(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 $26.0 \mathrm{~cm}$ in diameter to produce an electric field of magnitude $1500 \mathrm{~N} / \mathrm{C}$ just outside the surface of the sphere? (b) What is the electric field at a point $14.5 \mathrm{~cm}$ outside the surface of the sphere?

Key Concepts

Electric Field of a Point Charge

The electric field generated by a point charge diminishes with the square of the distance from the charge, as described by Coulomb's law. When dealing with a uniformly charged sphere, spherical symmetry lets us treat the sphere as a point charge located at its center for regions outside the sphere. This concept is essential for determining the field at points beyond the surface of the sphere.

Charge Quantization

Charge quantization refers to the fact that electric charge exists in discrete amounts, typically in multiples of the elementary charge (approximately 1.602×10?¹? coulombs for an electron). This concept is important when converting between a total charge calculated using electric field relations and the number of excess electrons required to produce that total charge.

Gauss's Law

Gauss's Law is a fundamental principle in electromagnetism that relates the electric flux through a closed surface to the total charge enclosed by that surface. In problems involving symmetrical charge distributions, such as uniformly charged spheres, Gauss's Law greatly simplifies the calculation of the electric field by allowing us to choose a symmetrical Gaussian surface over which the electric field is constant in magnitude and direction.

Spherical Symmetry

Spherical symmetry is a key concept when dealing with charge distributions that are uniform over the volume or surface of a sphere. This property ensures that the electric field depends only on the radial distance from the center of the sphere. It allows the problem to be treated as if all the charge were concentrated at the center when calculating the electric field at any point outside the sphere.

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