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Chapter 23: Q55P (page 683)

A charge distribution that is spherically symmetric but not uniform radially produces an electric field of magnitudeE=Kr4, directed radially outward from the center of the sphere. Here ris the radial distance from that center, andKis a constant. What is the volume density r of the charge distribution?

Short Answer

Expert verified

The volume charge density is6Kε0r3

Step by step solution

01

Listing the given quantities

E=Kr4

02

Understanding the concept of charge density and electric field

Using the formula of volume charge density and electric field determine the volume charge density

03

Explanation

We use,

E=qenc4πε0r2=14πε0r2∫01ρr4πr2dr

To solve for ρrand obtain,

role="math" localid="1657353658777" ρr=ε0r2ddrr2Er=ε0r2ddrKr6=6Kε0r3

The volume charge density is6Kε0r3

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Most popular questions from this chapter

The cube in Fig. 23-31 has edge length 1.40mand is oriented as shown in a region of uniform electric field. Find the electric flux through the right face if the electric field, in Newton per coulomb, is given by (a) 6.00i^,(b) -2.00j^, and -3.00i^+4.00k^(c). (d) What is the total flux through the cube for each field?

In Fig. 23-44, two large, thin metal plates are parallel and close to each other. On their inner faces, the plates have excess surface charge densities of opposite signs and magnitude7.0×10-22C/m2. In unit-vector notation, what is the electric field at points (a) to the left of the plates, (b) to the right of them, and (c) between them?

In Fig. 23-56, a non conducting spherical shell of inner radius a=2.00cm and outer radius b=2.40cm has (within its thickness) a positive volume charge density r=A/r , where Ais a constant and ris the distance from the center of the shell. In addition, a small ball of charge q=45.0fC is located at that center. What is value should Ahave if the electric field in the shell ( a≤r≤b) is to be uniform?

Figure 23-27 shows four solid spheres, each with charge Quniformly distributed through its volume. (a) Rank the spheres according to their volume charge density, greatest first. The figure also shows a point for each sphere, all at the same distance from the center of the sphere. (b) Rank the spheres according to the magnitude of the electric field they produce at point P, greatest first.

When a shower is turned on in a closed bathroom, the splashing of the water on the bare tub can fill the room’s air with negatively charged ions and produce an electric field in the air as great as 1000N/C. Consider a bathroom with dimensions2.5m×3.0m×2.0m. Along the ceiling, floor, and four walls, approximate the electric field in the air as being directed perpendicular to the surface and as having a uniform magnitude of600N/C. Also, treat those surfaces as forming a closed Gaussian surface around the room’s air. What are (a) the volume charge density r and (b) the number of excess elementary charges eper cubic meter in the room’s air?
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