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Difficulty: MediumSeries Circuits

A technician is testing a direct-current (DC) control circuit connected to a 20 V20\text{ V} power supply. An ammeter in the loop indicates a steady current of 0.5 A0.5\text{ A}. The circuit contains three resistors wired in series: two have fixed values of 12 Ω12\ \Omega and 16 Ω16\ \Omega, while the third is an unlabelled component. What is the resistance of the unlabelled resistor?

  1. A
    28 Ω28\ \Omega
  2. 12 Ω12\ \OmegaAnswer
  3. C
    10 Ω10\ \Omega
  4. D
    40 Ω40\ \Omega

Answer

12 Ω12\ \Omega
The correct answer of 12 Ω12\ \Omega is found by first calculating the total circuit resistance using Ohm's Law (RT=20 V0.5 A=40 ΩR_T = \frac{20\text{ V}}{0.5\text{ A}} = 40\ \Omega). In a series circuit, total resistance is the sum of all individual components (RT=R1+R2+R3R_T = R_1 + R_2 + R_3). Subtracting the two known values (12 Ω+16 Ω=28 Ω12\ \Omega + 16\ \Omega = 28\ \Omega) from 40 Ω40\ \Omega leaves exactly 12 Ω12\ \Omega for the unlabelled resistor.

Step-by-Step Solution

1
Calculate the total resistance (RTR_T) of the series circuit using Ohm's Law.
RT=VtotalI=20 V0.5 A=40 ΩR_T = \frac{V_{\text{total}}}{I} = \frac{20\text{ V}}{0.5\text{ A}} = 40\ \Omega
The total resistance of a DC circuit is equal to the total applied voltage divided by the total circuit current.
2
Sum the resistances of the two known series resistors.
Rknown=12 Ω+16 Ω=28 ΩR_{\text{known}} = 12\ \Omega + 16\ \Omega = 28\ \Omega
In a series circuit, individual resistances add directly to form the total equivalent resistance.
3
Subtract the sum of the known resistors from the total circuit resistance to find the unknown resistance (R3R_3).
R3=RT−Rknown=40 Ω−28 Ω=12 ΩR_3 = R_T - R_{\text{known}} = 40\ \Omega - 28\ \Omega = 12\ \Omega
Because RT=R1+R2+R3R_T = R_1 + R_2 + R_3, the unlabelled resistor's value is the remaining resistance needed to reach 40 Ω40\ \Omega.

Key Concept

In a DC series circuit, total resistance is the sum of all individual resistances (RT=R1+R2+⋯+RnR_T = R_1 + R_2 + \dots + R_n), and the total current is determined by Ohm's Law (I=VTRTI = \frac{V_T}{R_T}).
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Topics

Series Circuits

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