An alternating current (AC) sinusoidal voltage with a peak value of 10\text{ V} delivers t… — Armed Services Vocational Aptitude Battery (ASVAB)

An alternating current (AC) sinusoidal voltage with a peak value of $10\text{ V}$ delivers the same electrical power to a resistive heating element as a steady $10\text{ V}$ direct current (DC) supply.

Answer: Answer

Answer

False. An alternating current (AC) sine wave with a peak voltage of

10\text{ V}

delivers an effective (RMS) voltage of approximately

7.07\text{ V}

, which provides less electrical power to a resistor than a constant

10\text{ V}

direct current (DC) supply.

Direct current (DC) maintains a continuous, constant voltage and unidirectional current flow over time. In contrast, sinusoidal alternating current (AC) fluctuates, reaching its maximum peak voltage only momentarily and crossing zero volts twice per cycle. The effective value of an AC voltage—representing the DC voltage that would deliver the exact same average heating power across an identical resistor—is defined by its Root-Mean-Square (RMS) value (

V_{\text{RMS}} = 0.707 \times V_{\text{peak}}

). For a

10\text{ V}

peak AC supply, the effective voltage is only

7.07\text{ V}

, resulting in half the power output of a steady

10\text{ V}

DC supply.

Step-by-Step Solution

Identify the relationship between peak voltage and effective (RMS) voltage for alternating current.

V_{\text{RMS}} = \frac{V_{\text{peak}}}{\sqrt{2}} \approx 0.707 \times V_{\text{peak}} = 0.707 \times 10\text{ V} = 7.07\text{ V}

Direct current supplies a continuous, constant potential, whereas alternating current oscillates, meaning the effective power output of AC must be calculated using its RMS value.

Compare the power dissipated by the

10\text{ V}

DC source and the

10\text{ V}

peak AC source across a resistor

R

P_{\text{DC}} = \frac{(10\text{ V})^2}{R} = \frac{100}{R}

, whereas

P_{\text{AC}} = \frac{(V_{\text{RMS}})^2}{R} = \frac{(7.07\text{ V})^2}{R} = \frac{50}{R}

Power delivered to a resistor is proportional to the square of the applied voltage; the

10\text{ V}

peak AC source delivers only

50\%

of the power delivered by the

10\text{ V}

DC source.

Evaluate the truth value of the stem statement.

The statement is false.

Equal peak and DC voltage values do not result in equal power output due to the cyclic nature and lower effective RMS value of AC.

Key Concept

AC RMS Voltage vs. DC Equivalent Heating Power

Estimated Time:35s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic