(a) How many excess electrons must be distributed uniformly...

(a) How many excess electrons must be distributed uniformly within the volume of an isolated plastic sphere $30.0 \mathrm{~cm}$ in diameter to produce an electric field of $1150 \mathrm{~N} / \mathrm{C}$ just outside the surface of the sphere? (b) What is the electric field at a point $10.0 \mathrm{~cm}$ outside the surface of the sphere?

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 by that surface. It is especially useful in solving problems with high symmetry, such as spherical distributions, where it simplifies the calculation of the electric field both on the surface and outside the charged object.

Uniform Charge Distribution

A uniform charge distribution means that the total charge is spread evenly throughout the volume of an object. This assumption allows for the calculation of a constant charge density within the object, which, when combined with Gauss’s law, simplifies the analysis of the resultant electric field.

Electric Field of a Spherical Charge Distribution

For a spherically symmetric charge distribution, the electric field outside the sphere behaves as if all the charge were concentrated at its center, following an inverse-square law. This concept is central to determining the electric field at any point outside the sphere, whether at its surface or at a specified distance away.

Excess Electrons and Charge Conversion

Excess electrons contribute a negative net charge, and determining the total charge involves converting the number of electrons to coulombs by multiplying by the elementary charge. This conversion is crucial in bridging the gap between a discrete count of particles and a continuous charge value used in calculating electric fields.

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