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Difficulty: MediumSeries Circuits

A marine communication console operates on a 36 V36\text{ V} DC power bus. The circuit consists of three components connected in series: a status indicator resistor R1=18 ΩR_1 = 18\ \Omega, an internal filter resistor R2=24 ΩR_2 = 24\ \Omega, and an unknown current-limiting resistor R3R_3. An inline ammeter measures the total circuit current as 0.5 A0.5\text{ A}. What is the resistance of resistor R3R_3?

  1. A
    18 Ω18\ \Omega
  2. 30 Ω30\ \OmegaAnswer
  3. C
    24 Ω24\ \Omega
  4. D
    72 Ω72\ \Omega

Answer

The resistance of resistor R3R_3 is 30 Ω30\ \Omega.
Using Ohm's Law, the total series resistance is RT=VTI=36 V0.5 A=72 ΩR_T = \frac{V_T}{I} = \frac{36\text{ V}}{0.5\text{ A}} = 72\ \Omega. Since resistances in a series circuit are additive (RT=R1+R2+R3R_T = R_1 + R_2 + R_3), the unknown resistance is found by subtracting the sum of the known resistors (18 Ω+24 Ω=42 Ω18\ \Omega + 24\ \Omega = 42\ \Omega) from the total: R3=72 Ω−42 Ω=30 ΩR_3 = 72\ \Omega - 42\ \Omega = 30\ \Omega.

Step-by-Step Solution

1
Calculate the total series circuit resistance (RTR_T) using Ohm's Law.
RT=VTIT=36 V0.5 A=72 ΩR_T = \frac{V_T}{I_T} = \frac{36\text{ V}}{0.5\text{ A}} = 72\ \Omega
In a series circuit, total resistance equals total applied voltage divided by the common loop current.
2
Sum the resistances of the known series components.
R1+2=R1+R2=18 Ω+24 Ω=42 ΩR_{1+2} = R_1 + R_2 = 18\ \Omega + 24\ \Omega = 42\ \Omega
Resistances add directly in a series configuration.
3
Subtract the sum of the known resistances from the total circuit resistance to isolate R3R_3.
R3=RT−(R1+R2)=72 Ω−42 Ω=30 ΩR_3 = R_T - (R_1 + R_2) = 72\ \Omega - 42\ \Omega = 30\ \Omega
The total resistance is the sum of all individual resistors: RT=R1+R2+R3R_T = R_1 + R_2 + R_3.

Key Concept

In a series circuit, current remains constant throughout the entire loop, and total resistance equals the sum of all individual resistances (RT=R1+R2+R3R_T = R_1 + R_2 + R_3). Ohm's Law (R=VIR = \frac{V}{I}) relates total voltage to total resistance and current.
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Topics

Series Circuits

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