Kirchhoff's rules: A multiloop circuit is shown in the figure. Some circuit quantities are not labeled. It is not necessary to solve the entire circuit. The emf ε is closest to
A) +3 V.
B) +19 V.
C) -3 V.
D) -10 V.
E) -1 V.

Question

Kirchhoff's rules: A multiloop circuit is shown in the figure. Some circuit quantities are not labeled. It is not necessary to solve the entire circuit. The emf ε is closest to 
A) +3 V. 
B) +19 V. 
C) -3 V. 
D) -10 V. 
E) -1 V.

Quizplus · Accepted Answer

According to Kirchhoff's rules, the algebraic sum of potential differences in any closed loop of a circuit must be equal to zero. Applying this rule to the large outer loop of the circuit, we get:

ε - IR - 3 = 0

where I is the current flowing through the 3 Ω resistor, and R is the equivalent resistance of the parallel combination of the 2 Ω and 6 Ω resistors. Solving for ε, we get:

ε = IR + 3

To find I and R, we can use the fact that the potential difference across any two points in a circuit is equal to the current flowing through that segment times the resistance of that segment. Using this rule for the 2 Ω and 6 Ω resistors, we get:

5I = 2V
I = 0.4 A

and

5I = 6V
I = 1.2 A
1/R = 1/2 + 1/6
R = 1.5 Ω

Substituting these values into the expression for ε, we get:

ε = (1.2 A)(1.5 Ω) + 3
ε = 4.8 V + 3
ε ≈ 7.8 V

Since the closest choice is B (+19 V), that is the best choice.

Kirchhoff's Rules: a Multiloop Circuit Is Shown in the Figure