Premed · Premed · Physics 2
Lecture 7: DC Circuits: Kirchhoff's Laws
Physics II — Electromagnetism, Optics & Modern Physics
Learning Objectives
By the end of this lecture, students will be able to:
- Calculate equivalent resistance for resistors in series and parallel
- State and apply Kirchhoff's junction rule (current conservation) and loop rule (voltage conservation)
- Systematically solve multi-loop DC circuits using Kirchhoff's laws
- Analyze circuits containing multiple batteries and resistors
- Understand the operation of ammeters, voltmeters, and Wheatstone bridges
Lecture Content
I. Resistors in Series and Parallel
In a series combination, the same current flows through all resistors. The voltages across individual resistors add to give the total voltage: V_total = V_1 + V_2 + V_3 + ... The equivalent resistance is R_eq = R_1 + R_2 + R_3 + ..., so a series combination always increases the total resistance. The current in the circuit is I = V_total / R_eq.
In a parallel combination, all resistors share the same voltage. The currents through individual resistors add to give the total current: I_total = I_1 + I_2 + I_3 + ... The equivalent resistance satisfies 1/R_eq = 1/R_1 + 1/R_2 + 1/R_3 + ..., so a parallel combination always decreases the total resistance. For two resistors in parallel, R_eq = R_1 R_2 / (R_1 + R_2), which is always less than the smaller individual resistance.
Two useful special cases arise frequently. A voltage divider consists of two resistors R_1 and R_2 in series, where the voltage across each is V_1 = V_total x R_1 / (R_1 + R_2) and V_2 = V_total x R_2 / (R_1 + R_2). A current divider consists of two resistors R_1 and R_2 in parallel, where the current through R_1 is I_1 = I_total x R_2 / (R_1 + R_2). More current always flows through the smaller resistance.
II. Kirchhoff's Junction Rule (KCL)
At any junction (node) in a circuit, the sum of currents entering equals the sum of currents leaving: Sum of I_in = Sum of I_out. Equivalently, the algebraic sum of all currents at a junction is zero. This rule is a direct statement of conservation of charge, since charge cannot accumulate at a junction under steady-state conditions.
To apply this rule, assign a current variable and an assumed direction to each branch of the circuit. If the assumed direction turns out to be wrong, the calculated current will simply be negative, which is perfectly valid. Write one junction equation for each independent node.
III. Kirchhoff's Loop Rule (KVL)
Around any closed loop in a circuit, the sum of all voltage changes is zero: Sum of Delta V = 0. This rule is a statement of conservation of energy, since a charge that travels around a complete loop and returns to its starting point must return to the same potential.
Applying the loop rule requires consistent sign conventions. When traversing a loop through a battery from the negative to the positive terminal, the voltage increases by +epsilon; traversing from positive to negative gives -epsilon. When traversing a resistor in the direction of current flow, the voltage drops by -IR; traversing against the current gives +IR. Choose a consistent direction (clockwise or counterclockwise) for each loop.
<image>A two-loop circuit containing a battery epsilon_1 in the left loop and a battery epsilon_2 in the right loop, with three resistors: R_1 in the left branch, R_2 in the shared middle branch, and R_3 in the right branch. Current directions I_1 (clockwise in left loop), I_2 (downward in middle branch), and I_3 (clockwise in right loop) are labeled with arrows. Junction points A and B are marked where branches meet. Kirchhoff's junction equation at point A: I_1 = I_2 + I_3 is shown. Loop equations for both loops are written below with proper sign conventions indicated.</image>
IV. Systematic Method for Solving Kirchhoff's Problems
A systematic procedure ensures success with even complex circuits. First, label all currents by assigning a variable and assumed direction to each branch. Second, identify independent junctions and write (N - 1) junction equations for N nodes. Third, identify independent loops and write enough loop equations so that the total number of equations equals the number of unknowns. The number of independent loops equals the number of branches minus the number of nodes plus one. Fourth, write the equations using the sign conventions consistently. Fifth, solve the system of simultaneous equations. Sixth, interpret the results: any negative current value means the actual direction of flow is opposite to the assumed direction.
For a circuit with b branches and n nodes, you need b equations in total: (n - 1) from the junction rule and (b - n + 1) from the loop rule.
V. Measuring Instruments
An ammeter measures current and must be connected in series with the element being measured. To minimize its effect on the circuit, an ammeter must have very low internal resistance (ideally zero). It is constructed from a galvanometer with a small shunt resistor in parallel.
A voltmeter measures the potential difference across a circuit element and must be connected in parallel. To avoid drawing significant current from the circuit, a voltmeter must have very high internal resistance (ideally infinite). It is constructed from a galvanometer with a large multiplier resistor in series.
In practice, non-ideal instruments introduce loading effects. A non-ideal ammeter reduces the current in the circuit due to its finite resistance, while a non-ideal voltmeter draws current and reduces the measured voltage below its true value. Understanding these effects is essential for accurate measurements.
VI. Wheatstone Bridge and Other Applications
The Wheatstone bridge is a circuit designed for precisely measuring unknown resistances. Four resistors are arranged in a diamond configuration with a galvanometer across the middle. When the galvanometer reads zero current (the balanced condition), the unknown resistance is given by R_unknown = R_3 (R_2 / R_1). This technique is extremely sensitive and is used in strain gauges, temperature sensors, and precision measurements.
In multi-battery circuits, when batteries oppose each other, the net EMF determines the direction of current flow. Current flows from the higher EMF to the lower EMF through the external circuit, and the lower EMF battery is effectively being charged.
Grounding a circuit defines a reference point where V = 0. A ground connection does not change the current flow in the circuit; it merely establishes an absolute voltage reference against which all other potentials in the circuit can be measured.
<image>A Wheatstone bridge circuit diagram. Four resistors (R_1, R_2, R_3, and R_x for the unknown) are arranged in a diamond/bridge configuration. A battery with EMF epsilon is connected across one diagonal (top and bottom nodes). A galvanometer G is connected across the other diagonal (left and right nodes). Current directions are labeled in each branch. The balance condition R_x = R_3(R_2/R_1) is displayed below, along with the note that at balance, no current flows through the galvanometer (I_G = 0).</image>
