# 4.04 Kirchhoff's voltage law, (Kirchhoff's second law).

Kirchhoff's Voltage law states, that the sum of the e.m.f.s applied to any loop in a circuit, is equal to the sum of the voltage drops around the loop. The diagrams below show two circuits each with a single e.m.f. E.

In the first diagram there is only one possible loop for the current to flow around. In this loop two voltage drops occur, (one across each resistor (V1 and V2)).
Therefore E = V1 + V2.

In the second diagram, there are two possible loops that the current can flow around. In the first loop there are two voltage drops (V1 and V2) and in the second loop there are three voltage drops (V3, V4 and V5).
Kirchhoff's voltage law applies to each loop, so E = V1 + V2 and also E = V3 + V4 + V5.  As stated above Kirchhoff's voltage law applies to each loop in a circuit.
Consider a unit of charge flowing around the cct., (using conventional current flow). It begins at the negative terminal of the battery, which determines it's initial potential. As it moves through the battery it's potential is raised. Then it travels around the circuit, passing through the resistors which lower its potential. Eventually it returns to the negative terminal and back down to the same potential.

Important.
In these examples there is only one e.m.f. in each loop. In practice there may be more than one and if so we must add them together to find the total e.m.f. However we must take into account their direction!. Voltage sources such as batteries can be connected either way round in a circuit! Sometimes the e.m.f.s will be acting in the same direction, so we simply add their individual values together, i.e E1 + E2. However sometimes they will be acting in opposite directions! In this case we subtract one e.mf from the other, i.e. E1 - E2 etc.
(voltage drops are always simply added together).

## Comparison of Kirchhoff's voltage law, with gravitational potential.

Imagine a large building with two elevators and several different stairways which link different floors. Imagine that you travel to the top of the building, using the lifts and then down any combination of stairways, back to the ground. The reduction in height (S1, S2 etc) as you travel down the stairs, is obviously equal to the increase in height (E1, E2 etc.) initially provided by the lifts.

Note that changing height changes your gravitational potential, so the last statement is very similar to Kirchhoff's voltage law, which applies to electrical potential.

E1 + E2 = S1 + S2 +S3.
E1 + E2 = S1 + S4 +S5.

Note that the arrows on the stairways point down to the lower end of each stairway, to show the direction of travel. In a cct. diagram, the arrows for voltage drops, always point to the end of the resistor at the higher potential Kirchhoff's voltage law is an example of the law of conservation of energy. The law of conservation of energy states that energy cannot be created or destroyed. The e.m.f. voltage is a measure of the electrical potential, that is supplied by the power supply, (i.e. the Joules of energy per Coulomb of charge). The voltage drops are a measure of the reduction in potential, as the current flows through a resistor, (i.e. the Joules of energy per Coulomb of charge, that are being converted into heat by the resistor). Therefore if Kirchhoff's voltage law was not true, then the electrical energy being released as the charge moves round the cct., would not be equal to the energy being supplied and the law of conservation of energy would be violated!