Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 17: Problem 100

A parallel plate capacitor has a charge of \(0.020 \mu \mathrm{C}\) on each plate with a potential difference of \(240 \mathrm{V}\). The parallel plates are separated by \(0.40 \mathrm{mm}\) of air. (a) What is the capacitance for this capacitor? (b) What is the area of a single plate? (c) At what voltage will the air between the plates become ionized? Assume a dielectric strength of \(3.0 \mathrm{kV} / \mathrm{mm}\) for air.

Short Answer

Expert verified
Answer: The capacitance of the capacitor is \(8.33 \times 10^{-11} \text{F}\), the area of a single plate is \(3.76 \times 10^{-3} \text{m\)^2\(}\), and the voltage at which the air between the plates becomes ionized is \(1.2 \text{kV}\).

Step by step solution

01

Find the capacitance

Using the formula for the capacitance of a parallel plate capacitor: \(C = \frac{Q}{V}\) Where \(C\) is the capacitance, \(Q\) is the charge, and \(V\) is the potential difference. We have \(Q = 0.020 \mu \mathrm{C}\) and \(V = 240 \mathrm{V}\). Plug these values in the formula and solve for \(C\): \(C = \frac{0.020 \times 10^{-6} \text{C}}{240 \text{V}}\)
02

Calculate the capacitance

Calculate the capacitance: \(C = \frac{0.020 \times 10^{-6} \text{C}}{240 \text{V}} = 8.33 \times 10^{-11} \text{F}\) The capacitance of this capacitor is \(8.33 \times 10^{-11} \text{F}\).
03

Find the area of a single plate

To find the area of a single plate, we use the formula for the capacitance of a parallel plate capacitor with a dielectric constant: \(C = \frac{\epsilon_0 A}{d}\) Where \(C\) is the capacitance, \(\epsilon_0\) is the vacuum permittivity \(8.85 \times 10^{-12} \text{F/m}\), \(A\) is the area of a single plate, and \(d\) is the separation distance between the plates. We have \(C = 8.33 \times 10^{-11} \text{F}\) and \(d = 0.40 \mathrm{mm}\). Plug these values into the formula and solve for \(A\): \(A = \frac{C \times d}{\epsilon_0} = \frac{8.33 \times 10^{-11} \text{F} \times 0.40 \times 10^{-3} \text{m}}{8.85 \times 10^{-12} \text{F/m}}\)
04

Calculate the area of a single plate

Calculate the area of a single plate: \(A = \frac{8.33 \times 10^{-11} \text{F} \times 0.40 \times 10^{-3} \text{m}}{8.85 \times 10^{-12} \text{F/m}} = 3.76 \times 10^{-3} \text{m\)^2\(}\) The area of a single plate is \(3.76 \times 10^{-3} \text{m\)^2\(}\).
05

Find the voltage at which the air becomes ionized

Given the dielectric strength of air is \(3.0 \text{kV/mm}\), we can find the voltage at which the air becomes ionized by multiplying the dielectric strength by the separation distance: \(V_{ionization} = \text{Dielectric strength} \times \text{Separation distance}\) We have the dielectric strength as \(3.0 \text{kV/mm}\) and the separation distance as \(0.40 \mathrm{mm}\). Multiply them to get: \(V_{ionization} = 3.0 \text{kV/mm} \times 0.40 \mathrm{mm}\)
06

Calculate the ionization voltage

Calculate the ionization voltage: \(V_{ionization} = 3.0 \text{kV/mm} \times 0.40 \mathrm{mm} = 1.2 \text{kV}\) The voltage at which the air between the plates becomes ionized is \(1.2 \text{kV}\). To summarize the results: a) The capacitance of the capacitor is \(8.33 \times 10^{-11} \text{F}\). b) The area of a single plate is \(3.76 \times 10^{-3} \text{m\)^2\(}\). c) The voltage at which the air between the plates becomes ionized is \(1.2 \text{kV}\).

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

It is believed that a large electric fish known as Torpedo occidentalis uses electricity to shock its victims. A typical fish can deliver a potential difference of \(0.20 \mathrm{kV}\) for a duration of \(1.5 \mathrm{ms}\). This pulse delivers charge at a rate of \(18 \mathrm{C} / \mathrm{s} .\) (a) What is the rate at which work is done by the electric organs during a pulse? (b) What is the total amount of work done during one pulse?
Draw some electric field lines and a few equipotential surfaces outside a positively charged conducting cylinder. What shape are the equipotential surfaces?
A shark is able to detect the presence of electric fields as small as $1.0 \mu \mathrm{V} / \mathrm{m} .$ To get an idea of the magnitude of this field, suppose you have a parallel plate capacitor connected to a 1.5 - \(V\) battery. How far apart must the parallel plates be to have an electric field of $1.0 \mu \mathrm{V} / \mathrm{m}$ between the plates?
A defibrillator is used to restart a person's heart after it stops beating. Energy is delivered to the heart by discharging a capacitor through the body tissues near the heart. If the capacitance of the defibrillator is $9 \mu \mathrm{F}\( and the energy delivered is to be \)300 \mathrm{J},$ to what potential difference must the capacitor be charged?
A parallel plate capacitor is attached to a battery that supplies a constant voltage. While the battery is still attached, a dielectric of dielectric constant \(\kappa=3.0\) is inserted so that it just fits between the plates. What is the energy stored in the capacitor after the dielectric is inserted in terms of the energy \(U_{0}\) before the dielectric was inserted?
See all solutions

Recommended explanations on Physics Textbooks

Electricity and Magnetism

Read Explanation

Quantum Physics

Read Explanation

Physics of Motion

Read Explanation

Particle Model of Matter

Read Explanation

Scientific Method Physics

Read Explanation

Classical Mechanics

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