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Electricity and Magnetism

Kirchhoff's Laws

Kirchhoff's laws are fundamental circuit theorems used to analyze complex electrical networks containing multiple loops and junctions. They are direct consequences of the conservation of charge and energy.

UP TGT ScienceUP PGT PhysicsStudy Notes + Practice

Scope and prerequisites

TGT core

Application of Kirchhoff's Current Law (KCL) and Kirchhoff's Voltage Law (KVL) to simple resistor networks, nodal analysis, and basic Wheatstone bridge balancing.

PGT extension

Application to multi-loop circuits involving complex impedances, active sources, transient growth/decay in L-R/R-C circuits, and network theorem applications (Thevenin, Norton).

Prerequisites

Ohm's Law and resistance in series and parallel
Conservation of electric charge and conservation of energy
Sign conventions for potential drops across resistors and sources

Kirchhoff's Current Law (KCL / First Law)

States that the algebraic sum of electric currents meeting at any junction in a closed circuit is zero.
Mathematically expressed as \sum I = 0, representing the principle of conservation of electric charge.
Currents flowing towards a junction are taken as positive, while currents flowing away from the junction are taken as negative.

Kirchhoff's Voltage Law (KVL / Second Law)

States that the algebraic sum of potential changes around any closed loop in a network is zero.
Mathematically expressed as \sum \Delta V = 0, representing the principle of conservation of energy for electrostatic fields.
Potential drops in the direction of assumed current are taken as negative, while potential rises from negative to positive terminals of sources are positive.

Sign Conventions for Circuit Elements

For a resistor, traversing in the direction of current yields a potential drop of -IR.
For a resistor, traversing opposite to the current yields a potential rise of +IR.
For a DC voltage source (battery), traversing from the negative to the positive terminal gives a potential rise of +E.

Application to DC Network Reduction

Used systematically to reduce complex ladder networks or bridge circuits into solvable systems of linear equations.
Reduces to N-1 independent junction equations for N junctions and B - N + 1 independent loop equations for B branches.
Applicable to linear, bilateral networks containing independent and dependent voltage and current sources.

Derivations

Wheatstone Bridge Balancing Condition via Kirchhoff's Laws

Conditions: Steady state DC bridge network with four resistors P, Q, R, S and a galvanometer of resistance G.

Apply KCL at junctions B and D to assign branch currents: Let main current be I, currents through arms AB, AD be I_1 and I_2, and through galvanometer be I_g.
Apply KVL to closed loop ABDA: I_1P + I_gG - $I_2R = 0$.
Apply KVL to closed loop BCDB: (I_1 - I_g)Q - (I_2 + I_g)S - $I_gG = 0$.
At balance condition, the galvanometer shows zero deflection, hence $I_g = 0$.
Substitute $I_g = 0 into loop equations$ giving $I_1P = I_2R$ and $I_1Q = I_2S$.
Divide the two resulting equations to eliminate currents: P / $Q = R / S$.

Result: \frac{P}{Q} = \frac{R}{S}

Two-Loop Circuit Current Determination

Conditions: Two independent loops with EMFs E_1, E_2 and internal resistances r_1, r_2 connected to load resistors.

Assign loop currents I_1 and I_2 to two adjacent meshes sharing a common branch resistor r_3 carrying current (I_1 - I_2).
Write KVL for Mesh 1: E_1 - I_1 R_1 - (I_1 - $I_2)r_3 = 0$.
Write KVL for Mesh 2: E_2 - I_2 R_2 - (I_2 - $I_1)r_3 = 0$.
Rearrange into simultaneous linear equations in terms of unknown mesh currents I_1 and I_2.
Solve the linear system using Cramer's rule or algebraic substitution to find branch currents.

Result: $I_1 = \frac{E_1(R_2 + r_3) + E_2 r_3}{(R_1 + r_3)(R_2 + r_3) - r_3^2}$

Formula sheet

Formula / law	Equation	Variables and units	Conditions
Kirchhoff's Current Law (KCL)	$$\sum_{k=1}^{n} I_k = 0$$	$I_k = Branch currents entering or leaving a node$ Ampere (A)	Applicable at any circuit node in steady or transient state
Kirchhoff's Voltage Law (KVL)	$$\sum_{k=1}^{m} V_k = \sum \Delta V = 0$$	$V_k = Potential drops$ and EMFs around a closed loop Volt (V)	Applicable around any closed loop
Ohmic Drop Term	$$V_{drop} = I R$$	I = Current through resistor, R = Resistance Volt (V)	Linear, ohmic resistors
Wheatstone Bridge Balance Condition	$$\frac{P}{Q} = \frac{R}{S}$$	P, Q, R, S = Arm resistances of the bridge Dimensionless ratio	Galvanometer current $I_g = 0$
Power Balance in a Network	$$\sum E I = \sum I^2 R$$	E = Source EMF, I = Current, R = Resistance Watt (W)	Total power generated equals total power dissipated
Nodal Equation form	$$V_j \sum \frac{1}{R_i} - \sum \frac{V_m}{R_m} = \sum I_{source}$$	$V_j = Node potential$, $R_i = Connected resistances$ Ampere (A)	Applied at non-reference nodes
Mesh Current Matrix Form	$$[R][I] = [V]$$	[R] = Resistance matrix, [I] = Mesh currents, [V] = Mesh EMFs Various	Planar networks
Conservation Expression for Charge at Node	$$\sum I_{in} = \sum I_{out}$$	I_{in}, $I_{out} = Incoming$ and outgoing currents Ampere (A)	Steady state DC current continuity

Graphs and experiments

Meter Bridge Experiment Setup (Practical Application of KVL/Balanced Bridge)

Draw: A uniform resistance wire of length 100 cm mounted on a wooden board, connected to a battery, resistance box, unknown resistance, and a jockey via a galvanometer.

Interpret: At balance point length l, the ratio of lengths substitutes for resistance ratios: R / $S = l / (100 - l)$.

Exam use: Tested for calculating unknown resistance and error minimization in resistance measurements.

Potential Drop Profile Across a Closed Loop

Draw: A step-up and step-down potential profile graph plotting electrical potential V against path traversal distance along a closed loop.

Interpret: Shows that traversing a complete closed loop returns the potential to its original value, confirming conservative nature of electrostatic potential field.

Exam use: Conceptual verification of KVL conservation of energy.

Galvanometer Deflection vs. Jockey Position Graph

Draw: A V-shaped curve of galvanometer deflection against the position of the jockey along the meter bridge wire, crossing zero at the null point.

Interpret: The sharp zero-crossing allows precise identification of the balance length where bridge network is balanced.

Exam use: Practical sensitivity analysis of bridge circuits.

Solved numericals

TGT core

Find the current I in a single loop circuit containing an EMF source of 12 V, internal resistance 1 \Omega, and an external load resistor of 5 \Omega.

Apply KVL to the single loop: E - I r - I R = 0.
Substitute values: 12 - I(1) - I(5) = 0.
Solve for I: 12 - 6I = 0 \Rightarrow I = 2 A.

Answer: 2 A

TGT core

At a junction, three currents meet: $I_1 = 4 A entering$, $I_2 = 3 A entering$, and I_3 leaving the junction. Find the magnitude and direction of I_3.

Apply KCL at the junction: \sum $I_{in} = \sum I_{out}$.
Substitute values: 4 A + 3 $A = I_3$.
Calculate result: $I_3 = 7 A leaving the junction$.

Answer: 7 A leaving the junction

PGT extension

In a two-loop network, mesh equations are given by 4I_1 - 2$I_2 = 10$ and -2I_1 + 6$I_2 = 0$. Find the mesh currents I_1 and I_2.

From the second equation, express I_1 in terms of I_2: 2$I_1 = 6I_2 \Rightarrow I_1 = 3I_2$.
Substitute I_1 into the first equation: 4(3I_2) - 2$I_2 = 10$.
Simplify: 12I_2 - 2$I_2 = 10 \Rightarrow 10I_2 = 10 \Rightarrow I_2 = 1 A$.
Calculate I_1: $I_1 = 3(1) = 3 A$.

Answer: $I_1 = 3 A$, $I_2 = 1 A$

Assumptions and limitations

Kirchhoff's Voltage Law assumes negligible magnetic flux variation through the closed loop; if changing magnetic fields are present, induced electromotive forces (Faraday's law) must be included.
Lumped-element circuit model assumes circuit dimensions are much smaller than the electromagnetic wavelength of the operating signals.

Common misconceptions

Common error: KVL is based on the conservation of electric charge.

Correct: KVL is based on the conservation of energy.

Potential difference represents work done per unit charge, connecting directly to electrostatic potential energy conservation.

Common error: Sign conventions for KVL depend on the absolute direction of current flow.

Correct: Sign conventions depend on the chosen traversal direction of the loop, regardless of assumed current directions.

Current directions are initially assumed; correct signs emerge during algebraic solution of the simultaneous equations.

Quick revision

KCL stems from charge conservation (\sum I = 0) and KVL stems from energy conservation (\sum \Delta V = 0).
Kirchhoff's Current Law (KCL): \su$m_{k=1}^{n} I_k = 0$
Kirchhoff's Voltage Law (KVL): \su$m_{k=1}^{m} V_k = \sum \Delta V = 0$

Must remember

Kirchhoff's Current Law (KCL) is based on the principle of conservation of electric charge ($\sum I = 0$).
Kirchhoff's Voltage Law (KVL) is based on the principle of conservation of energy ($\sum \Delta V = 0$).
When traversing a resistor in the direction of the assumed current, the potential change is $-IR$.
When traversing a DC voltage source from negative to positive terminal, the potential change is $+E$.
The Wheatstone bridge balance condition is given by $\frac{P}{Q} = \frac{R}{S}$ when galvanometer current is zero.

Glossary

Kirchhoff's Current Law (KCL)States that the algebraic sum of electric currents meeting at any junction is zero ($\sum I = 0$).

Kirchhoff's Voltage Law (KVL)States that the algebraic sum of potential changes around any closed loop is zero ($\sum \Delta V = 0$).

NodeA point in a circuit where two or more circuit elements meet.

MeshA closed loop that does not contain any inner loops.

Wheatstone BridgeAn electrical circuit used to measure an unknown electrical resistance by balancing two legs of a bridge circuit.

KCL vs KVL

Feature	Kirchhoff's Current Law (KCL)	Kirchhoff's Voltage Law (KVL)
Basic Principle	Conservation of Charge	Conservation of Energy
Mathematical Form	$\sum I = 0$	$\sum \Delta V = 0$
Application Point	Junctions or Nodes	Closed Loops or Meshes
Primary Variable	Electric Current ($I$)	Potential Difference ($V$ or $E$)

Practice Quiz

Choose an option to reveal the explanation.

1. Which fundamental conservation law is the basis for Kirchhoff's Current Law (KCL)?

Explanation: KCL states that $\sum I = 0$ at any junction, which fundamentally represents the conservation of electric charge at a node.

2. Which fundamental conservation law is the basis for Kirchhoff's Voltage Law (KVL)?

Explanation: KVL states that $\sum \Delta V = 0$ around a closed loop, representing the conservation of electrostatic potential energy per unit charge.

3. When traversing a resistor in the direction opposite to the assumed current flow, what is the potential change?

Explanation: Moving opposite to the current direction across a resistor encounters a potential rise, represented as $+IR$.

4. What is the potential change when traversing a DC voltage source from its negative terminal to its positive terminal?

Explanation: Moving from the negative to the positive terminal of a battery represents a potential rise of $+E$.

5. For a bridge network with four arm resistances $P$, $Q$, $R$, and $S$, what is the condition for the Wheatstone bridge to be balanced?

Explanation: The balance condition for a Wheatstone bridge occurs when the galvanometer current is zero, leading to the ratio $\frac{P}{Q} = \frac{R}{S}$.

6. How many independent junction equations can be formed for a network with $N$ junctions using KCL?

Explanation: For $N$ junctions in a network, there are $N - 1$ independent junction equations because one junction serves as the reference (ground).

7. In a network with $B$ branches and $N$ junctions, how many independent loop equations are required?

Explanation: The number of independent loop equations in a connected planar network is given by $B - N + 1$.

8. In KCL calculations, currents flowing towards a junction are conventionally assigned which sign?

Explanation: By convention, currents flowing towards a junction are taken as positive, while currents flowing away are taken as negative.

9. If a network contains active sources and complex impedances, what extension of KCL/KVL is typically applied in PGT-level analysis?

Explanation: For complex AC or multi-loop DC/transient networks, systematic mesh and nodal matrix formulations using linear equations are employed.

10. What is the unit of the Ohmic drop term $V_{drop} = I R$?

Explanation: Potential drop is measured in Volts (V), calculated as the product of current in Amperes and resistance in Ohms.

11. In a network power balance, the total power generated by sources equals:

Explanation: By the conservation of energy in electrical circuits, the total power supplied by sources ($\sum E I$) equals the total power dissipated in resistors ($\sum I^2 R$).

12. Which of the following is a correct statement regarding the sign convention for KVL loops?

Explanation: When traversing in the direction of the assumed current through a resistor, the potential change is a drop, expressed as $-IR$.

13. What does a nodal equation at a non-reference node equate the sum of currents leaving the node to?

Explanation: Nodal analysis applies KCL where the sum of currents leaving through connected conductances equals the sum of independent current sources entering the node.

14. Which network theorem utilizes successive application of KCL and KVL to replace a linear active network with an equivalent voltage source and series resistance?

Explanation: Thevenin's theorem simplifies linear networks into an equivalent voltage source ($V_{th}$) and series resistance ($R_{th}$) derived using circuit laws.

15. In transient $L-R$ circuits, how are Kirchhoff's laws applied initially at $t = 0^+$?

Explanation: At $t = 0^+$, an inductor opposes sudden changes in current, acting as an open circuit ($I = 0$), which sets initial boundary conditions for KVL equations.

PYQ Patterns

These are pattern-based practice questions. Authentic year labels are only used when verified from a source.

1. In a closed circuit loop, the algebraic sum of products of current and resistance in each part of the circuit plus the algebraic sum of emfs is equal to:

Explanation: This is a direct statement of Kirchhoff's Voltage Law ($\sum \Delta V = 0$), which states that the sum of potential drops and emfs around any closed loop is zero.

2. At a junction in an electrical network where five wires meet carrying currents $2\text{ A}$, $3\text{ A}$, $-1\text{ A}$, and $4\text{ A}$ (with incoming positive and outgoing negative), what is the value of the fifth current $I_5$?

Explanation: According to KCL, $\sum I = 0$. Thus $2 + 3 + (-1) + 4 + I_5 = 0 \implies 8 + I_5 = 0 \implies I_5 = -8\text{ A}$.

3. Which of the following circuit theorems or laws is directly tied to the principle of conservation of energy?

Explanation: KVL is based on conservation of energy because potential difference represents work done per unit charge around a closed path.

4. In applying KVL to a single-loop circuit containing a battery of EMF $E$ and internal resistance $r$ connected to an external load $R$, the current $I$ is given by:

Explanation: Applying KVL around the loop: $E - Ir - IR = 0 \implies I = \frac{E}{R + r}$.

5. What is the primary purpose of formulating mesh current equations in planar networks?

Explanation: Mesh analysis using KVL automatically satisfies KCL at junctions, reducing the number of simultaneous linear equations to be solved.

6. In a Wheatstone bridge circuit, if the galvanometer shows zero deflection when $P = 100\,\Omega$, $Q = 100\,\Omega$, and $R = 50\,\Omega$, what is the unknown resistance $S$?

Explanation: Using the balance condition $\frac{P}{Q} = \frac{R}{S}$, we get $\frac{100}{100} = \frac{50}{S} \implies S = 50\,\Omega$.

Timed Topic Test

Attempt these questions under a ten-minute limit using a separate timer.

1. State which law is violated if energy is created out of nothing in a closed electrical loop analysis.

Explanation: KVL represents conservation of energy; a non-zero net potential change around a closed loop without sources would violate conservation of energy.

2. In a nodal analysis setup, if node 1 has potential $V_1$ connected via resistors $R_1, R_2, R_3$ to nodes of potentials $V_2, V_3$ and ground ($0\text{ V}$), what is the coefficient of $V_1$?

Explanation: From the nodal equation form $V_j \sum \frac{1}{R_i} - \sum \frac{V_m}{R_m} = \sum I_{source}$, the self-conductance attached to node $V_1$ is the sum of inverse resistances connected to it.

3. Correct coefficient of $V_1$ in the nodal conductance sum is:

Explanation: The multiplier for node potential $V_j$ in a nodal equation is the sum of conductances ($\frac{1}{R_i}$) connected directly to that node.

4. A battery of EMF $12\text{ V}$ and internal resistance $2\,\Omega$ is connected across a load resistor $10\,\Omega$. What is the terminal voltage across the load?

Explanation: Current $I = \frac{12}{10 + 2} = 1\text{ A}$. Terminal voltage $V = E - Ir = 12 - (1)(2) = 10\text{ V}$.

5. When applying KVL to an active multi-loop circuit, how many independent loop equations are needed for a circuit with 4 branches and 3 nodes?

Explanation: Using $B - N + 1 = 4 - 3 + 1 = 2$ independent loop equations.

6. Which of the following statements about KCL is incorrect?

Explanation: KCL assumes no accumulation of charge at a node (steady-state conservation of charge), meaning charge does not accumulate indefinitely.

7. In matrix form $[R][I] = [V]$ for mesh analysis, what does the off-diagonal element $R_{ij}$ ($i \neq j$) represent?

Explanation: In mesh resistance matrices, off-diagonal terms represent the common resistance shared between adjacent loops $i$ and $j$.

8. If all resistances in a balanced Wheatstone bridge are doubled, what happens to the balance condition?

Explanation: The balance condition $\frac{P}{Q} = \frac{R}{S}$ depends on resistance ratios; scaling all resistances by a factor of 2 leaves the ratio unchanged.

9. What is the power dissipated in a $5\,\Omega$ resistor carrying a current of $2\text{ A}$?

Explanation: Power $P = I^2 R = (2)^2 \times 5 = 4 \times 5 = 20\text{ W}$.

10. In a complex circuit, if an ideal current source is present in a branch, how does it affect mesh analysis formulation?

Explanation: An ideal current source fixes the branch current, requiring the combination of adjacent meshes into a 'supermesh' to bypass the unknown voltage across the source when applying KVL.