27.6:
Kirchoff’s Rules: Application
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.
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 generates a set of linear equations that allow us to find the unknown values in circuits. These may be currents, voltages, or resistances.
When applying Kirchhoff's first rule, the junction rule, label the current in each branch and decide its direction. If the chosen direction is wrong, it will have the correct magnitude, although the current will be negative. When applying Kirchhoff's second rule, the loop rule, identify a closed loop and decide the direction to go around it, clockwise or counterclockwise. In many circuits, it will be necessary to construct more than one loop. When traversing each loop, be consistent with the sign of the change in potential.
In the case of the voltage source, the emf is considered positive when we travel through the source in the direction from a negative to a positive terminal. It is considered to be negative when we travel in the reverse direction. When encountering a resistor in the loop, if the travel direction of the loop through the resistor is the same as that of the assumed current direction, the potential drop across the resistor is negative. However, the potential drop across the resistor is taken as positive when the travel direction is opposite to the assumed current direction.
In conclusion, Kirchhoff's method of analysis requires the following procedure:
- Recognize all the circuit elements, identify the polarity of each emf source, and imagine the current's directions in the emf.
- Assign labels to all the known quantities and symbols to the unknown quantities. Assign the current direction in each part of the circuit.
- Apply Kirchhoff's junction and loop rule.
- Obtain the number of independent equations equal to the number of unknowns and simultaneously solve the equations for unknowns.
Suggested Reading
- Young, H.D. and Freedman, R.A. (2012). University Physics with Modern Physics. San Francisco, CA: Pearson. Pp. 856–858.
- OpenStax. (2019). University Physics Vol. 2. [Web version]. Retrieved from https://openstax.org/details/books/university-physics-volume-2; section 10.4; Pp. 455–462.