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Chapter 24: Q. 39 (page 684)

The earth has a vertical electric field at the surface, pointing down, that averages100N/C. This field is maintained by various atmospheric processes, including lightning. What is the excess charge on the surface of the earth?

Short Answer

Expert verified

The excess charge on the surface of the earthQinis-4.5×105C.

Step by step solution

01

Formula for electric flux and charge

The electric flux is that the amount of electrical field that flows through a closed surface.

Gauss' law states that the electrical field passing through a surface is proportional to the charge within the surface.

Electric flux is,

Φe=EA=Qinϵo

Qin=ϵoEA

localid="1648750984200" Qin=4πϵoEr2...1

02

Calculation for excess charge on the surface of the world

The radius of the gaussian surface is up to the radius of the earth's outer surface,

r=6.37×106m

As it enters the world, the electrical field is negative.

E=-100N/C

Substitute values in equation localid="1648752090427" 1,

we get,

Qin=4πϵoEr2

=4π8.85×10-12C2/N·m2(-100N/C)6.37×106m2

=-4.5×105C

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Most popular questions from this chapter

An infinite cylinder of radius Rhas a linear charge density λ. The volume charge density C/m3within the cylinder (r≤R)is ρ(r)=rρ0/R, where ρ0is a constant to be determined.

a. Draw a graph of ρversus localid="1648911863544" xfor an x-axis that crosses the cylinder perpendicular to the cylinder axis. Let xrange from −2Rto 2R.

b. The charge within a small volume dVis dq=ρdV. The integral of ρdVover a cylinder of length localid="1648848405768" Lis the total charge Q=λLwithin the cylinder. Use this fact to show that ρ0=3λ/2πR2.

Hint: Let dVbe a cylindrical shell of length L, radius r, and thickness dr. What is the volume of such a shell?

c. Use Gauss's law to find an expression for the electric field strength Einside the cylinder, localid="1648889098349" r≤R, in terms of λand R.

d. Does your expression have the expected value at the surface, localid="1648889146353" r=R? Explain.

Suppose you have the uniformly charged cube in FIGURE Q24.1. Can you use symmetry alone to deduce the shape of the cube’s electric field? If so, sketch and describe the field shape. If not, why not?

A sphere of radius Rhas total charge Q. The volume charge Calc density role="math" localid="1648722354966" Cm3within the sphere is ρr=Cr2, whereC is a constant to be determined.
a. The charge within a small volume dVis dq=ρdV. The integral of ρdVover the entire volume of the sphere is the total chargeQ. Use this fact to determine the constant Cin terms of QandR .
Hint: Let dVbe a spherical shell of radiusr and thicknessdr. What is the volume of such a shell?
b. Use Gauss's law to find an expression for the electric field strengthE inside the sphere, ,r≤R in terms of QandR.
c. Does your expression have the expected value at the surface,r=R ? Explain.

The electric field strength just above one face of a copper penny is 2000N/C. What is the surface charge density on this face of the penny?

A tetrahedron has an equilateral triangle base with20-cm-long edges and three equilateral triangle sides. The base is parallel to the ground, and a vertical uniform electric field of strength 200N/C passes upward through the tetrahedron. a. What is the electric flux through the base? b. What is the electric flux through each of the three sides?
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