Kirchhoff’s law of current and voltage (KCL & KVL)

Kirchhoff’s law is one of the most fundamental and useful law in current electricity. There are two laws given by Kirchhoff. These are Kirchhoff’s law of current or the Kirchhoff’s current law (KCL) and Kirchhoff’s law of voltage or the Kirchhoff’s voltage law (KVL). In another article, we have discussed Ohm’s law of electricity and in this article, we are going to discuss the Kirchhoff’s law of current and voltage.

Contents in this article:

  1. Kirchhoff’s current law (KCL)
  2. Kirchhoff’s Voltage law (KVL)
  3. Application of Kirchhoff’s law
  4. Limitation of Kirchhoff’s law

Kirchhoff’s law of current (KCL)

Statement of Kirchhoff’s current law(KCL)

Kirchhoff’s Current Law gives the relation between the currents passing through a junction point. This law states that the algebraic sum of the total current passing through (enters and leaves) a junction point is zero.

Explanation of KCL

Circuit diagram to explain KCL
Circuit diagram to explain KCL

we consider a junction point J where four branches meet. Here i1, i2, i3 and i4 are the four currents passing through the junction point J via the four branches. Then according to Kirchhoff’s Current Law,

\sum i=0 …………….(1) This is the mathematical form of KCL.

Now, currents entering the junction point are the positive currents and the currents leaving the junction point are the negative currents. Then

or, i1– i2– i3-i4 = 0

  or,   i1= i2+ i3+ i4  …………. (2)

KCL is consistent with law of conservation of electric charge

From equation (2) one can see that the amount of current entering the junction point is equal to the total current leaving the junction point. Hence, amount of charge entering the junction point is equal to the total charge leaving the junction point. So, there is no accumulation of charges at the junction. Thus, Kirchhoff’s current law (KCL) is consistent with the law of conservation of charge.

Kirchhoff’s law of Voltage (KVL)

Statement of Kirchhoff’s voltage law (KVL)

KVL gives the relation between total EMF of the batteries and total voltage drops across the resistances in a closed circuit. This Law states that the algebraic sum of the total EMF in a closed loop is equal to the algebraic sum of the voltage drops across each resistances in the loop.

Explanation of KVL

Circuit Diagram to explain KVL
Circuit Diagram to explain KVL

We consider a closed loop consisting of two batteries of EMFs E1 and E2 respectively (E1 >E2 ) and two resistances R1 and R2 respectively. Since, the batteries are connected in opposite polarities, the algebraic sum of the EMFs is (E1 -E2). If the current through the loop be I, then voltage drops across the resistances are V1=IR1  and  V2=IR2  respectively. Then according to Kirchhoff’s Voltage Law,

\small \sum E=\sum V

or, \small \sum E=\sum IR

or, (E1 -E2)=(IR1+IR2) ……………. (3)

This is the mathematical form of the KVL.

KVL is consistent with law of conservation of electric Energy

If we multiply by I in both sides of equation-(3) then we get,

I(E1 -E2)=(I2R1+I2R2) …………….. (4)

So, the power delivered by battery is equal to the power consumed in resistances. Since, the electric power is defined the change in energy per unit time, then from equation-(4) one can say that the Kirchhoff’s voltage law is consistent with the law of conservation of energy.

Application of Kirchhoff’s laws

  1. KCL can be used to find the unknown current at any junction point.
  2. Using KVL one can determine the potential drop across a resistance.
  3. One can determine the current through any resistance or branch by using Kirchhoff’s laws.
  4. Another important use of KVL is to find the unknown resistance in a loop.

Limitations of Kirchhoff’s law

There are some limitations of Kirchhoff’s laws. These are –

  1. Kirchhoff’s law are not applicable in non-linear circuit where the non-ohmic circuit elements like diode, transistor etc. are used.
  2. Current flow produces electric field around the circuit. Also electrons in the circuit produce electric field around it. These electric and magnetic field may vary due to some causes. But while applying Kirchhoff’s law (KVL) it is assumed that electric and magnetic field around the loop is constant.
  3. Kirchhoff’s law (KVL) is not consistent with Faraday’s law of electromagnetic Induction as it do not consider the change in magnetic field during its application. If KVL includes the change in magnetic field around the circuit then there will be some loss in electric energy due to the induced EMF in the loop. Hence, some additional term will arrive in equation-(4) to satisfy the law of conservation of energy.

This is all from this article on KCL and KVL and applications of KCL and KVL. If you have any doubt on this topic you can ask me in the comment section below.

Thank you!

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  3. Ohm’s law experiment
  4. Electronic circuit components – use of capacitor, inductor, Resistor, diode etc.