Potential distribution along the nonuniform circuit unit

Fig. 13 – Potential distribution on terminals of a sourse of the EMF

The nonuniform subcircuit terms the part of an electric circuit containing a source of the EMF. We will consider distribution of a potential on different circuit units with extraneous forces.

At first we will consider a nonuniform electric circuit unit 1-2 consisting only one source of the EMF (see Fig. 13).

In open-circuit (idle) mode the current through a source does not flow I =0, then all work of extraneous force A _EXTR is used only for separation of charges in a source, that is to say for creation of EMF. Thus an open-circuit voltage on the source equals to greatest possible potential difference on source terminals 1-2 (dashed line on fig. 13):

; [e]= V, (53)

Thus, the EMF of a source e is a greatest possible voltage on its terminals at no-load condition, and numerically is equal to a work done by extraneous forces during a transition of electric charge unit through the source. The EMF of a source is measured in volts.

When through a source e the current I ¹ 0 flows, then part of work A _EXTR of extraneous forces will be used for overcoming of internal resistance of the source r. As a result, the source heats up. This part of work according to Joule-Lentz law is equal:

A_RES=I ² r D t=q×Ir.

Then voltage on the source terminals 1-2 will be less, than in idle mode conditions (53) on value of voltage drop Ir on internal resistance of the source:

. (54)

On a Fig. 13 this voltage drop Ir on internal resistance of the source r is denoted by an arrow downwards.

Fig. 14 – Potential distribution along a nonuniform circuit unit

Let's consider a nonuniform circuit unit 1-3, which contains the source e and an external (load) resistor R (see Fig. 14). Then part of work A _EXTR of extraneous forces will be used for overcoming of internal resistance of the source r and resistance of load resistor R. As a result, the source and the external resistor heat up. Therefore, this part of work we can obtain using a Joule-Lentz law:

A_RES=I ²(r+R)D t=q×I (r+R).

Then voltage on circuit unit terminals 1-3 in comparison with 1-2 (54) is being decreased by value ofvoltage drop Ir, and, in addition, by value of voltage drop IR on external resistor R:

. (55)

On a Fig. 14 both voltage drops Ir on internal resistance of the source r and on load resistor R are denoted by arrows downwards.

Fig. 15 – Potential distribution along a nonuniform circuit unit, which contain opposite connection of EMF

Let's consider a circuit unit 1-4, which contains the source e₁, which action of an extraneous forces coincides with a current direction, external resistance R, and a source e₂, which action of an extraneous forces does not coincide with a current direction (see Fig. 15).

In case e₂<e₁, it means, that the source e₂ will be charged by a current, created by the source e₁. Then part of work A _EXTR of extraneous forces will be used for overcoming of internal resistance of the source r ₁, for overcoming of resistance of load resistor R, for overcoming of internal resistance of the second source r ₂:

A_RES=I ²(r ₁ +R + r ₂)D t=q×I (r ₁ +R + r ₂),

and, in addition, for source e₂ charging:

A _CH = e₂ q.

Therefore voltage on circuit unit terminals 1-4 in comparison with 1-3 (55) is being decreased by value ofvoltage drop Ir ₂on internal resistance of second source, and in addition by value of it EMFe₂:

. (56)

As we can see, e₁ is positive, because direction of extraneous forces work (from “–“ to “+” inside the source) coincides with direction of current I, but e₂ is negative, because direction of extraneous forces work noncoincides with direction of current I.

Рис. 16 – Potential distribution along a closed circuit

If terminals 1 and 3 of nonuniform circuit unit 1-3 that is shown on Fig, 14 are closed by a conductor, then potentials of points 1 and 3 will be equalized:

j₁=j₃,

and we obtain the elementary variant of closed electric circuit (see Fig. 16). In this case all work A _EXTR=e q of extraneous forces will be used for overcoming of internal resistance of the source r and resistance of load resistor R:

A_EXTR = A _RES= q×I (r+R).

From (55) we obtain, then all EMF of the source e drops on internal resistance of the source r and on external resistor R:

. (57)

From formulas (53) - (56) we can see, that according to definition the voltage U –is a potential difference on terminals of a circuit unit.

However, it is necessary to note, that voltage U_r on terminals of an EMF source (54) is being accepted with plus sign, and is being defined as work of extraneous (internal) forces on transition of a unit positive charge. Thus, from a Fig.14 we can see, that it is equal U_r= j₂–j₁ to a difference of a terminating potential j₂ (greater), and an initial potential j₁ (smaller). Voltage on any nonuniform circuit unit (53) - (56) is being analogously defined.

If we define a voltage U_R on terminals of the homogeneous circuit unit 2-3, which is not containing an EMF source (Fig. 14), it will be positive, when we consider it as a work of field forces (external) on transition of a unit positive charge. Thus, from a Fig. 14 we can see, that it is equal U_R= j₂–j₃ to a difference of an initial potential (greater) j₂, and a terminating potential (smaller) j₃:

. (58)

Therefore for the homogeneous circuit unit AB from fig. 10 Ohm's law write down in the form of the formula (20).