Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 11: Problem 1575

Three concentric spherical shells have radii a, \(b\) and \(c(a

Short Answer

Expert verified
(D) \(\mathrm{V}_{\mathrm{C}}=\mathrm{V}_{\mathrm{A}} \neq \mathrm{V}_{\mathrm{B}}\)

Step by step solution

01

Write the expression for potential

For a spherical shell with a radius r and surface charge density σ, the potential V can be written as: \[V = \frac{1}{4 \pi \epsilon_0} \frac{Q}{r}\] where Q is the total charge on the shell, and \(\epsilon_0\) is the vacuum permittivity. For our problem, we will denote the total charge for each shell as \(Q_A\), \(Q_B\), and \(Q_C\).
02

Find the total charge on each shell

The total charge on a spherical shell is given by: \[Q = \sigma \times \text{Surface Area}\] For each shell, their total charges are: - \(Q_A = \sigma \times 4\pi a^2\) - \(Q_B = -\sigma \times 4\pi b^2\) - \(Q_C = \sigma \times 4\pi (a+b)^2\)
03

Calculate the potentials for each shell

Now, we can substitute the corresponding total charges into the expression for potential to calculate the potentials for each shell: - \(V_A = \frac{1}{4 \pi \epsilon_0} \frac{\sigma \times 4\pi a^2}{a} = \frac{\sigma a}{\epsilon_0}\) - \(V_B = \frac{1}{4 \pi \epsilon_0} \frac{-\sigma \times 4\pi b^2}{b} = -\frac{\sigma b}{\epsilon_0}\) - \(V_C = \frac{1}{4 \pi \epsilon_0} \frac{\sigma \times 4\pi (a+b)^2}{(a + b)} = \frac{\sigma (a+b)}{\epsilon_0}\)
04

Compare the potentials

Now, we can compare the potentials of the shells: - \(V_C = \frac{\sigma (a+b)}{\epsilon_0} = \frac{\sigma a}{\epsilon_0} + \frac{\sigma b}{\epsilon_0} = V_A - V_B\) From this result, we can see that \(V_C = V_A \neq V_B\), which corresponds to option (D). Therefore, the correct answer is: (D) \(\mathrm{V}_{\mathrm{C}}=\mathrm{V}_{\mathrm{A}} \neq \mathrm{V}_{\mathrm{B}}\)

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

Point charges \(q_{1}=2 \mu c\) and \(q_{2}=-1 \mu c\) care kept at points \(\mathrm{x}=0\) and \(\mathrm{x}=6\) respectively. Electrical potential will be zero at points ..... (A) \(\mathrm{x}=-2, \mathrm{x}=2\) (B) \(\mathrm{x}=1, \mathrm{x}=5\) (C) \(\mathrm{x}=4, \mathrm{x}=12\) (D) \(\mathrm{x}=2, \mathrm{x}=9\)

Let $\mathrm{P}(\mathrm{r})\left[\mathrm{Q} /\left(\pi \mathrm{R}^{4}\right)\right] \mathrm{r}$ be the charge density distribution for a solid sphere of radius \(\mathrm{R}\) and total charge \(\mathrm{Q}\). For a point ' \(\mathrm{P}\) ' inside the sphere at distance \(\mathrm{r}_{1}\) from the centre of the sphere the magnitude of electric field is (A) $\left[\mathrm{Q} /\left(4 \pi \epsilon_{0} \mathrm{r}_{1}^{2}\right)\right]$ (B) $\left[\left(\mathrm{Qr}_{1}^{2}\right) /\left(4 \pi \in{ }_{0} \mathrm{R}^{4}\right)\right]$ (C) $\left[\left(\mathrm{Qr}_{1}^{2}\right) /\left(3 \pi \epsilon_{0} \mathrm{R}^{4}\right)\right]$

Two positive point charges of \(12 \mu \mathrm{c}\) and \(8 \mu \mathrm{c}\) are \(10 \mathrm{~cm}\) apart each other. The work done in bringing them $4 \mathrm{~cm}$ closer is .... (A) \(5.8 \mathrm{~J}\) (B) \(13 \mathrm{eV}\) (C) \(5.8 \mathrm{eV}\) (D) \(13 \mathrm{~J}\)

A long string with a charge of \(\lambda\) per unit length passes through an imaginary cube of edge \(\ell\). The maximum possible flux of the electric field through the cube will be ....... (A) \(\sqrt{3}\left(\lambda \ell / \in_{0}\right)\) (B) \(\left(\lambda \ell / \in_{0}\right)\) (C) \(\sqrt{2}\left(\lambda \ell / \in_{0}\right)\) (D) \(\left[\left(6 \lambda \ell^{2}\right) / \epsilon_{0}\right]\)

Two metal plate form a parallel plate capacitor. The distance between the plates is \(\mathrm{d}\). A metal sheet of thickness \(\mathrm{d} / 2\) and of the same area is introduced between the plates. What is the ratio of the capacitance in the two cases? (A) \(4: 1\) (B) \(3: 1\) (C) \(2: 1\) (D) \(5: 1\)
See all solutions

Recommended explanations on English Textbooks

Pragmatics

Read Explanation

Discourse

Read Explanation

Essay Plan

Read Explanation

Sociolinguistics

Read Explanation

Key Concepts in Language and Linguistics

Read Explanation

Research and Composition

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