential difference across the series combination, V = V1 + V2 + V3, holds true. The equivalent resistance Rᵢ for resistors in series can be calculated using the formula Rᵢ = R1 + R2 + R3.

Imagine a long line of people passing a message from one to another. Each person (resistor) has to then pass the message (electric current) along to the next. The same message flows through everyone in sequence, just like current flows through each resistor in series. If one person stops passing the message (as in a faulty resistor), the entire process halts, showing that the entire series depends on every individual part functioning.

In Activity 11.4, insert a voltmeter across the ends X and Y of the series combination of three resistors. Plug the key in the circuit and note the voltmeter reading. It gives the potential difference across the series combination of resistors. Now measure the potential difference across the two terminals of the battery.

Connecting a voltmeter in parallel to the series combination of resistors allows us to measure the total voltage across all resistors at once. Then, by measuring the voltage across each individual resistor, you can see how the voltage divides across each component. If you note the voltages across each resistor, they will add up to equal the total voltage from the battery, confirming the relationship V = V1 + V2 + V3.

Think of this like water flowing through a series of taps (the resistors). The total amount of water pressure (voltage from the battery) needs to push equally against all taps before it flows into the next part of the system. Each tap uses some of that pressure, reflecting how voltage 'drops' across each resistor in the series.

It is possible to replace the three resistors joined in series by an equivalent single resistor of resistance Rᵢ, such that the potential difference V across it, and the current I through the circuit remains the same. Applying Ohm’s law to the entire circuit, we have V = I Rᵢ.

When resistors are connected in series, you can simplify the circuit by replacing the combination with a single equivalent resistor. This equivalent resistor will be equal to the sum of the resistances of the individual resistors. Thus, using Ohm's Law, knowing the total current and voltage allows you to find the equivalent resistance that would maintain the same electrical behavior as the original combination.

Imagine a long hallway (representing resistors) where you can only pass one person (the current) at a time. If you consolidate this hallway into a smaller one (the equivalent resistor), the same number of people can still move through, but it is more manageable and easier to analyze the flow.

Now, let us consider the arrangement of three resistors joined in parallel. Make a parallel combination XY of three resistors having resistances R1, R2, and R3, respectively. Connect it with a battery, a plug key and an ammeter.

In a parallel circuit, each resistor is connected across the same two points, which means that each one has the same voltage across it. The total current flowing from the battery is divided among the different paths available for current to flow through. Thus, if R1, R2, and R3 represent the individual resistances, the total current (I) is equal to the sum of the currents through each resistor: I = I1 + I2 + I3. The total equivalent resistance in parallel reduces overall current in the circuit, calculated by the formula 1/Rᵢ = 1/R1 + 1/R2 + 1/R3.

It's like a multi-lane highway where cars (the current) can choose different paths (the resistors). Each lane has an equal speed limit (voltage), and cars can take any lane, making traffic flow smoother and faster. The more lanes available, the easier it becomes for cars to move through the area.

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