Examkin

Question

Difficulty: HardParallel Circuits

A 60 V60\text{ V} direct-current power supply feeds a circuit containing three parallel branches. The total current drawn from the supply is 10 A10\text{ A}. If the first two branches have resistances of 20 Ω20\ \Omega and 30 Ω30\ \Omega, what is the resistance of the third branch?

  1. A
    6 Ω6\ \Omega
  2. 12 Ω12\ \OmegaAnswer
  3. C
    18 Ω18\ \Omega
  4. D
    50 Ω50\ \Omega

Answer

The resistance of the third branch is 12 Ω12\ \Omega.
Because the circuit is wired in parallel, the full supply voltage of 60 V60\text{ V} is applied across each branch. Using Ohm's Law (I=V/RI = V / R), the first branch draws 60 V/20 Ω=3 A60\text{ V} / 20\ \Omega = 3\text{ A} and the second branch draws 60 V/30 Ω=2 A60\text{ V} / 30\ \Omega = 2\text{ A}. The sum of these two branch currents is 5 A5\text{ A}. Since the total source current is 10 A10\text{ A}, the third branch must carry the remaining 10 A−5 A=5 A10\text{ A} - 5\text{ A} = 5\text{ A}. Applying Ohm's Law to the third branch yields R3=60 V/5 A=12 ΩR_3 = 60\text{ V} / 5\text{ A} = 12\ \Omega.

Step-by-Step Solution

1
Determine the voltage across each parallel branch.
Each branch receives the full source voltage of 60 V60\text{ V}.
In a parallel circuit, the voltage across every parallel branch is identical to the source voltage.
2
Calculate the individual currents flowing through the first two known branches using Ohm's Law (I=V/RI = V / R).
I1=60 V/20 Ω=3 AI_1 = 60\text{ V} / 20\ \Omega = 3\text{ A} and I2=60 V/30 Ω=2 AI_2 = 60\text{ V} / 30\ \Omega = 2\text{ A}.
Branch currents must be determined to find how much of the total current is accounted for by the known branches.
3
Find the current flowing through the third branch by subtracting the known branch currents from the total supply current (I3=Itotal−I1−I2I_3 = I_{\text{total}} - I_1 - I_2).
I3=10 A−(3 A+2 A)=5 AI_3 = 10\text{ A} - (3\text{ A} + 2\text{ A}) = 5\text{ A}.
Total current in a parallel circuit equals the sum of the individual branch currents.
4
Calculate the resistance of the third branch using Ohm's Law (R3=V/I3R_3 = V / I_3).
R3=60 V/5 A=12 ΩR_3 = 60\text{ V} / 5\text{ A} = 12\ \Omega.
Applying Ohm's Law to the third branch gives its required resistance value.

Key Concept

In a parallel circuit, voltage remains constant across all branches, the total current equals the sum of all individual branch currents (Itotal=I1+I2+⋯+InI_{\text{total}} = I_1 + I_2 + \dots + I_n), and each branch resistance satisfies Ohm's Law (Rn=V/InR_n = V / I_n).
Estimated Time:1m 30s
Rate this question

Topics

Parallel Circuits

More questions from this topic