phy1020.DVI

3. A third approach would be to use proportions to figure out how the current splits going through the
   two parallel resistors. Since the resistors areR 2 D5andR 3 D3, we know that^3 / 8 of currentI 1
   goes through the5resistor, and^5 / 8 goes through the3resistor. (Proportionally more current goes
   through the smaller resistor.) In general, fortworesistorsR 1 andR 2 in parallel, the currents will be

I 1 D

R 2

R 1 CR 2

I (22.13)

I 2 D

R 1

R 1 CR 2

I; (22.14)

whereIis the current going in to the parallel combination, before it splits intoI 1 going throughR 1

andI 2 going throughR 2. Knowing the currents through each resistor and the two resistances, we can

use Ohm’s law to solve for the voltages across each resistor. This proportion method is really only

useful for two resistors in parallel; for three or more in parallel, the formulæ become too complicated

to be practical.

Any of these three approaches will give the same results. The currents through and voltages across each
of the resistors of Fig. 22.3 is shown in Table 22-1.

Table 22-1. Results of circuit analysis of the simple circuit of Fig. 22.3.

Resistor R() I(A) V(V)

R 1 10 0.5053 5.0526

R 2 5 0.1895 0.9474

R 3 3 0.3158 0.9474

Note the following from this table:

• V 1 CV 2 DV 1 CV 3 D 6 V. Looping once around the circuit—taking either the path throughR 2 or the
  one throughR 3 —gives a total potential drop equal to the battery voltage.


• V 2 DV 3. When resistors are connected in parallel, they all have the same potential drop across them.


• I 2 CI 3 DI 1. When the the current splits at a junction, the sum of the currents leaving the junction
  equals the current going into the junction.


• I 2 D^3 = 8 I 1 ;I 3 D^5 = 8 I 1. When current splits at a junction, it divides in proportion to the resistance in
  each branch.

22.4 Circuit Analysis Principles.

We can summarize here a few basic principles to keep in mind:

• When making a complete loop around the circuit, the sum of the voltage rises (due to batteries) equals
  the sum of the voltage drops (due to resistors). This will be true of any loop you take around the circuit.


• When the current splits at a junction, the sum of the currents leaving the junction equals the sum of the
  currents entering the junction.


• Current will split at a junction in proportion to the resistance in each branch, with more current going
  through the branch of least resistance.