GUSTAV ROBERT KIRCHHOFF ( – ) E XAMPLE . [Eq. ( . ) between the positive terminal P and the negative terminal N; I here is the current flowing from N to P through the cell].
If, while labelling the current I through the cell one goes from P to N, then of course V = ε + I r ( . ) Having clarified labelling, we now state the rules and the proof: (a) Junction rule: At any junction, the sum of the currents entering the junction is equal to the sum of currents leaving the junction (Fig. . ).
This applies equally well if instead of a junction of several lines, we consider a point in a line. The proof of this rule follows from the fact that when currents are steady, there is no accumulation of charges at any junction or at any point in a line. Thus, the total current flowing in, (which is the rate at which charge flows into the junction), must equal the total current flowing out. (b) Loop rule: The algebraic sum of changes in potential around any closed loop involving resistors and cells in the loop is zero (Fig.
. ). This rule is also obvious, since electric potential is dependent on the location of the point. Thus starting with any point if we come back to the same point, the total change must be zero.
In a closed loop, we do come back to the starting point and hence the rule. Example . A battery of V and negligible internal resistance is connected across the diagonally opposite corners of a cubical network consisting of resistors each of resistance Ω (Fig. .
). Determine the equivalent resistance of the network and the current along each edge of the cube. FIGURE . FIGURE .
At junction a the current leaving is I + I and current entering is I . The junction rule says I = I + I . At point h current entering is I . There is only one current leaving h and by junction rule that will also be I .
For the loops ‘ahdcba’ and ‘ahdefga’, the loop rules give – I – I + = and – I + I – = .