## VOLTAGE AND KIRCHHOFF’S VOLTAGE LAW

Gustav Robert Kirchhoff (1824–1887)

Charge moving in an electric circuit gives rise to a current, as stated in the preceding section. Naturally, it must take some work, or energy, for the charge to move between two points in a circuit, say, from point a to point b. The total work per unit charge associated with the motion of charge between two points is called voltage. Thus, the units of voltage are those of energy per unit charge; they have been called volts in honor of Alessandro Volta:

The voltage, or potential difference, between two points in a circuit indicates the energy required to move charge from one point to the other. As will be presently shown, the direction, or polarity, of the voltage is closely tied to whether energy is being dissipated or generated in the process. The seemingly abstract concept of work being done in moving charges can be directly applied to the analysis of electrical circuits; consider again the simple circuit consisting of a battery and a light bulb.

Fig.1 Illustration of Kirchhoff’s voltage law: v1 = v2 Voltages around a circuit

The circuit is drawn again for convenience in Figure 1, with nodes defined by the letters a and b. A series of carefully conducted experimental observations regarding the nature of voltages in an electric circuit led Kirchhoff to the formulation of the second of his laws, Kirchhoff’s voltage law, or KVL.

The principle underlying KVL is that no energy is lost or created in an electric circuit; in circuit terms, the sum of all voltages associated with sources must equal the sum of the load voltages, so that the net voltage around a closed circuit is zero. If this were not the case; we would need to find a physical explanation for the excess (or missing) energy not accounted for in the voltages around a circuit. Kirchhoff’s voltage law may be stated in a form similar to that used for KCL:

where the vn are the individual voltages around the closed circuit. Making reference to Figure 1, we see that it must follow from KVL that the work generated by the battery is equal to the energy dissipated in the light bulb in order to sustain the current flow and to convert the electric energy to heat and light:

vab = − vba

v1 = v2

One may think of the work done in moving a charge from point a to point b and the work done moving it back from b to a as corresponding directly to the voltages across individual circuit elements. Let Q be the total charge that moves

around the circuit per unit time, giving rise to the current i. Then the work done in moving Q from b to a (i.e., across the battery) is

Wba = Q × 1.5 V

Fig.2 Concept of voltage as potential difference (The presence of a voltage, v2, across the open terminals a and b indicates the potential energy that can enable the motion of charge, once a closed circuit is established to allow current to flow)

Similarly, work is done in moving from a to b, that is, across the light bulb. Note that the word potential is quite appropriate as a synonym of voltage, in that voltage represents the potential energy between two points in a circuit: if we remove the light bulb from its connections to the battery, there still exists a voltage across the (now disconnected) terminals b and a. This is illustrated in Figure 2. A moment’s reflection upon the significance of voltage should suggest that it must be necessary to specify a sign for this quantity.

Consider, again, the same drycell or alkaline battery, where, by virtue of an electrochemically induced separation of charge, a 1.5-V potential difference is generated. The potential generated by the battery may be used to move charge in a circuit. The rate at which charge is moved once a closed circuit is established (i.e., the current drawn by the circuit connected to the battery) depends now on the circuit element we choose to connect to the battery. Thus, while the voltage across the battery represents the potential for providing energy to a circuit, the voltage across the light bulb indicates the amount of work done in dissipating energy. In the first case, energy is generated; in the second, it is consumed (note that energy may also be stored, by suitable circuit elements yet to be introduced). This fundamental distinction requires attention in defining the sign (or polarity) of voltages.

Fig.3 Sources and loads in an electrical circuit. (A symbolic representation of the battery–light bulb circuit of Figure 1)

In general, refer to elements that provide energy as sources, and to elements that dissipate energy as loads. Standard symbols for a generalized source-and-load circuit are shown in Figure 3.