Undergraduate
  1. Classical mechanics  1.1. dynamics  1.1.1. Motion in one dimension  1.1.2. Motion in two dimensions  1.1.3. projectile motion  1.1.4. Relative speed  1.1.5. Uniform circular motion  1.1.6. Non-uniform circular motion  1.1.7. Reference frames and transformations  1.2. Newton's Laws of Motion  1.2.1. First law of motion  1.2.2. Second law of motion  1.2.3. Third Law of Motion  1.2.4. Applications of Newton's Laws  1.2.5. Friction and its types  1.2.6. Free Body Diagram  1.2.7. Constraints and pseudo-forces  1.3. Work and Energy  1.3.1. work done by the force  1.3.2. Work–energy theorem  1.3.3. Kinetic energy  1.3.4. Potential energy in classical mechanics  1.3.5. Conservative and non-conservative forces  1.3.6. Energy Conservation  1.3.7. Power and efficiency  1.4. Speed and collisions  1.4.1. Linear momentum  1.4.2. Conservation of momentum  1.4.3. Impulse and impact force  1.4.4. Elastic and inelastic collision  1.4.5. Center of mass and speed  1.4.6. Rocket Propulsion  1.5. Rotational motion  1.5.1. Torque and Angular Momentum  1.5.2. Moment of inertia  1.5.3. Rotational Kinematics  1.5.4. Rotational Energy  1.5.5. Rolling Motion  1.5.6. Precession and gyroscopic motion  1.6. Gravitational force  1.6.1. Newton's law of universal gravitation  1.6.2. Gravitational potential energy  1.6.3. Orbital mechanics  1.6.4. Kepler's laws of planetary motion  1.6.5. Gravitational field and potential  1.7. Fluid mechanics  1.7.1. Pressure and Pascal's Principle  1.7.2. Buoyancy and Archimedes' principle  1.7.3. Fluid Dynamics and Bernoulli's Principle  1.7.4. Viscosity and Poiseuille's law  1.7.5. Surface tension and capillarity  1.8. Oscillations and waves  1.8.1. Simple Harmonic Motion  1.8.2. Damped and driven oscillations  1.8.3. Coupled oscillations and common modes  1.8.4. Wave properties and types  1.8.5. Sound Waves and the Doppler Effect  1.8.6. Wave interference and superposition
  2. Electromagnetism  2.1. Electrostatics  2.1.1. Electric charge and properties  2.1.2. Coulomb's law  2.1.3. Electric field and electric potential  2.1.4. Gauss's Law  2.1.5. Capacitance and Dielectric  2.1.6. Electric dipole and potential energy  2.2. Electric circuits  2.2.1. Current and Resistance  2.2.2. Ohm's law  2.2.3. Kirchhoff's Laws  2.2.4. RC Circuit  2.2.5. AC Circuits and Reactance  2.2.6. Electric power and energy  2.3. Magnetism  2.3.1. Magnetic Fields and Forces  2.3.2. Ampere's Law  2.3.3. Magnetic dipole moment  2.3.4. Biot-Savart law  2.3.5. Magnetic Materials and Hysteresis  2.4. Electromagnetic induction  2.4.1. Faraday's Law  2.4.2. Lenz's law  2.4.3. Self and mutual induction  2.4.4. Transformers and Inductive Circuits  2.5. Maxwell's equations  2.5.1. Gauss's law for electricity  2.5.2. Gauss's law for magnetism  2.5.3. Faraday's law of induction  2.5.4. Ampere-Maxwell law  2.5.5. Electromagnetic waves
  3. Thermodynamics  3.1. Laws of Thermodynamics  3.1.1. Zeroth law of thermodynamics  3.1.2. First law of thermodynamics  3.1.3. Second Law  3.1.4. Third Law  3.2. Heat and work  3.2.1. Heat transfer (conduction, convection, radiation)  3.2.2. Thermal expansion  3.2.3. Heat engines and refrigerators  3.2.4. Carnot cycle and efficiency  3.3. Statistical mechanics  3.3.1. Maxwell–Boltzmann distribution  3.3.2. Entropy and Probability  3.3.3. Partition function  3.3.4. Fermi–Dirac and Bose–Einstein statistics
  4. Optics  4.1. Geometrical Optics  4.1.1. Reflection and Refraction  4.1.2. Mirrors and lenses  4.1.3. Optical Instruments  4.1.4. Fermat's principle  4.2. Wave optics  4.2.1. Interference in wave optics  4.2.2. Diffraction  4.2.3. Polarization  4.2.4. Coherence and holography
  5. Quantum mechanics  5.1. Wave–particle duality  5.1.1. Blackbody radiation in wave–particle duality in quantum mechanics  5.1.2. Photoelectric effect  5.1.3. Compton scattering  5.1.4. De Broglie wavelength  5.2. Schrödinger Equation  5.2.1. Time-independent Schrödinger equation  5.2.2. Particle in a box  5.2.3. Quantum Tunneling  5.2.4. Potential wells and obstacles  5.3. Quantum States  5.3.1. Wave function  5.3.2. Quantum operators  5.3.3. Heisenberg uncertainty principle  5.3.4. Angular momentum and spin
  6. Relativity  6.1. Special relativity  6.1.1. Lorentz transformations  6.1.2. Time expansion and length contraction  6.1.3. Relativistic energy and momentum  6.2. General relativity  6.2.1. Principle of Equivalence  6.2.2. Schwarzschild metric  6.2.3. Gravitational waves  6.2.4. Black holes and event horizons
  7. Solid state physics  7.1. Crystal structure  7.1.1. Bravais lattices  7.1.2. X-ray diffraction  7.1.3. Band theory  7.1.4. Phonons and lattice vibrations  7.2. Electrical and Magnetic Properties  7.2.1. Conductors, Semiconductors and Insulators  7.2.2. Superconductivity  7.2.3. Hall effect
  8. Nuclear and particle physics  8.1. Atomic Structure  8.1.1. Atomic Model  8.1.2. Nuclear binding energy  8.2. Radioactivity  8.2.1. Alpha, beta, gamma decay  8.2.2. Half life  8.2.3. Nuclear Fission and Fusion  8.3. Particle physics  8.3.1. Standard Model  8.3.2. Quarks and Leptons  8.3.3. antimatter  8.3.4. Fundamental interactions
  9. Astrophysics and cosmology  9.1. Stellar evolution  9.1.1. Star formation  9.1.2. Black holes and neutron stars  9.1.3. White dwarfs and supernovae  9.2. Cosmology  9.2.1. The Big Bang Theory  9.2.2. Dark matter and dark energy  9.2.3. Cosmic microwave background

Undergraduate → Electromagnetism → Electric circuits ↓


Kirchhoff's Laws


Kirchhoff's laws are fundamental principles used in circuit analysis. These laws describe the conservation of current and energy in electrical circuits. They are named after German physicist Gustav Kirchhoff, who first explained them in the mid-19th century. Understanding these laws is essential for analyzing and designing complex circuits in both academic and practical situations.

Kirchhoff's current law (KCL)

Kirchhoff's current law, also known as the junction law, states that the sum of the currents entering a junction must equal the sum of the currents leaving the junction. This law is based on the conservation of charge. In simple terms, all the current flowing into a point must eventually flow out.

Mathematically, KCL is expressed as:

∑ I_in = ∑ I_out

Let us consider a simple circuit in which currents flow in and out of a junction.

         I1 I2 I3

According to KCL, at the junction we have:

I1 = I2 + I3

This means that the amount of current flowing into the junction (I1) is equal to the sum of the currents flowing out (I2 and I3).

Kirchhoff's voltage law (KVL)

Kirchhoff's voltage law, or loop rule, states that the sum of the electromotive forces (emf) and potential differences (voltages) around any closed loop or mesh is equal to zero. This law is based on the conservation of energy. It implies that after traveling around the loop, the net change in potential energy is zero.

Mathematically, KVL is expressed as:

∑ V = 0

Consider a simple circuit loop containing a battery and two resistors:

         V1 R1 R2

In this loop, KVL can be written as:

V1 - R1*I - R2*I = 0

Where V1 is the battery voltage, R1 and R2 are resistances, and I is the current flowing through the loop.

Practical examples using Kirchhoff's laws

Example 1: Series circuit

Consider a series circuit with a battery and three resistors connected one after the other. The circuit can be represented as:

         Vb R1 R2 R3

Using KVL to analyze the loop, we get:

Vb = I*R1 + I*R2 + I*R3

In this equation, Vb is the battery voltage, and I is the current flowing through all the components (since this is a series circuit).

Example 2: Parallel circuit

Now consider a parallel circuit where a single voltage source is connected to three parallel branches, each of which has a resistor:

         Vb R1 R2 R3

Using KCL on the positive side of the junction, we get:

I = I1 + I2 + I3

Where I is the total current from the voltage source, and I1, I2, and I3 are the currents through resistors R1, R2, and R3, respectively. The voltage across each resistor is the same and equal to Vb (the voltage provided by the battery).

Complex circuit example

Let us consider a more complex circuit where both series and parallel elements are present:

         V1 R1 R2 R3 R4 R5

In this circuit, apply KVL to several loops to find the current through each branch.

Loop 1 (upper loop):

V1 - I1*R1 - I2*R2 = 0

Loop 2 (lower parallel branch):

I2*R4 - I3*R3 = 0

Here, loop 1 focuses on the upper path through resistors R1 and R2 and loop 2 considers the lower path through resistors R3 and R4.

Use of KCL at junctions:

I1 = I3 + I5

Through a systematic approach to applying these rules, all unknown problems such as voltage and current can be solved for any type of circuit configuration.

Conclusion

Kirchhoff's laws, both KCL and KVL, are indispensable in the analysis and design of electronic circuits. They allow us to accurately predict the behavior of circuits and are the foundation of modern electrical engineering. By understanding and applying these rules, we can solve unknowns in circuits, ensuring that devices and systems work correctly.

Through numerous examples it is evident that whether dealing with simple or complex circuits, Kirchhoff's laws provide a systematic approach to solving circuit-related issues. Their application is not limited to theoretical fields but also extends to practical engineering, making them an important component of any electrical engineering academic curriculum.


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Kirchhoff's Laws

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