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Difficulty: Very hardParallel Circuits

A mobile radar subsystem is powered by a 24 V24\text{ V} direct-current source. Its internal circuitry contains a parallel configuration with three distinct branches.

The total equivalent resistance of this parallel section is exactly 3 Ω3\ \Omega.
A technician measures the current through the first branch to be 3 A3\text{ A}.
The second branch has a known resistance of 24 Ω24\ \Omega.

What is the resistance of the third branch in ohms?

Answer: 6 \Omega

Answer

The resistance of the third branch is 6 Ω6\ \Omega.
The correct value is derived by finding the total circuit current (8 A8\text{ A}) and subtracting the known branch currents (3 A3\text{ A} and 1 A1\text{ A}) to find the current in the third branch (4 A4\text{ A}). Dividing the common 24 V24\text{ V} source by this 4 A4\text{ A} current yields 6 Ω6\ \Omega.

Step-by-Step Solution

1
Calculate the total circuit current using the total voltage and equivalent resistance.
Itotal=24 V/3 Ω=8 AI_{total} = 24\text{ V} / 3\ \Omega = 8\text{ A}
Ohm's law applies to the entire circuit, linking total voltage, total current, and equivalent resistance.
2
Determine the current flowing through the second branch.
I2=24 V/24 Ω=1 AI_2 = 24\text{ V} / 24\ \Omega = 1\text{ A}
In a parallel circuit, the full source voltage drops across every branch. Ohm's law (I=V/RI = V/R) gives the branch current.
3
Calculate the current flowing through the third branch.
I3=8 A−(3 A+1 A)=4 AI_3 = 8\text{ A} - (3\text{ A} + 1\text{ A}) = 4\text{ A}
Kirchhoff's Current Law states that the total current is the sum of the individual branch currents (Itotal=I1+I2+I3I_{total} = I_1 + I_2 + I_3).
4
Calculate the resistance of the third branch.
R3=24 V/4 A=6 ΩR_3 = 24\text{ V} / 4\text{ A} = 6\ \Omega
Apply Ohm's law (R=V/IR = V/I) specifically to the third branch using the source voltage and the calculated branch current.

Key Concept

Applying Ohm's law to both individual branches and the entire circuit, combined with parallel circuit rules for voltage and current.

Alternative Method

Use the parallel resistance reciprocal formula: 1/Req=1/R1+1/R2+1/R31/R_{eq} = 1/R_1 + 1/R_2 + 1/R_3. First, find R1=V/I1=24 V/3 A=8 ΩR_1 = V/I_1 = 24\text{ V} / 3\text{ A} = 8\ \Omega. Then, substitute the known values: 1/3=1/8+1/24+1/R31/3 = 1/8 + 1/24 + 1/R_3. Finding a common denominator of 24 gives 8/24=3/24+1/24+1/R38/24 = 3/24 + 1/24 + 1/R_3, which simplifies to 8/24=4/24+1/R38/24 = 4/24 + 1/R_3. Therefore, 1/R3=4/24=1/61/R_3 = 4/24 = 1/6, meaning R3=6 ΩR_3 = 6\ \Omega.
Estimated Time:1m 30s
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Topics

Parallel Circuits

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