Electricity and Magnetism: electric circuits (internal resistance and series-parallel networks) – Week 3 focus
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Subject: Physical Sciences
Class: Grade 12
Term: 2nd Term
Week: 3
Theme: General lesson support
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This week, we delve into the fascinating world of electric circuits, focusing on the crucial aspects of internal resistance and series-parallel networks. Understanding these concepts is paramount not just for excelling in Physical Sciences but also for comprehending how everyday devices function and how electrical energy is distributed in our homes and communities. For example, load shedding affects homes, and understanding the internal resistance of batteries in backup systems (like inverters) is vital. Comprehending series-parallel circuits helps us troubleshoot faulty wiring in buildings.
2.1 Internal Resistance (r) Every real battery or power source has internal resistance (represented by 'r'). This internal resistance is due to the opposition to the flow of charge within the battery itself, caused by the chemical composition and physical structure of the battery. This means that not all the energy supplied by the battery is available to the external circuit. Effect on Terminal Potential Difference (V terminal ): The electromotive force (emf, ε) is the total potential difference a battery can supply.
However, because of internal resistance, the actual potential difference available to the external circuit, called the terminal potential difference (V terminal ), is always less than the emf when the battery is delivering current. The relationship between emf, internal resistance, current (I), and terminal potential difference is: V terminal = ε - Ir ε: Electromotive force (emf) – the total energy supplied per unit charge by the battery (measured in volts). I: Current in the circuit (measured in amperes). r: Internal resistance of the battery (measured in ohms).
V terminal : Potential difference across the terminals of the battery (measured in volts). Why? Imagine a water pump trying to fill a tank. The pump's "emf" is its total potential to push water. But, the pipe connecting the pump to the tank has friction (like internal resistance). Some of the pump's effort is lost overcoming this friction, so the water reaching the tank (V terminal ) is less than the pump's full potential.
Example 1: A battery with an emf of 12V has an internal resistance of 0.5Ω. When a 2.5Ω resistor is connected to the battery, what is the terminal potential difference?
Solution: Calculate the total resistance (R total ): R total = External Resistance + Internal Resistance = 2.5Ω + 0.5Ω = 3Ω Calculate the current (I): Using Ohm's Law (V = IR), but with emf instead of V: ε = IR total => I = ε / R total = 12V / 3Ω = 4A Calculate the terminal potential difference (V terminal ): V terminal = ε - Ir = 12V - (4A)(0.5Ω) = 12V - 2V = 10V Therefore, the terminal potential difference is 10V. 2.2 Series Resistor Networks Resistors in series are connected end-to-end, so the same current flows through each resistor.
Equivalent Resistance (R eq ): The total resistance of resistors in series is the sum of the individual resistances: R eq = R 1 + R 2 + R 3 + ...
Current (I): The current is the same through each resistor in series. I = V / R eq , where V is the total voltage across the series combination.
Potential Difference (V): The potential difference across each resistor is different and proportional to its resistance: V 1 = IR 1 , V 2 = IR 2 , etc. The sum of the potential differences across all resistors in series equals the total potential difference: V = V 1 + V 2 + V 3 + ... 2.3 Parallel Resistor Networks Resistors in parallel are connected side-by-side, so the potential difference across each resistor is the same.
Equivalent Resistance (R eq ): The reciprocal of the total resistance of resistors in parallel is the sum of the reciprocals of the individual resistances: 1/R eq = 1/R 1 + 1/R 2 + 1/R 3 + ...
Current (I): The current through each resistor is different and inversely proportional to its resistance: I 1 = V / R 1 , I 2 = V / R 2 , etc. The sum of the currents through all resistors in parallel equals the total current: I = I 1 + I 2 + I 3 + ...
Potential Difference (V): The potential difference is the same across each resistor in parallel. 2.4 Series-Parallel Resistor Networks These networks combine both series and parallel connections. To analyze them, you must systematically reduce the circuit by finding equivalent resistances for the series and parallel sections.
Example 2: Consider the following circuit: A 6Ω resistor is in series with a parallel combination of a 4Ω resistor and a 12Ω resistor. This whole arrangement is connected to a 12V battery with an internal resistance of 1Ω. Calculate the current through each resistor.
Solution: Find the equivalent resistance of the parallel section (R parallel ): 1/R parallel = 1/4Ω + 1/12Ω = 3/12Ω + 1/12Ω = 4/12Ω R parallel = 12Ω / 4 = 3Ω Find the equivalent resistance of the entire external circuit (R external ): R external = 6Ω + 3Ω = 9Ω Find the total resistance of the circuit, including internal resistance (R total ): R total = 9Ω + 1Ω = 10Ω Calculate the total current (I): I = ε / R total = 12V / 10Ω = 1.2A Calculate the potential difference across the parallel combination (V parallel ): Since the parallel combination is equivalent to a 3Ω resistor, V parallel = I R parallel = 1.2A 3Ω = 3.6V Calculate the current through the 4Ω resistor (I 4Ω ): I 4Ω = V parallel / 4Ω = 3.6V / 4Ω = 0.9A Calculate the current through the 12Ω resistor (I 12Ω ): I 12Ω = V parallel / 12Ω = 3.6V / 12Ω = 0.3A
Note: I 4Ω + I 12Ω = 0.9A + 0.3A = 1.2A, which is the total current, as expected.