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Chapter 21: Problem 2

The electric flux through a closed surface is zero. Must the electric field be zero on that surface? If not, give an example.

Short Answer

Expert verified
No, the electric field does not necessarily have to be zero on a surface even if the electric flux is zero. An example is a closed surface carrying equal amounts of positive and negative charges; the fields from each charge cancel out, resulting in a net flux of zero, but the field at any point is not zero.

Step by step solution

01

Understanding Electric Flux and Field

Electric flux is the measure of the electric field passing perpendicularly through a given area. For a closed surface, the net electric flux is given by the formula: \(\Phi_E = 鈭� E . ds\), where E is the electric field and ds is the differential surface area. Importantly, if there is no change in electric charge within the closed surface, the electric flux will be zero as per Gauss's law.
02

Interpretation of Zero Electric Flux

Zero electric flux through a closed surface means there is no net electric field leaving or entering the surface. It does not imply that the electric field itself is zero everywhere on the surface. Inside a closed conducting surface, the field lines enter and leave the surface by equal amounts, so the net flux is zero.
03

Providing an Example

Imagine a closed surface with an equal amount of positive and negative charges. The electric fields produced by the positive and negative charges cancel each other out. Although the electric field is not zero at any given point on the surface, the net flux is zero because the inward and outward fields counterbalance each other.

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Most popular questions from this chapter

A point charge is located a fixed distance outside of a uniformly charged sphere. If the sphere shrinks in size without losing any charge, what happens to the force on the point charge?

The charge density within a charged sphere of radius \(R\) is given by \(\rho=\rho_{0}-a r^{2},\) where \(\rho_{0}\) and \(a\) are constants and \(r\) is the distance from the center. Find an expression for \(a\) such that the electric field outside the sphere is zero.

A point charge \(q\) is at the center of a spherical shell of radius \(R\) carrying charge \(2 q\) spread uniformly over its surface. Write expressions for the electric field strength at (a) \(\frac{1}{2} R\) and \((b) 2 R.\)

A 15 -nC point charge is at the center of a thin spherical shell of radius \(10 \mathrm{cm},\) carrying -22 nC of charge distributed uniformly over its surface. Find the magnitude and direction of the electric field (a) \(2.2 \mathrm{cm},\) (b) \(5.6 \mathrm{cm},\) and (c) \(14 \mathrm{cm}\) from the point charge.

Why can't you use Gauss's law to determine the field of a uniformly charged cube? Why couldn't you use a cubical Gaussian surface?
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