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Difficulty: MediumParallel Circuits

An emergency egress lighting system operates from a 50 V50\text{ V} direct-current (DC) power source connected across two parallel branch circuits. Branch 1 contains an LED driver with an effective resistance of 25 Ω25\ \Omega. If the total current drawn from the power source is 3 A3\text{ A}, what is the resistance, in ohms (Ω\Omega), of Branch 2?

Answer: 50 ohms

Answer

The resistance of Branch 2 is 50 Ω50\ \Omega.
In a parallel circuit, each branch receives the full source voltage of 50 V50\text{ V}. Branch 1 draws I1=50 V25 Ω=2 AI_1 = \frac{50\text{ V}}{25\ \Omega} = 2\text{ A}. Since the total current is 3 A3\text{ A}, Branch 2 must draw I2=3 A−2 A=1 AI_2 = 3\text{ A} - 2\text{ A} = 1\text{ A}. Applying Ohm's Law to Branch 2 gives R2=50 V1 A=50 ΩR_2 = \frac{50\text{ V}}{1\text{ A}} = 50\ \Omega.

Step-by-Step Solution

1
Determine the voltage across each parallel branch.
V1=V2=50 VV_1 = V_2 = 50\text{ V}
In any parallel circuit configuration, the full source voltage appears equally across every individual parallel branch.
2
Calculate the current flowing through Branch 1 using Ohm's Law (I=VRI = \frac{V}{R}).
I1=50 V25 Ω=2 AI_1 = \frac{50\text{ V}}{25\ \Omega} = 2\text{ A}
Ohm's Law allows solving for branch current when branch voltage and resistance are known.
3
Calculate the remaining current flowing through Branch 2 by subtracting Branch 1 current from total line current.
I2=3 A−2 A=1 AI_2 = 3\text{ A} - 2\text{ A} = 1\text{ A}
According to Kirchhoff's Current Law, total current entering a parallel network must equal the sum of all individual branch currents (Itotal=I1+I2I_{\text{total}} = I_1 + I_2).
4
Calculate the resistance of Branch 2 using Ohm's Law (R=VIR = \frac{V}{I}).
R2=50 V1 A=50 ΩR_2 = \frac{50\text{ V}}{1\text{ A}} = 50\ \Omega
Dividing the branch voltage by the branch current yields the branch's effective resistance.

Key Concept

Parallel circuit branch voltage equality, Kirchhoff's Current Law (Itotal=I1+I2I_{\text{total}} = I_1 + I_2), and Ohm's Law (R=VIR = \frac{V}{I}).
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Topics

Parallel Circuits

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