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Chapter 31: Problem 71

The signal from the oscillating electrode is fed into an amplifier, which reports the measured voltage as an rms value, 1.5 nV. What is the potential difference between the two extremes? (a) 1.5 nV; (b) 3.0 nV; (c) 2.1 nV; (d) 4.2 nV.

Short Answer

Expert verified
The potential difference is 4.2 nV (d).

Step by step solution

01

Understanding RMS Voltage

The RMS (Root Mean Square) voltage is a measure of the effective value of an AC voltage. It is defined as the square root of the mean of the squares of the instantaneous values over a period. It provides a way to represent a varying voltage as an equivalent constant value.
02

Calculate Peak Voltage from RMS

To find the peak voltage from the RMS voltage for a sinusoidal waveform, we use the formula: \[ V_{peak} = V_{rms} \times \sqrt{2} \] Given that the RMS voltage \( V_{rms} \) is 1.5 nV, we insert this value into the formula: \[ V_{peak} = 1.5 \times \sqrt{2} \approx 2.12 \text{ nV} \]
03

Determining the Potential Difference

The potential difference between the two extremes of the waveform is twice the peak voltage because it goes from \(-V_{peak}\) to \(+V_{peak}\):\[ \Delta V = 2 \times V_{peak} = 2 \times 2.12 \approx 4.24 \text{ nV} \] Rounding to two significant figures, \( \Delta V \) is 4.2 nV.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Peak Voltage
Peak Voltage is the maximum voltage reached by an alternating current (AC) waveform. It represents the highest point of the wave鈥檚 oscillation. Understanding peak voltage is crucial because it shows the amplitude of the waveform.
In a sinusoidal waveform, the peak voltage is related to the RMS (Root Mean Square) voltage. The formula connecting RMS and peak voltage for a sinusoidal waveform is:
  • \[ V_{peak} = V_{rms} \times \sqrt{2} \]
This formula tells us that the peak voltage is the RMS voltage times the square root of two. For example, if the RMS voltage is 1.5 nV, the peak voltage will be approximately 2.12 nV. The peak voltage is important for designing circuits. It helps in understanding how much maximum potential is applied at any point without exceeding the safe limits.
Potential Difference
Potential Difference, often referred to simply as voltage, is the difference in electric potential between two points. It is the driving force that pushes the electrical current through a circuit. In an AC context, the potential difference varies with time.
The potential difference between the maximum positive and maximum negative values of an AC waveform, known as the "amplitude range," is twice the peak voltage. This is because the wave stretches from \(-V_{peak}\) to +\(V_{peak}\).
  • The potential difference between extremes is calculated as:\[ \, \Delta V = 2 \times V_{peak}\,\]
For instance, if the peak voltage is 2.12 nV, the potential difference becomes approximately 4.24 nV. Understanding this concept is key when working with AC systems so that components can be adequately chosen to withstand the voltage swings.
AC Voltage
AC Voltage refers to the voltage type where the current direction alternates, typically in a sinusoidal pattern. This is different from DC (Direct Current) voltage, where the flow is in one constant direction.
AC voltage is widely used for power distribution because it is easy to transform and has less power loss over long distances. Devices like transformers rely on AC voltage to adjust levels up or down.
In AC circuits:
  • Voltage and current vary with time, creating a sine wave.
  • The RMS value of voltage is crucial as it provides a measure of power delivered by the AC, allowing it to be equivalent to a DC voltage of the same value.
  • AC systems are used in household outlets and power grids, making understanding of its principles fundamental for electricity-related work.
AC voltage powers most of our everyday devices and is integral to modern electrical systems.

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Most popular questions from this chapter

At a frequency \(\omega_1\) the reactance of a certain capacitor equals that of a certain inductor. (a) If the frequency is changed to \(\omega_2\) = \(2\omega_1\), what is the ratio of the reactance of the inductor to that of the capacitor? Which reactance is larger? (b) If the frequency is changed to \(\omega3\)= \(\omega_1\)/3 , what is the ratio of the reactance of the inductor to that of the capacitor? Which reactance is larger? (c) If the capacitor and inductor are placed in series with a resistor of resistance R to form an \(L-R-C\) series circuit, what will be the resonance angular frequency of the circuit?

What is the dc impedance of the electrode, assuming that it behaves as an ideal capacitor? (a) 0; (b) infinite; (c) \(\sqrt{2}\times10^4\Omega\); (d) \(\sqrt{2}\times10^6\Omega\).

An \(L-R-C\) series circuit draws 220 W from a 120-V (rms), 50.0-Hz ac line. The power factor is 0.560, and the source voltage leads the current. (a) What is the net resistance \(R\) of the circuit? (b) Find the capacitance of the series capacitor that will result in a power factor of unity when it is added to the original circuit. (c) What power will then be drawn from the supply line?

An \(L-R-C\) series circuit is constructed using a 175-\(\Omega\) resistor, a 12.5-\(\mu\)F capacitor, and an 8.00-mH inductor, all connected across an ac source having a variable frequency and a voltage amplitude of 25.0 V. (a) At what angular frequency will the impedance be smallest, and what is the impedance at this frequency? (b) At the angular frequency in part (a), what is the maximum current through the inductor? (c) At the angular frequency in part (a), find the potential difference across the ac source, the resistor, the capacitor, and the inductor at the instant that the current is equal to one-half its greatest positive value. (d) In part (c), how are the potential differences across the resistor, inductor, and capacitor related to the potential difference across the ac source?

An \(L-R-C\) series circuit with \(L\) = 0.120 H, \(R\) = 240 \(\Omega\), and \(C\) = 7.30 \(\mu\)F carries an rms current of 0.450 A with a frequency of 400 Hz. (a) What are the phase angle and power factor for this circuit? (b) What is the impedance of the circuit? (c) What is the rms voltage of the source? (d) What average power is delivered by the source? (e) What is the average rate at which electrical energy is converted to thermal energy in the resistor? (f) What is the average rate at which electrical energy is dissipated (converted to other forms) in the capacitor? (g) In the inductor?
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