Kirchhoff’s laws are used to solve complicated electric circuits. Along with Ohm’s law, they are the first line of attack to solve any given circuit. However, bear in mind that these laws are valid only in the lumped matter discipline, i.e. when the circuit is in steady state.

Kirchhoff’s Voltage Law

The net potential drop across a loop is zero. In other words, if you end up where you started the sum of voltages along the loop must be zero.

Kirchhoff’s Voltage Law

    \[\sum_{loop}V_{i}\ = 0\]

The law derives from Faraday’s law of Induction which states that the voltage induced in a closed circuit is proportional to the rate of change of the magnetic flux it encloses:


    \[\oint \vec{E}.\vec{dl} = -\frac{\partial \Phi _{B}}{\partial t} = \sum_{loop}V_{i}\]

Since we have a steady circuit, there is no changing magnetic flux. No changing magnetic flux means no net voltage (potential difference) induced in the loop:

    \[\Phi _{B} = 0 \Rightarrow \sum_{loop}V_{i}\ = 0\]


Kirchhoff’s Current Law

The total current entering a junction is the same as the total current leaving it.

    \[\sum_{node}I_{i}\ = 0\]

Kirchhoff’s Current Law

This law is basically a statement of conservation of charge because current is nothing but charges in motion and since there is no charge generation (or recombination) at the junction, it must be constant. More formally, this law derives from the Gauss’s Law:

    \[\oint \vec{J}.\vec{ds} = -\frac{\partial q}{\partial t}\]

Since charge is constant, ∂q/∂t = 0 and

    \[\sum_{node}I_{i}\ = 0\]

If you are still confused, go watch this video by Crash Course Physics. By the way, all of their videos are great.

If you are really into Electric circuits and or Electronics, there is a series of courses by MIT on edx. The level is equivalent to the first semester in Electrical Engineering degree programme.

Kirchhoff’s laws
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