AC Voltage RMS and Peak Relationship

Let's carefully examine the provided statements regarding AC voltage and current. We will analyze Assertion (A) and Reason (R) individually and then determine the relationship between them.

Assertion Analysis: 5V AC Supply and Peak Voltage

Assertion (A) states: "In everyday use, a 5V AC supply means 5 V RMS with the peak voltage about 7 V."

• RMS (Root Mean Square) Voltage: For an alternating current (AC) voltage, the RMS value is the effective value that produces the same heating effect as a direct current (DC) voltage of the same magnitude. Most AC voltmeters measure the RMS value.
• Relationship between RMS and Peak Voltage: For a sinusoidal AC waveform, the relationship between the peak voltage ($\text{V}_{\text{peak}}$) and the RMS voltage ($\text{V}_{\text{RMS}}$) is given by the formula:
  
  \[ \text{V}_{\text{RMS}} = \frac{\text{V}_{\text{peak}}}{\sqrt{2}} \]
  Alternatively, to find the peak voltage from the RMS voltage:
  
  \[ \text{V}_{\text{peak}} = \text{V}_{\text{RMS}} \times \sqrt{2} \]
• Calculation: If the RMS voltage ($\text{V}_{\text{RMS}}$) is 5 V, then the peak voltage ($\text{V}_{\text{peak}}$) would be:
  
  \[ \text{V}_{\text{peak}} = 5 \text{ V} \times \sqrt{2} \] \[ \text{V}_{\text{peak}} = 5 \text{ V} \times 1.414 \] \[ \text{V}_{\text{peak}} \approx 7.07 \text{ V} \]

Therefore, a 5V AC supply indeed refers to 5 V RMS, and its corresponding peak voltage is approximately 7 V. This makes Assertion (A) a true statement.

Reason Analysis: Why RMS Values are Used

Reason (R) states: "In everyday use, AC voltages (and currents) are always given as RMS values because this allows a sensible comparison to be made with steady DC voltages (and currents), such as from a battery."

• Power Equivalence: The primary reason for using RMS values for AC is that they represent the effective voltage or current that delivers the same amount of power to a resistive load as a DC voltage or current of the same magnitude.
• Sensible Comparison: For example, a 12 V RMS AC supply will dissipate the same amount of heat in a given resistor as a 12 V DC supply. This makes it straightforward to compare the "strength" or "effect" of an AC source with a DC source. Without RMS values, comparing peak AC voltage to DC voltage would be misleading in terms of power delivery.

Thus, Reason (R) accurately explains why RMS values are the standard for specifying AC voltages and currents in practical applications. This makes Reason (R) a true statement.

Relationship Between Assertion (A) and Reason (R)

We have established that both Assertion (A) and Reason (R) are true. Now, let's determine if Reason (R) is a correct explanation for Assertion (A).

• Assertion (A) describes the common practice of specifying AC voltage in terms of its RMS value and the resulting peak voltage.
• Reason (R) provides the fundamental justification for this common practice: using RMS values allows for direct and meaningful comparison with DC voltages regarding power dissipation and overall effect.

Because RMS values are used to allow sensible comparisons with DC (as stated in R), it follows that when an AC supply is rated (e.g., "5V AC"), it refers to the RMS value (as stated in A). The peak voltage then follows directly from this RMS value. Therefore, Reason (R) provides the underlying rationale for the convention described in Assertion (A).

Conclusion

Both Assertion (A) and Reason (R) are true, and Reason (R) is a correct explanation of Assertion (A).