27.6
Kirchoff's Rules: Application
Kirchhoff's rules quantify the current flowing through a circuit and the voltage variations around the loop in a circuit. Applying Kirchhoff's rules g…
An 18 Volt battery is used to charge a run-down battery of 2 ohm internal resistance and lights a bulb of 3 ohms. The current through the run-down battery and the bulb is 1 ampere and 2 amperes, respectively, in the shown direction.
What will be the current and internal resistance of the power supplying battery and the emf of the run-down battery?
Since there are three unknown quantities, three equations will be required.
The first equation is obtained by applying Kirchhoff's first rule at junction "a". This gives the current flowing through the power supplying battery.
The second equation is obtained by applying Kirchhoff's second rule, along the outer loop of the circuit.
Solving this expression results in the value of internal resistance of the power supplying battery.
The third equation is obtained by applying the loop rule in the left-hand loop. Solving this equation gives the emf of the run-down battery.
The negative sign indicates that the actual polarity of the battery is opposite to the assumed polarity in the circuit.
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Q1: How do you apply Kirchhoff's junction rule to find unknown currents?
Kirchhoff's junction rule states that the sum of currents entering a junction equals the sum leaving it. Label the current in each branch and assign a direction. If your chosen direction is incorrect, the calculated current will be negative, but its magnitude remains accurate. This rule generates the first equation needed to solve for unknown values in multi-loop circuits.
Q2: What is the correct sign convention when traversing a voltage source in a loop?
When applying Kirchhoff's loop rule, the electromotive force is positive when traveling through a voltage source from its negative to positive terminal. It becomes negative when traveling in the reverse direction. Consistent sign conventions ensure accurate equations and correct solutions for unknown voltages and currents in the circuit.
Q3: How do you determine the potential drop sign across a resistor in a loop?
When traversing a resistor, the potential drop is negative if your loop direction matches the assumed current direction through it. The potential drop is positive when your loop direction opposes the assumed current direction. Applying this sign convention consistently across all resistors in each loop ensures correct linear equations for solving circuit unknowns.
Q4: Why might a calculated battery emf be negative in Kirchhoff's analysis?
A negative emf result indicates that the actual polarity of the battery is opposite to the polarity you initially assumed in the circuit diagram. The magnitude of the negative value is correct; the sign simply tells you to reverse the battery's terminals. This outcome is normal and helps verify your circuit assumptions during analysis.
Q5: How many independent equations do you need to solve a circuit with three unknowns?
You need exactly three independent equations to solve for three unknowns. Apply Kirchhoff's junction rule at appropriate junctions and Kirchhoff's loop rule around distinct closed loops until you have enough equations. The number of independent equations must equal the number of unknowns to obtain a unique solution for currents, voltages, or resistances.
Q6: What is the first step in applying Kirchhoff's method to analyze a circuit?
The first step is to recognize all circuit elements and identify the polarity of each emf source. Then assign labels to all known quantities and symbols to all unknowns. Next, imagine and assign current directions in each part of the circuit. This systematic preparation ensures you have all necessary information before applying Kirchhoff's junction and loop rules.
Q7: How does Kirchhoff's loop rule help solve for unknown internal resistance?
Applying Kirchhoff's loop rule around an outer loop generates an equation containing the unknown internal resistance. By traversing the loop and accounting for all emf sources and resistive drops with correct signs, you create a linear equation. Solving this equation algebraically yields the internal resistance value, which cannot be found using the junction rule alone.