27.14
Power Dissipated in a Circuit: Problem Solving
The equivalent resistance of a combination of resistors depends on their values and how they are connected.
The simplest combinations of resistors are…
Consider three resistors of one ohm, two ohms, and four ohms in a parallel configuration connected to a battery of emf four volts. What is the total power supplied by the battery and the power dissipated by each resistor?
First, using the formula for the equivalent resistance, its value is found to be 0.57 ohms.
Substituting this value in Ohm's law, the total current supplied by the battery to the circuit is determined to be seven amperes.
The total power supplied by the battery equals the product of the total current and the source voltage. Its value is calculated to be 28 watts.
The power dissipated by each resistor can be determined using the relation of power with the resistance and voltage. As the voltage received by each resistor is the same, the value of the power dissipated by the first, second, and third resistors are calculated to be 16 watts, eight watts, and four watts.
So, the total power dissipated by the individual resistors equals the power supplied by the battery.
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Q1: How do you calculate the total power supplied by a battery in a parallel circuit?
Total power supplied by the battery equals the product of total current and source voltage. First, find equivalent resistance using the parallel resistance formula, then apply Ohm's law to determine total current. Multiply this current by the battery's emf to get total power in watts. This value should equal the sum of power dissipated by individual resistors, confirming energy conservation.
Q2: Why does each resistor in a parallel circuit receive the same voltage?
In a parallel circuit, all resistor leads connect directly to the same two points, creating identical potential differences across each resistor. This equal voltage distribution is a fundamental characteristic of parallel configurations. However, the current through each resistor varies based on its resistance value, determined by Ohm's law applied individually to each branch.
Q3: What is the relationship between equivalent resistance and individual resistor values in a parallel circuit?
In a parallel circuit, equivalent resistance is smaller than any individual resistor value. The reciprocal of equivalent resistance equals the sum of reciprocals of individual resistances. This inverse relationship means adding resistors in parallel decreases total resistance, allowing more current to flow from the battery compared to series configurations.
Q4: How does power dissipation differ among resistors with different values in a parallel circuit?
Power dissipated by each resistor depends on its resistance value and the voltage across it. Since all resistors receive the same voltage in parallel, lower resistance values dissipate more power. Using P = V²/R, a one-ohm resistor dissipates more power than a four-ohm resistor at identical voltage, demonstrating the inverse relationship between resistance and power dissipation.
Q5: Why must total power dissipated by resistors equal power supplied by the battery?
Energy conservation requires that all power supplied by the battery is dissipated as heat in the resistors. The battery converts chemical energy into electrical energy, which resistors convert to thermal energy. Summing individual power dissipations yields the total battery power, confirming no energy is lost or created in the circuit, only transformed.
Q6: How do you find the current through each resistor in a parallel circuit?
Apply Ohm's law to each resistor individually: I = V/R. Since all resistors experience the same voltage in parallel, divide that voltage by each resistor's resistance to find its current. The sum of all individual currents equals the total current supplied by the battery, which can be verified using the equivalent resistance and total voltage.
Q7: What role does Ohm's law play in solving parallel circuit power problems?
Ohm's law (V = IR) is essential for finding current, voltage, and resistance relationships throughout the circuit. It determines total current from equivalent resistance and battery voltage, calculates individual resistor currents from shared voltage, and enables power calculations using P = VI or P = V²/R. This foundational relationship connects all circuit quantities needed for power analysis.