Kirchhoff’s Law

In ordinary conditions, ohm’s law is used for analyzing and measuring the current and voltage of the circuit. But, there are several complex cases, in which Ohm’s Law cannot be used to measure the current and voltage. In such circuit, Kirchhoff’s law is used.

There are two laws of Kirchhoff, which are described below:

Kirchhoff’s first law: The algebraic sum of current at a junction of electric current is always zero.

$$ \text{i.e.,}\sum \text{I = 0}$$

Sign convention: The incoming current is taken as positive and outgoing current is taken as negative.

At a junction, sum of incoming current is always equal to sum of outgoing current.  So, from figure, we can write:

I1 + I2 – I3 – I4 – I5 = 0

or, I1 + I2 = I3 + I4 + I5

Kirchhoff’s second law: In a closed loop, the sum of emf is equal to sum of product of current and resistance. This law is also known as loop law or voltage law.

$$\text{i.e.,}   \sum \text{E} = \sum \text{IR}$$

Sign convention: In the direction of loop, emf and current have positive sign otherwise negative sign is written.

Let us consider a complex circuit as shown in figure below. We can use Kirchhoff’s law is used to find the current in different part of the circuit. Here, the direction of emf and current flow in anticlockwise direction is taken as positive and that in anticlockwise direction is taken as negative.

Now, applying Kirchhoff’s Law in loop ABCFA, we get:

$$\sum \text{E} = \sum \text{IR}$$

$$\text{or, (-E}_1) + \text{(-E}_2) = \text{(+I}_1)\text{R}_1 + \text{(-I}_2)\text{R}_2 $$
$$\text{or, E}_1 – \text{E}_2 = \text{I}_1\text{R}_1 – \text{I}_2\text{R}_2……(i)$$

In loop FCDEF, we get:

$$\sum \text{E} = \sum \text{IR}$$
$$\text{or, (-E}_2) + \text{(-E}_3) = \text{(+I}_2)\text{R}_2 + \text{(-I}_3) \text{R}_3 $$
$$\text{or, E}_2 – \text{E}_3 = \text{I}_2\text{R}_2 – \text{I}_3\text{R}_3……(ii)$$

Applying Kirchhoff’s first law in at junction F, we get:
$$\sum \text{I} = 0$$

$$\text{(or, +I}_1) + \text{(+I}_2) – \text{I}_3 = 0$$

$$\text{or, I}_1 + \text{I}_2 = \text{I}_3……(iii)$$

We can find the value of I1 + I2 + I3 from the above three equations.

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