Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 23: Problem 43

An electric field is established in a nonuniform rod. A voltmeter is used to measure the potential difference between the left end of the rod and a point a distance \(x\) from the left end. The process is repeated, and it is found that the data are described by the relationship \(\Delta V=270 x^{2},\) where \(\Delta V\) has the units \(\mathrm{V} / \mathrm{m}^{2}\). What is the \(x\) -component of the electric field at a point \(13 \mathrm{~cm}\) from the left end?

Short Answer

Expert verified
Answer: The x-component of the electric field at a point 13 cm from the left end is -7.02 × 10² V/m.

Step by step solution

01

Write down the given relationship

We are given the relationship between the potential difference \(\Delta V\) and the distance \(x\) as \(\Delta V = 270x^2\), where \(\Delta V\) has units of V/m².
02

Recall the relationship between electric field and potential

The relationship between the electric field \(E\) and the potential difference \(\Delta V\) is given by \(\Delta V = -\int E dx\), where the \(x\)-component of the electric field is \(E_x\).
03

Differentiate the potential difference with respect to x

To find the electric field at a specific point, we need to differentiate the potential difference with respect to distance \(x\). Let \(\frac{d(\Delta V)}{dx} = \frac{d(270x^2)}{dx}\). Differentiating \(270x^2\) with respect to \(x\), we get \(\frac{d(\Delta V)}{dx} = 540x\).
04

Find the electric field at the given point

Now, we can find the \(x\)-component of the electric field at the given point by using the result from step 3 as the \(x\)-component of the electric field is given by the negative of the derivative of potential difference with respect to x: \(E_x = -\frac{d(\Delta V)}{dx}\). At a point 13 cm from the left end, \(x = 13 \times 10^{-2}\) meters. Plug this value into our result from step 3: \(E_x = -540x = -540(13 \times 10^{-2}) = -7.02 \times 10^2 \mathrm{V/m}\) The \(x\)-component of the electric field at a point \(13 \mathrm{~cm}\) from the left end is \(-7.02 \times 10^2 \mathrm{V/m}\).

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

Nuclear fusion reactions require that positively charged nuclei be brought into close proximity, against the electrostatic repulsion. As a simple example, suppose a proton is fired at a second, stationary proton from a large distance away. What kinetic energy must be given to the moving proton to get it to come within \(1.00 \cdot 10^{-15} \mathrm{~m}\) of the target? Assume that there is a head-on collision and that the target is fixed in place.

A particle with a charge of \(+5.0 \mu C\) is released from rest at a point on the \(x\) -axis, where \(x=0.10 \mathrm{~m}\). It begins to move as a result of the presence of a \(+9.0-\mu C\) charge that remains fixed at the origin. What is the kinetic energy of the particle at the instant it passes the point \(x=0.20 \mathrm{~m} ?\)

A positive charge is released and moves along an electric field line. This charge moves to a position of a) lower potential and lower potential energy. b) lower potential and higher potential energy. c) higher potential and lower potential energy. d) higher potential and higher potential energy.

The electric potential in a volume of space is given by \(V(x, y, z)=x^{2}+x y^{2}+y z\). Determine the electric field in this region at the coordinate (3,4,5) .

Derive an expression for electric potential along the axis (the \(x\) -axis) of a disk with a hole in the center, as shown in the figure, where \(R_{1}\) and \(R_{2}\) are the inner and outer radii of the disk. What would the potential be if \(R_{1}=0 ?\)
See all solutions

Recommended explanations on Physics Textbooks

Medical Physics

Read Explanation

Force

Read Explanation

Kinematics Physics

Read Explanation

Energy Physics

Read Explanation

Radiation

Read Explanation

Dynamics

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