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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=p3蔚0d.

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+=p3蔚0r+

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-=p3蔚0r-

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 p3蔚0r+for E+and p3蔚0r-for E+in the equation.

E=p3蔚0r+-p3蔚0r-

From the diagram d=r+-r-.

Solve further as,

E=p3蔚0d

Therefore, the electric field in the overlapped region is E=p3蔚0d.

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Most popular questions from this chapter

Consider two concentric spherical shells, of radiiaand b.Suppose the inner one carries a charge q ,and the outer one a charge -q(both of them uniformly distributed over the surface). Calculate the energy of this configuration, (a) using Eq. 2.45, and (b) using Eq. 2.47 and the results of Ex. 2.9.

We know that the charge on a conductor goes to the surface, but just

how it distributes itself there is not easy to determine. One famous example in which the surface charge density can be calculated explicitly is the ellipsoid:

x2a2+y2b2+z2c2=1

In this case15

蟽=Q4蟺abc(x2a4+y2b4+z2c4=1)-1/2

(2.57) where is the total charge. By choosing appropriate values for a,band c. obtain (from Eq. 2.57):

(a) the net (both sides) surface charge a (r)density on a circular disk of radius R;(b) the net surface charge density a (x) on an infinite conducting "ribbon" in the xyplane, which straddles theyaxis from x=-ato x=a(let A be the total charge per unit length of ribbon);

(c) the net charge per unit length 位(x) on a conducting "needle," running from x= -ato x= a . In each case, sketch the graph of your result.

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 蟽at 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?

Find the potential on the rim of a uniformly charged disk (radius R002C

charge density u).

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.
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