Consider the circuit in the figure, with the current directions defined as shown

Question

Consider the circuit in the figure, with the current directions defined as shown. There are four resistors in this circuit (R1, R2, R3, and R4) and four batteries with emfs ?1 = 15.5 V, ?2 = 3.5 V, ?3 = 11.5 V, and ?4 = 20.5 V, each with internal resistance given by ri as marked in the figure.

Randomized Variables
?1 = 15.5 V
?2 = 3.5 V
?3 = 11.5 V
?4 = 20.5 V

(a) Calculate the current I2 in amps.

(b) Calculate the current I3 in amps.

(c) Calculate the current I1 in amps.

Explanation / Answer

a)

at node b

i1 = i2 + i3 eq-1

Using KVL in loop abefa

i1R1 + i3R2 - E2 + i3 r2 + i1 R4 - E1 + i1r1 = 0

i1 (20) + i3 (6) - 3.5 + i3(0.25) + i1 15 - 15.5 + i1 (0.5) = 0

35.5 i1 + 6.25 i3 = 19                            

i3 = (19 - 35.5 i1 ) / 6.25                              eq-2

Using KVL in loop acdefa

i1R1 + i2 R3 - E3 + i2r3 + i2 r4 + E4 + i1 R4 - E1 + i1r1 = 0

i1 (20) + 8 i2 - 11.5 + 0.5 i2 + 0.75 i2 + 20.5 + i1 15 - 15.5 + i1 (0.5) = 0

35.5 i1 + 9.25 i2 = 6.5

i2 = (6.5 - 35.5 i1 ) / 9.25                       eq-3

Using eq-1 , eq-2   and eq-3

i1 = (6.5 - 35.5 i1 ) / 9.25  + (19 - 35.5 i1 ) / 6.25

i1 = 0.36

i2 = (6.5 - 35.5 (0.36) ) / 9.25 = - 0.68

i3 = (19 - 35.5 (0.36) ) / 6.25 = 0.995

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