/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 34 Express the displacement current... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 16: Problem 34

Express the displacement current in a capacitor in terms of the capacitance and the rate of change of the voltage across the capacitor.

Short Answer

Expert verified
The displacement current (Id) in a capacitor is given by the expression \(Id = C\frac{dV}{dt}\), where C is the capacitance and \(\frac{dV}{dt}\) is the rate of change of the voltage across the capacitor.

Step by step solution

01

Find the Capacitance Formula

Recall that the capacitance of a capacitor is given by the formula: C = \(\frac{Q}{V}\), where C is the capacitance, Q is the charge on the capacitor, and V is the voltage across the capacitor.
02

Write the Displacement Current Formula

The displacement current (Id) through the dielectric in a capacitor is given by Maxwell's equation: Id = \(\frac{dQ}{dt}\), where Id is the displacement current and \(\frac{dQ}{dt}\) is the rate of change of charge with respect to time.
03

Express Charge in Terms of Voltage and Capacitance

From the capacitance formula in Step 1, we have: Q = CV. Now, replace the charge Q in the displacement current formula from Step 2 with the expression in terms of capacitance and voltage: Id = \(\frac{d(CV)}{dt}\).
04

Apply the Chain Rule

Since capacitance remains constant while voltage changes with time, we can apply the chain rule to differentiate the product of capacitance and voltage with respect to time: Id = C\(\frac{dV}{dt}\), where \(\frac{dV}{dt}\) is the rate of change of voltage across the capacitor.
05

Express the Displacement Current

Now, we have the expression for the displacement current in terms of the capacitance and the rate of change of the voltage across the capacitor: Id = C\(\frac{dV}{dt}\). The displacement current in a capacitor is equal to the product of the capacitance and the rate of change of the voltage across the capacitor.

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.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Capacitance
Capacitance is a fundamental property of capacitors. It is defined as the ability of a capacitor to store electrical charge. The measure of capacitance tells us how much charge a capacitor can hold for each volt applied to it. This relationship is expressed by the formula:\[ C = \frac{Q}{V} \]Here, \( C \) is the capacitance measured in Farads (F), \( Q \) is the charge in Coulombs, and \( V \) is the voltage across the capacitor in Volts.
  • Capacitance depends on factors like the physical size of the capacitor plates and the distance between them.
  • Materials between the plates (the dielectric) also affect capacitance, as certain materials can increase the capacitance.
Understanding capacitance helps us determine how efficiently a capacitor can function in storing and releasing electrical energy.
Voltage across capacitor
The voltage across a capacitor plays a crucial role in its operation. Voltage is the electric potential difference between two points鈥攊n this case, the potential difference between the two plates of a capacitor.When a capacitor is connected to a power source, it charges up, and an electric field develops across its plates. The voltage across a capacitor can be defined by the formula:\[ V = \frac{Q}{C} \]
  • In this formula, \( V \) is the voltage across the capacitor, \( Q \) is the charge stored, and \( C \) is the capacitance.
  • The voltage increases as charge is stored, up to the dielectric breakdown voltage limit, beyond which the capacitor can fail.
Voltage across a capacitor must be controlled to ensure safe and efficient operation of electronic circuits.
Rate of change of voltage
The rate of change of voltage is important when analyzing and designing circuits involving capacitors. It describes how quickly the voltage across a capacitor changes over time.In the context of displacement current, the rate of change of voltage \( \frac{dV}{dt} \) is significant because it directly influences the current flowing through a capacitor. When the voltage changes, the displacement current is given by:\[ I_d = C \frac{dV}{dt} \]
  • This formula highlights the fact that a rapidly changing voltage results in a larger displacement current.
  • Engineers need to ensure that the rate of change does not exceed the capacitor's ratings to prevent damage.
Monitoring the rate of voltage change is essential for the stability and performance of various electronic applications.

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

What property of light corresponds to loudness in sound?

A parallel-plate capacitor with plate separation \(d\) is connected to a source of emf that places a time-dependent voltage \(V(t)\) across its circular plates of radius \(r_{0}\) and area \(A=\pi r_{0}^{2}(\text { see below })\) (a) Write an expression for the time rate of change of energy inside the capacitor in terms of \(V(t)\) and \(d V(t) / d t\) (b) Assuming that \(V(t)\) is increasing with time, identify the directions of the electric field lines inside the capacitor and of the magnetic field lines at the edge of the region between the plates, and then the direction of the Poynting vector \(\overrightarrow{\mathbf{S}}\) at this location. (c) Obtain expressions for the time dependence of \(E(t),\) for \(B(t)\) from the displacement current, and for the magnitude of the Poynting vector at the edge of the region between the plates. (d) From \(\overrightarrow{\mathbf{S}}\), obtain an expression in terms of \(V(t)\) and \(d V(t) / d t\) for the rate at which electromagnetic field energy enters the region between the plates. (e) Compare the results of parts (a) and (d) and explain the relationship between them.

The Poynting vector describes a flow of energy whenever electric and magnetic fields are present. Consider a long cylindrical wire of radius \(r\) with a current \(I\) in the wire, with resistance \(R\) and voltage \(V\). From the expressions for the electric field along the wire and the magnetic field around the wire, obtain the magnitude and direction of the Poynting vector at the surface. Show that it accounts for an energy flow into the wire from the fields around it that accounts for the Ohmic heating of the wire.

What is the physical significance of the Poynting vector?

How many helium atoms, each with a radius of about 31 pm, must be placed end to end to have a length equal to one wavelength of \(470 \mathrm{nm}\) blue light?
See all solutions

Recommended explanations on Physics Textbooks

Linear Momentum

Read Explanation

Medical Physics

Read Explanation

Mechanics and Materials

Read Explanation

Wave Optics

Read Explanation

Circular Motion and Gravitation

Read Explanation

Electric Charge Field and Potential

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.