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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

What is the minimum-energy configuration for a system ofNequal

point charges placed on or inside a circle of radius R? Because the charge on

a conductor goes to the surface, you might think theNcharges would arrange

themselves (uniformly) around the circumference. Show (to the contrary) that for

N = 12 it is better to place 11 on the circumference and one at the center. How about for N = 11 (is the energy lower if you put all 11 around the circumference, or if you put 10 on the circumference and one at the center)? [Hint: Do it numerically-you'll need at least 4 significant digits. Express all energies as multiples of q24蟺蔚0R]

Two large metal plates (each of area A) are held a small distance d

a part. Suppose we put a chargeQon each plate; what is the electrostatic pressure on the plates?

Imagine that new and extraordinarily precise measurements have revealed an error in Coulomb's law. The actual force of interaction between two point charges is found to be

F=14蟺蔚0q1q2r2(1+r位)e鈭�(r位)r^

where 位 is a new constant of nature (it has dimensions of length, obviously, and is a huge number鈥攕ay half the radius of the known universe鈥攕o that the correction is small, which is why no one ever noticed the discrepancy before). You are charged with the task of reformulating electrostatics to accommodate the new discovery. Assume the principle of superposition still holds.

a. What is the electric field of a charge distribution 蟻 (replacing Eq. 2.8)?

b. Does this electric field admit a scalar potential? Explain briefly how you reached your conclusion. (No formal proof necessary鈥攋ust a persuasive argument.)

c. Find the potential of a point charge q鈥攖he analog to Eq. 2.26. (If your answer to (b) was "no," better go back and change it!) Use 鈭� as your reference point.

d. For a point charge q at the origin, show that

鈭�SE.da+1位2鈭�V痴诲蟿=1蔚0q

where S is the surface, V the volume, of any sphere centered at q.

e. Show that this result generalizes:

鈭�SE.da+1位2鈭�V痴诲蟿=1蔚0Qenc

for any charge distribution. (This is the next best thing to Gauss's Law, in the new "electrostatics.鈥�)

f. Draw the triangle diagram (like Fig. 2.35) for this world, putting in all the appropriate formulas. (Think of Poisson's equation as the formula for 蟻 in terms of V, and Gauss's law (differential form) as an equation for 蟻 in terms of E.)

g. Show that some of the charge on a conductor distributes itself (uniformly!) over the volume, with the remainder on the surface. [Hint: E is still zero, inside a conductor.]

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 radius R and the total charge Q.

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

charge density u).
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