Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 2: Q18P (page 76)

Calculate the divergence of the following vector functions:

Two spheres, each of radius R and carrying uniform volume charge densities +p and -p , respectively, are placed so that they partially overlap (Fig. 2.28). Call the vector from the positive center to the negative center d. Show that the field in the region of overlap is constant, and find its value. [Hint: Use the answer to Prob. 2.12.]

Short Answer

Expert verified

The electric field in the overlapped region isE=p3ε0d.

Step by step solution

01

Determine the expression for the electric field in the two sphere.

Consider the diagram for the sphere that has radius R and carries the uniform charge such that they are overlapped with the distance from the center as d.

Consider the equation for electric field inside the positive sphere is,

E+=p3ε0r+

Here, ris the radius of the positive centered sphere, pis the uniform charge density.

Consider the equation for electric field inside the negative sphere is,

E-=p3ε0r-

Here, r- is the radius of the positive centered sphere.

02

Solve for the electric filed in the overlapping region.

Consider the expression for the resultant electric field as,

E=E++E-

Substitute p3ε0r+for E+and p3ε0r-for E+in the equation.

E=p3ε0r+-p3ε0r-

From the diagram d=r+-r-.

Solve further as,

E=p3ε0d

Therefore, the electric field in the overlapped region is E=p3ε0d.

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

Use Gauss's law to find the electric field inside a uniformly charged solid sphere (charge density p) Compare your answer to Prob. 2.8.

Find the net force that the southern hemisphere of a uniformly charged solid sphere exerts on the northern hemisphere. Express your answer in terms of the radiusR and the total charge Q.

A metal sphere of radius R ,carrying charge q ,is surrounded by a

thick concentric metal shell (inner radius a,outer radius b,as in Fig. 2.48). The

shell carries no net charge.

(a) Find the surface charge density σat R ,at a ,and at b .

(b) Find the potential at the center, using infinity as the reference point.

(c) Now the outer surface is touched to a grounding wire, which drains off charge

and lowers its potential to zero (same as at infinity). How do your answers to (a) and (b) change?

In a vacuum diode, electrons are "boiled" off a hot cathode, at potential zero, and accelerated across a gap to the anode, which is held at positive potential V0. The cloud of moving electrons within the gap (called space charge) quickly builds up to the point where it reduces the field at the surface of the cathode to zero. From then on, a steady current I flows between the plates.

Suppose the plates are large relative to the separation (A>>d2in Fig. 2.55), so

that edge effects can be neglected. Then V,ρand v (the speed of the electrons) are all functions of x alone.

  1. Write Poisson's equation for the region between the plates.

  1. Assuming the electrons start from rest at the cathode, what is their speed at point x , where the potential is V(x)?

  1. In the steady state, I is independent of x. What, then, is the relation between p and v?

  1. Use these three results to obtain a differential equation for V, by eliminating ρand v.

  1. Solve this equation for Vas a function of x, V0and d. Plot V(x), and compare it to the potential without space-charge. Also, find ρand v as functions of x.

  1. Show that
    I=kV03/2

and find the constant K. (Equation 2.56 is called the Child-Langmuir law. It holds for other geometries as well, whenever space-charge limits the current. Notice that the space-charge limited diode is nonlinear-it does not obey Ohm's law.)

Find the electric field a distance zabove the center of a flat circular disk of radius R(Fig. 2.1 0) that carries a uniform surface charge a.What does your formula give in the limit R→∞? Also check the case localid="1654687175238" z≫R.
See all solutions

Recommended explanations on Physics Textbooks

Fundamentals of Physics

Read Explanation

Radiation

Read Explanation

Oscillations

Read Explanation

Space Physics

Read Explanation

Particle Model of Matter

Read Explanation

Energy Physics

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