In fair weather, over flat ground, there is a downward...

In fair weather, over flat ground, there is a downward electric field of about $150 \mathrm{N} / \mathrm{C}$. Assume that Earth is a conducting sphere with charge on its surface. If the electric field just outside is $150 \mathrm{N} / \mathrm{C}$ pointing radially inward, calculate the total charge on Earth and the charge per unit area.

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Key Concepts

Gauss's Law

Gauss's Law is a fundamental principle in electromagnetism that relates the electric flux through a closed surface to the charge enclosed within that surface. It is particularly useful for computing electric fields when symmetric charge distributions are present, as it simplifies the integration process by allowing one to select an appropriate Gaussian surface.

Electric Field of a Spherical Charge Distribution

For a spherical charge distribution, especially when dealing with a conductor like Earth, the electric field outside the sphere behaves as if all the charge were concentrated at its center. This property, derived from Gauss's Law, simplifies the analysis by allowing the use of the inverse square law to relate the electric field magnitude to the total charge and the distance from the center.

Surface Charge Density

Surface charge density is defined as the amount of electric charge per unit area on a surface. For conductors, where charges reside on the exterior surface, this concept is crucial as it relates the total charge on the conductor to its geometrical surface area. Understanding surface charge density helps in analyzing electric fields at surfaces and in solving problems involving distributed charges.

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