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Chapter 21: Problem 7

In Gauss's law, \(\oint \vec{E} \cdot d \vec{A}=q / \epsilon_{0},\) does the field \(\vec{E}\) necessarily arise only from charges within the closed surface?

Short Answer

Expert verified
No, the electric field \(\vec{E}\) does not necessarily arise only from charges within the closed surface according to Gauss's law. The field can also be due to charges located outside the surface.

Step by step solution

01

Understanding Gauss's Law

According to Gauss's law, the electric field flux through a closed surface is proportional to the charge enclosed by the surface. The law is represented as \(\oint \vec{E} \cdot d \vec{A}=q / \epsilon_{0}\), where \(\vec{E}\) is the electric field vector, \(d \vec{A}\) is the area element, \(q\) is the charge enclosed within the surface and \(\epsilon_{0}\) is the permittivity of free space. It's important to understand that the law talks about the relation between charge enclosed by the surface and resulting electric field flux, and not the origin of the electric field.
02

Deducing Implication from Gauss's Law

From Gauss's law, we can deduce that the electric field \(\vec{E}\) is related to the charge \(q\) enclosed by the Gaussian surface. However, it doesn't specify where the field arises from. It can be due to the charges inside the Gaussian surface or due to other charges outside of it.
03

Giving a final verdict

After understanding and analyzing Gauss's law, we can conclude that it does not stipulate only the charges inside the Gaussian surface contribute to the electric field. The field can also be due to charges located outside the surface. Hence, the electric field \(\vec{E}\) does not necessarily arise only from charges within the closed surface according to Gauss's law.

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Most popular questions from this chapter

What is the electric field strength just outside the surface of a conducting sphere carrying surface charge density \(1.4 \mu \mathrm{C} / \mathrm{m}^{2} ?\)

A thick, spherical shell of inner radius \(a\) and outer radius \(b\) carries a uniform volume charge density \(\rho .\) Find an expression for the electric field strength in the region \(a< r< b,\) and show that your result is consistent with Equation 21.5 when \(a=0.\)

A spherical shell of radius \(15 \mathrm{cm}\) carries \(4.8 \mu \mathrm{C}\) distributed uniformly over its surface. At the center of the shell is a point charge. If the electric field at the sphere's surface is \(750 \mathrm{kN} / \mathrm{C}\) and points outward, what are (a) the point charge and (b) the field just inside the shell?

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?

A study shows that mammalian red blood cells (RBCs) carry electric charge resulting from 4.4 million (for rabbit cells) to 15 million (for human cells) excess electrons spread over their surfaces. Approximating rabbit and human RBCs as spheres with radii \(30 \mu \mathrm{m}\) and \(36 \mu \mathrm{m},\) respectively, find the electric field strengths at the cells' surfaces.
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