# Kirchhoff’s first law

What is Kirchhoff’s first law : Kirchhoff’s circuit laws are two analogies that deal with current and potential difference (commonly known as voltage) in lumped element models of electrical circuits . They were first described by the German physicist Gustav Kirchhoff in 1845. [1] It generalized the work of George Ohm and preceded the work of James Clerk Maxwell . Widely used in electrical engineering , they are also called Kirchhoff’s laws or simply Kirchhoff’s laws, These laws can be applied in the time and frequency domain and form the basis fornetwork analysis .

Both Kirchhoff’s laws can be understood as consequences of Maxwell’s equations in the low-frequency limit. They are accurate for DC circuits and for AC circuits at frequencies where the wavelength of electromagnetic radiation is much larger than in the circuit itself.

## Kirchhoff’s current law

This law, also called Kirchhoff’s first law , Kirchhoff’s point law , or Kirchhoff’s junction law (or nodal law ), states that, for any node (junction) in an electric circuit , the current flowing in that node The sum of the currents is equal to the sum of the currents flowing through that node; or equivalent:

The algebraic sum of currents in a network of conductors meeting at a point is zero.

Recalling that current is a signed (positive or negative) quantity indicating the direction towards or away from a node, this principle can be summed up as:

\sum _{k=1}^{n}{I}_{k}=0

where n is the total number of branches in which currents flow towards or away from the node.

The law is based on the conservation of charge where charge (measured in coulombs) is the product of current (in amperes) and time (in seconds). If the net charge in a field is constant, then the law of current will apply to the boundaries of that field. [2] [3] This means that the law of current depends on the fact that the net charge in wires and components is constant.

### Use

A matrix version of Kirchhoff’s current law is the basis of most circuit simulation software , such as SPICE . Current law along with Ohm’s law is used to perform nodal analysis .

The existing law applies to any lumped network, irrespective of the nature of the network; Whether unilateral or bilateral, active or passive, linear or non-linear.

## Kirchhoff’s voltage law

This law, also called Kirchhoff’s second law , Kirchhoff’s loop (or mesh ) law , or Kirchhoff’s second law , states the following:

The directed sum of the potential difference (voltage) around any closed loop is zero.

Similarly to Kirchhoff’s current law, the voltage law can be stated as:

\sum _{k=1}^{n}V_{k}=0


Here, n is the total number of voltages measured.

Derivation of Kirchhoff’s voltage law A similar derivation can be found in The Feynman Lectures on Physics, Volume II, Chapter 22: AC Circuits . Consider some arbitrary circuit. Approximate the circuit with lumped elements, so that the (time-varying) magnetic field is contained in each component and the field outside the circuit is negligible. Based on this assumption, the Maxwell–Faraday equation shows that

{\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}=\mathbf {0} }

on the outskirts of. If each component has a finite volume, then the external field is simply connected , and thus the electric field in that region is conservative . Therefore, for any loop in the circuit, we get

{\displaystyle \sum V_{i}=-\sum \int _{{\mathcal {P}}_{i}}\mathbf {E} \cdot \mathrm {d} \mathbf {l} =\oint \mathbf {E} \cdot \mathrm {d} \mathbf {l} =0}

where is the path around the exterior of each component , from one terminal to the other. pi

### Generalization

In the low-frequency range, the voltage drop around any loop is zero. It consists of imaginary loops arranged arbitrarily in space – but not limited to loops depicted by circuit elements and conductors. In the low-frequency limit, this is a consequence of Faraday’s law of induction (which is one of Maxwell’s equations ).

It has practical application in situations involving ” static electricity “.

### Borders

Kirchhoff’s circuit laws are a result of the lumped-element model and both depend on the model being applied to the circuit. When the model doesn’t apply, the laws don’t apply.

The current law is based on the assumption that the net charge in any wire, junction, or knotted component is constant. This cannot happen whenever the electric field between parts of a circuit is negligible, such as when two wires are connected capacitively . This occurs in high-frequency AC circuits, where the lumped element model is no longer applicable. [4] For example, in a transmission line , the charge density in the conductor will oscillate continuously.

On the other hand, the voltage law relies on the fact that the action of time-varying magnetic fields is confined to individual components, such as inductors. In fact, the induced electric field generated by an inductor is not limited, but the leaking fields are often negligible.

### Real circuit modeling with lumped elements

The knotted element approximation for a circuit is accurate at low frequencies. At high frequencies, leaking fluxes and different charge densities in the conductors become significant. To an extent,

it is still possible to model such circuits using parasitic components . If the frequencies are very high, it may be more appropriate to directly simulate the field using finite element modeling or other techniques .

In order to model the circuit so that both laws can still be used, it is important to understand the difference between physical circuit elements and ideal lumped elements. For example, a wire is not an ideal conductor. Unlike an ideal conductor, wires can inductively and capacitively connect to (and to themselves) each other, and have a limited propagation delay. Real conductors can be modeled in terms of lumped elements by considering the distributed parasitic capacitance between conductors for model capacitive coupling , or parasitic (mutual) inductance for model inductive coupling . [4] Wiring also has some self-inductance, which is why it is necessary to isolate capacitors.

#### Example

Assume an electrical network consisting of two voltage sources and three resistors.

According to the first law:

i_{1}-i_{2}-i_{3}=0\,

Applying the second law to a closed circuit 1 , and substituting the voltage using Ohm’s law, lets:

-R_{2}i_{2}+{\mathcal {E}}_{1}-R_{1}i_{1}=0


The second law, again combined with Ohm’s law, applies to closed circuit 2 :

-R_{3}i_{3}-{\mathcal {E}}_{2}-{\mathcal {E}}_{1}+R_{2}i_{2}=0


This produces a system of linear equations in 1 , 2 , 3 :

{\begin{cases}i_{1}-i_{2}-i_{3}&=0\\-R_{2}i_{2}+{\mathcal {E}}_{1}-R_{1}i_{1}&=0\\-R_{3}i_{3}-{\mathcal {E}}_{2}-{\mathcal {E}}_{1}+R_{2}i_{2}&=0\end{cases}}


which is equal to

{\displaystyle {\begin{cases}i_{1}+(-i_{2})+(-i_{3})&=0\\R_{1}i_{1}+R_{2}i_{2}+0i_{3}&={\mathcal {E}}_{1}\\0i_{1}+R_{2}i_{2}-R_{3}i_{3}&={\mathcal {E}}_{1}+{\mathcal {E}}_{2}\end{cases}}}


considering that

{\displaystyle R_{1}=100\Omega ,\ R_{2}=200\Omega ,\ R_{3}=300\Omega }

{\displaystyle {\mathcal {E}} _ {1} = 3 {\text {V}}, {\mathcal {E}} _ {2} = 4 {\text {V}}}

the solution is

{\displaystyle {\begin{cases}i_{1}={\frac {1}{1100}}{\text{A}}\\[6pt]i_{2}={\frac {4}{275}}{\text{A}}\\[6pt]i_{3}=-{\frac {3}{220}}{\text{A}}\end{cases}}}


Section 3 has a negative sign which means that the assumed direction of 3 was wrong and 3 is actually flowing in the opposite direction of the red arrow labeled 3 . Current in R3 flows from left to right.

## What is Kirchhoff’s second law?

Kirchhoff’s law of potential difference / KVL)
This law is also called ‘Kirchhoff’s second law’, Kirchhoff’s law of loops (or meshes). T about a loop: the algebraic sum of all potential differences is zero. That is, here, n is equal to the number of some potential difference in the loop.

## What is the rule of the loop?

In other words, “the algebraic sum of all the potential differences in a loop is zero.” It is also called Kirchhoff’s ‘Loop Law’. It is based on the law of conservation of energy.

## What is the consequence of Kirchhoff’s Treaty Law?

Kirchhoff’s junction law is a consequence of conservation of electric charge.

## How many Kirchhoff’s laws are there?

Scientists Kirchhoff gave two laws regarding current and voltage, which are called Kirchhoff’s law.

## What do you understand by electric circuit?

The interconnection of electrical components (voltage source, resistance, inductance, capacitor and keys etc.) and electromechanical components (switches, motors, speakers etc.) is called electric circuit or electrical network. … Usually a network consisting of one or more closed loops is called an electric circuit.

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