As the flow of current is considered as flow of quantity, at any point in the circuit the total current enters, is exactly equal to the total current leaves the point. The point may be considered anywhere in the circuit.
Suppose the point is on the conductor through which the current is flowing, then the same current crosses the point which can alternatively said that the current enters at the point, will leave the point. As we said the point may be anywhere on the circuit, so it can also be a junction point in the circuit. So, total quantity of current enters at the junction point must be exactly equal to total quantity of current that leaves the junction. This is the very basic thing about flowing of current and fortunately Kirchhoff Current law says the same. The law is also known as Kirchhoff First Law and this law stated that, at any junction point in the electrical circuit, the summation of all the branch currents is zero. If we consider all the currents enter in the junction are considered as positive current, then convention of all the branch currents leaving the junction are negative. Now if we add all these positive and negative signed currents, obviously, we will get result of zero.
The mathematical form of Kirchhoff's Current Law is as follows,
We have a junction where n number of beaches meet together.
Lets, The currents in branches 1, 2, 3 .... m are entering to the junction.
Whereas currents in branches are leaving from the junction.
So the currents in the branches 1, 2, 3 .... m may be considered as positive as per general convention and similarly the currents in the branches may be considered as negative.
Hence all the branch currents in respect of the said junction are - Now, the summation of all currents at the junction is- This is equal to zero according to Kirchhoff Current Law. Therefore, The mathematical form of Kirchhoff First Law is ∑ I = 0 at any junction of electrical network.
If we consider a closed loop conventionally, if we consider all the voltage gains along the loop are positive then all the voltage drops along the loop should be considered as negative. The summation of all these voltages in a closed loop is equal to zero. Suppose n numbers of back to back connected elements form a closed loop. Among these circuit elements m number elements are voltage source and n - m number of elements drop voltage such as resistors.
The voltages of sources are And voltage drops across the resistors respectively, As it is said that the voltage gain conventionally considered as positive, and voltage drops are considered as negative, the voltages along the closed loop are - Now according to Kirchhoff Voltage law, the summation of all these voltages results to zero. So accordingly Kirchhoff Second Law, ∑V = 0.