Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 21: Problem 2

A European outlet supplies \(220 \mathrm{V}\) (rms) at \(50 \mathrm{Hz}\). How many times per second is the magnitude of the voltage equal to $220 \mathrm{V} ?$

Short Answer

Expert verified
Answer: The magnitude of the voltage reaches 220V 100 times per second.

Step by step solution

01

Express the voltage as a function of time

We know that the voltage of a sinusoidal wave can be represented as: $$v(t) = V_m \cdot \sin(\omega t)$$ Where \(V_m\) is the peak voltage, \(\omega\) is the angular frequency and \(t\) is the time. To find the peak voltage, we can use the formula: $$V_\text{rms} = \frac{V_m}{\sqrt{2}}$$ Given \(V_\text{rms} = 220V\), we can find the peak voltage as follows: $$V_m = V_\text{rms} \cdot \sqrt{2} = 220V \cdot \sqrt{2}$$ And the angular frequency can be found using: $$\omega = 2\pi f$$ Where \(f\) is the frequency. For this problem, \(f=50Hz\). This gives us: $$\omega = 2\pi \cdot 50Hz$$ Now we can write our sinusoidal voltage function: $$v(t) = 220\sqrt{2} \cdot \sin(100\pi t)$$
02

Solve the equation for when the magnitude is 220V

We need to find the points in time when the magnitude of \(v(t)\) equals the given voltage magnitude of \(220V\). We can set up the equation as follows: $$|220\sqrt{2} \cdot \sin(100\pi t)| = 220V$$ To solve the equation, we can divide both sides by \(220V\): $$|\sin(100\pi t)| = \frac{1}{\sqrt{2}}$$ Since sine has a maximum value of 1, it can achieve a value of \(\frac{1}{\sqrt{2}}\) twice in each period. Therefore, the value of \(220V\) is reached twice in each cycle.
03

Find the total number of times \(220V\) is reached per second

We know that the frequency is \(50Hz\), meaning there are 50 cycles of the signal in a second. Since the voltage reaches \(220V\) twice in one cycle, it will do so \(50 \times 2 = 100\) times in a second. So the magnitude of the voltage is equal to \(220V\) 100 times per second.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A \(150-\Omega\) resistor is in series with a \(0.75-\mathrm{H}\) inductor in an ac circuit. The rms voltages across the two are the same. (a) What is the frequency? (b) Would each of the rms voltages be half of the rms voltage of the source? If not, what fraction of the source voltage are they? (In other words, \(V_{R} / \ell_{m}=V_{L} / \mathcal{E}_{m}=?\) ) (c) What is the phase angle between the source voltage and the current? Which leads? (d) What is the impedance of the circuit?
An \(R L C\) series circuit is connected to an ac power supply with a \(12-\mathrm{V}\) amplitude and a frequency of \(2.5 \mathrm{kHz}\) If $R=220 \Omega, C=8.0 \mu \mathrm{F},\( and \)L=0.15 \mathrm{mH},$ what is the average power dissipated?
A television set draws an rms current of \(2.50 \mathrm{A}\) from a \(60-\mathrm{Hz}\) power line. Find (a) the average current, (b) the average of the square of the current, and (c) the amplitude of the current.
A 40.0 -mH inductor, with internal resistance of \(30.0 \Omega\) is connected to an ac source $$\varepsilon(t)=(286 \mathrm{V}) \sin [(390 \mathrm{rad} / \mathrm{s}) t]$$ (a) What is the impedance of the inductor in the circuit? (b) What are the peak and rms voltages across the inductor (including the internal resistance)? (c) What is the peak current in the circuit? (d) What is the average power dissipated in the circuit? (e) Write an expression for the current through the inductor as a function of time.
Suppose that an ideal capacitor and an ideal inductor are connected in series in an ac circuit. (a) What is the phase difference between \(v_{\mathrm{C}}(t)\) and \(v_{\mathrm{L}}(t) ?\) [Hint: since they are in series, the same current \(i(t)\) flows through both. \(]\) (b) If the rms voltages across the capacitor and inductor are \(5.0 \mathrm{V}\) and \(1.0 \mathrm{V}\), respectively, what would an ac voltmeter (which reads rms voltages) connected across the series combination read?
See all solutions

Recommended explanations on Physics Textbooks

Famous Physicists

Read Explanation

Electricity and Magnetism

Read Explanation

Electrostatics

Read Explanation

Fundamentals of Physics

Read Explanation

Fields in Physics

Read Explanation

Magnetism and Electromagnetic Induction

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free