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Difficulty: HardSeries Circuits

An emergency communications console is powered by a 120 V120\text{ V} DC source connected in series with three resistive elements: R1=15 ΩR_1 = 15\ \Omega, R2=25 ΩR_2 = 25\ \Omega, and an unknown load resistor R3R_3. If a voltmeter placed across R1R_1 reads 18 V18\text{ V}, what is the resistance value of R3R_3?

  1. A
    85 Ω85\ \Omega
  2. 60 Ω60\ \OmegaAnswer
  3. C
    100 Ω100\ \Omega
  4. D
    104 Ω104\ \Omega

Answer

The resistance value of R3R_3 is 60 Ω60\ \Omega.
In a series circuit, the current is identical at all points. Using Ohm's Law on the first resistor gives a loop current of I=18 V15 Ω=1.2 AI = \frac{18\text{ V}}{15\ \Omega} = 1.2\text{ A}. The total circuit resistance is RT=VTI=120 V1.2 A=100 ΩR_T = \frac{V_T}{I} = \frac{120\text{ V}}{1.2\text{ A}} = 100\ \Omega. Since series resistances are additive (RT=R1+R2+R3R_T = R_1 + R_2 + R_3), the unknown resistance is R3=100 Ω−(15 Ω+25 Ω)=60 ΩR_3 = 100\ \Omega - (15\ \Omega + 25\ \Omega) = 60\ \Omega.

Step-by-Step Solution

1
Calculate the common series circuit current using the known voltage drop and resistance of the first resistor.
I=V1R1=18 V15 Ω=1.2 AI = \frac{V_1}{R_1} = \frac{18\text{ V}}{15\ \Omega} = 1.2\text{ A}
In a series circuit, current remains constant through every component.
2
Calculate the total equivalent resistance of the entire circuit using Ohm's Law with the total source voltage.
RT=VTI=120 V1.2 A=100 ΩR_T = \frac{V_T}{I} = \frac{120\text{ V}}{1.2\text{ A}} = 100\ \Omega
Total resistance equals total supply voltage divided by the total circuit current.
3
Determine the resistance of the unknown resistor by subtracting the known resistors from the total series resistance.
R3=RT−R1−R2=100 Ω−15 Ω−25 Ω=60 ΩR_3 = R_T - R_1 - R_2 = 100\ \Omega - 15\ \Omega - 25\ \Omega = 60\ \Omega
The total resistance in a series circuit is the linear sum of all individual resistances: RT=R1+R2+R3R_T = R_1 + R_2 + R_3.

Key Concept

Series Circuit Analysis and Ohm's Law Multi-Step Synthesis
Estimated Time:1m 30s
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Topics

Series Circuits

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