OCR Specification focus:
‘Kirchhoff’s second law: sum of emfs equals sum of potential drops around a closed loop.’
Kirchhoff’s second law is a fundamental principle in circuit analysis, expressing how energy is conserved in electrical systems. It ensures total supplied energy equals total energy used.
Kirchhoff’s Second Law: The Foundation of Energy Conservation
Kirchhoff’s second law, also known as the loop rule, arises from the principle of conservation of energy. It states that in any closed electrical loop, the total electromotive force (e.m.f.) equals the total potential difference (p.d.) across all components. Essentially, this means that electrical energy supplied by sources is entirely converted into other forms of energy (such as heat or light) by components in the circuit.
Understanding the Principle
In an electric circuit, charges move under the influence of electric fields. As they complete a closed loop, they gain energy when passing through sources of e.m.f. and lose energy when passing through resistors or other components. Kirchhoff’s second law quantifies this energy balance.
Kirchhoff’s Second Law: In any closed loop, the sum of the electromotive forces is equal to the sum of the potential drops across all elements in that loop.
This relationship ensures no energy is lost or gained overall; the circuit obeys energy conservation.
The Concept of a Closed Loop
A closed loop refers to any continuous path within a circuit that starts and ends at the same point. Multiple loops may exist in complex circuits, and Kirchhoff’s second law applies independently to each one.
When analysing a circuit, students move around a loop, adding the e.m.f.s (energy supplied) and subtracting the p.d.s (energy used). The algebraic sum of all energy changes in a loop is zero.
EQUATION
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Kirchhoff’s Second Law (Loop Equation): ΣE = ΣV
E = Electromotive force supplied (volts, V)
V = Potential difference across components (volts, V)
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This equation means the total e.m.f. around the loop equals the total potential drop, or equivalently, ΣE − ΣV = 0.
A labelled closed loop showing Kirchhoff’s voltage law: the algebraic sum of potential rises and drops around any loop is zero. This neatly encodes energy conservation in a circuit. The figure focuses on voltage terms without extra component detail, matching the A-Level depth. Source.
Between analysing sources and loads, it’s crucial to assign consistent directions for current and voltage. Sign conventions determine whether each term is added or subtracted when applying the law.
Electromotive Force (e.m.f.) and Potential Difference
To apply Kirchhoff’s second law correctly, one must understand what e.m.f. and potential difference represent.
Electromotive Force (e.m.f.): The energy supplied per unit charge by a source as it moves charge around a complete circuit.
Potential Difference (p.d.): The energy converted or dissipated per unit charge as charge passes through a component.
Both e.m.f. and p.d. are measured in volts (V). While e.m.f. is associated with energy supplied, potential difference is associated with energy used.
Direction Conventions
When applying Kirchhoff’s second law:
Moving through a source from negative to positive terminal represents a rise in potential (+E).
Moving through a resistor in the direction of current represents a drop in potential (−IR).
The sum of rises and drops around the loop is zero.
This systematic approach helps maintain consistency and prevents sign errors.
Diagram showing traversal arrows and polarity choices when applying Kirchhoff’s second law across a battery and series resistors. It illustrates when to record +ε and when to record −IR. The extra labels about traversal direction go slightly beyond the bare statement of the law but are essential for accurate sign-keeping. Source.
Application of Kirchhoff’s Second Law in Circuit Analysis
Kirchhoff’s second law is essential for analysing circuits, especially when multiple loops or sources are present. The method involves combining it with Kirchhoff’s first law (the junction rule) to determine unknown currents and voltages.
Procedure for Applying Kirchhoff’s Second Law
When analysing a circuit using Kirchhoff’s laws:
Identify all independent loops within the circuit.
Assign a direction (clockwise or anticlockwise) for current flow in each loop.
Write down an equation for each loop, summing e.m.f.s and potential drops.
Schematic of a single closed loop with an ideal source and resistors, each segment labelled by its potential difference. It is intended for writing the loop equation by summing segment voltages to zero. The figure includes node labels (a–d) that are standard schematic aids and do not add material beyond the syllabus scope. Source.
Use Ohm’s law (V = IR) to express potential drops in terms of current and resistance.
Solve the resulting simultaneous equations for the unknowns.
Energy Conservation Perspective
Kirchhoff’s second law reflects the conservation of energy principle. The total energy gained by charge carriers as they pass through sources equals the total energy lost as they pass through resistive or other elements.
For example, if a 12 V battery drives current through resistors in a closed loop, the sum of all potential differences across the resistors must equal 12 V. If it doesn’t, energy would appear to be lost or created, violating conservation principles.
Significance and Practical Use
Kirchhoff’s second law is used widely in electrical engineering, electronics, and physics. It enables accurate predictions of how voltages distribute across components, making it fundamental in designing circuits that work safely and efficiently.
Key Implications
Energy Balance: Confirms that all supplied electrical energy is accounted for as useful or wasted energy within components.
Multiple Sources: In circuits with several batteries or power supplies, the algebraic sum of all e.m.f.s still equals the sum of all voltage drops.
Complex Networks: Allows analysis of non-trivial networks that cannot be simplified using simple series or parallel rules.
Points to Remember
Kirchhoff’s second law is independent of the circuit’s geometry; it applies universally.
It is derived from the law of conservation of energy, not from Ohm’s law.
The law applies instantaneously, assuming steady-state conditions (no time-varying magnetic fields).
If time-varying magnetic fields exist, induced e.m.f.s must also be included in the loop equation.
Connection to Energy Conservation in Physics
Kirchhoff’s second law aligns with the broader concept of energy conservation across physics. Whether in mechanical, thermal, or electrical systems, energy cannot be created or destroyed — only transformed. The electrical version of this law provides a quantitative link between supplied and dissipated energy, confirming that energy losses such as heat in resistors correspond exactly to the energy delivered by e.m.f. sources.
Understanding this relationship gives students a deeper appreciation of how electrical circuits exemplify fundamental physical laws governing all energy systems. Kirchhoff’s second law therefore serves not only as a powerful analytical tool but also as a direct expression of one of nature’s most important conservation principles.
FAQ
When you reverse the direction of traversal, the sign of each term in the loop equation changes.
Potential rises (across sources from negative to positive) become potential drops, and potential drops (across resistors in the direction of current) become rises.
The overall relationship remains valid, because the equation ΣE − ΣV = 0 is symmetrical — multiplying both sides by −1 gives the same result.
Kirchhoff’s second law assumes the electric field is conservative, meaning the potential difference depends only on the endpoints.
When a magnetic field through a loop changes with time, a non-conservative induced e.m.f. appears (from Faraday’s law). This induced e.m.f. produces a circulating electric field that cannot be expressed as simple potential differences, so the sum of potential changes around the loop is no longer zero.
Yes, it still applies perfectly. Internal resistance simply introduces an extra potential drop within the power source.
When applying the law:
The e.m.f. (E) represents the total energy supplied per coulomb.
The terminal voltage (V) equals the e.m.f. minus the lost volts (Ir) due to internal resistance.
Thus, ΣE = ΣV still holds if all resistive losses, including those inside the cell, are included in the loop equation.
Kirchhoff’s second law provides the energy balance around a loop, while Ohm’s law gives the relationship between voltage, current, and resistance for each component.
When used together:
Kirchhoff’s law supplies the framework for the whole loop (ΣE = ΣIR).
Ohm’s law allows substitution for each voltage drop (V = IR).
Together, they allow simultaneous equations to be written and solved for unknown currents and voltages in multi-loop circuits.
It reveals how electrical energy is distributed among components in a network.
By applying the law, students can trace how energy supplied by sources divides into different resistive paths.
It also shows that even in complex circuits, charge carriers never gain or lose energy overall after a complete loop — confirming that electrical systems obey the same universal conservation principles governing all forms of energy.
Practice Questions
Question 1 (2 marks)
State Kirchhoff’s second law and explain briefly what physical principle it is based on.
Mark Scheme:
1 mark for correctly stating Kirchhoff’s second law: The sum of the electromotive forces (e.m.f.s) around a closed loop equals the sum of the potential drops.
1 mark for identifying the underlying principle: It is based on the conservation of energy — energy supplied equals energy used in a closed loop.
Question 2 (5 marks)
A circuit consists of a 12 V battery connected in series with two resistors, R1 = 3.0 Ω and R2 = 5.0 Ω.
(a) Using Kirchhoff’s second law, write the equation representing the balance of e.m.f. and potential drops around the loop. (2 marks)
(b) Calculate the current in the circuit. (1 mark)
(c) Determine the potential difference across each resistor and explain how this demonstrates Kirchhoff’s second law. (2 marks)
Mark Scheme:
(a)
1 mark for writing the correct loop equation: 12 = V1 + V2
1 mark for substituting V = IR for each resistor: 12 = I(3.0) + I(5.0)
(b)
1 mark for correct current: I = 12 / (3.0 + 5.0) = 1.5 A
(c)
1 mark for calculating voltage across each resistor:
V1 = 1.5 × 3.0 = 4.5 V, V2 = 1.5 × 5.0 = 7.5 V
1 mark for explanation: The total potential drop (4.5 + 7.5 = 12 V) equals the total e.m.f. (12 V), showing that energy supplied equals energy used, consistent with Kirchhoff’s second law.
