A series circuit consists of a 48\text{ V} direct-current (DC) power source and three iden… — Armed Services Vocational Aptitude Battery (ASVAB)

A series circuit consists of a $48\text{ V}$ direct-current (DC) power source and three identical $16\,\Omega$ resistors labeled $R_1$ , $R_2$ , and $R_3$ . Under normal operation, each resistor drops $16\text{ V}$ . During troubleshooting, a technician measures $24\text{ V}$ across $R_1$ , $0\text{ V}$ across $R_2$ , and $24\text{ V}$ across $R_3$ , while current continues to flow through the circuit. Which of the following conditions explains this circuit fault?

A
Resistor $R_2$ has developed an open circuit.
Resistor $R_2$ has developed a short circuit.Answer
C
Resistor $R_2$ has doubled its internal resistance value.
D
The power supply voltage has decreased by half.

Answer

Resistor

R_2

has developed a short circuit.

When a component becomes shorted in a series circuit, its effective resistance drops to zero, resulting in a measured voltage drop of

0\text{ V}

across its terminals. The total circuit resistance decreases from

48\,\Omega

32\,\Omega

, increasing the circuit current to

1.5\text{ A}

. This higher current causes the remaining two

16\,\Omega

resistors to each drop

1.5\text{ A} \times 16\,\Omega = 24\text{ V}

, exactly matching the technician's meter readings.

Step-by-Step Solution

Analyze the voltage drop across the faulted component (

R_2

The voltage across

R_2

is measured at

0\text{ V}

A component with zero resistance (a short circuit) drops zero volts according to Ohm's Law (

V = I \times R = I \times 0 = 0\text{ V}

Evaluate the remaining circuit voltages across

R_1

and

R_3

The sum of voltages is

24\text{ V} + 24\text{ V} = 48\text{ V}

, matching the source voltage.

Kirchhoff's Voltage Law confirms the full source voltage is distributed across the remaining active components in the loop.

Calculate the total circuit resistance and current under shorted conditions.

R_{\text{total}} = 16\,\Omega + 0\,\Omega + 16\,\Omega = 32\,\Omega

, producing a current of

I = \frac{48\text{ V}}{32\,\Omega} = 1.5\text{ A}

A shorted component decreases total circuit resistance, which increases total current flow through the remaining healthy components.

Key Concept

Effect of a short circuit in a series circuit

Alternative Method

Use Kirchhoff's Voltage Law (

V_{\text{source}} = V_1 + V_2 + V_3

). Substituting the measured values gives

48\text{ V} = 24\text{ V} + 0\text{ V} + 24\text{ V}

. Because

V_2 = 0\text{ V}

while current is actively flowing,

R_2

must have zero resistance (

R_2 = \frac{0\text{ V}}{I} = 0\,\Omega

), which is the definition of a short circuit.

Estimated Time:1m 0s

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Answer

Step-by-Step Solution

Key Concept

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Alternative Method

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