A solid insulating sphere has total charge Q and radius R....

A solid insulating sphere has total charge $Q$ and radius $R$. The sphere's charge is distributed uniformly throughout its volume. Let $r$ be the radial distance measured from the center of the sphere. If $E=800 \mathrm{~N} / \mathrm{C}$ at $r=R / 2,$ what is $E$ at $r=2 R ?$

Key Concepts

Gauss's Law

Gauss's Law is a fundamental principle in electromagnetism that relates the electric flux emanating from a closed surface to the total charge enclosed within that surface. It is especially useful in calculating electric fields when there is a high degree of symmetry, such as spherical or cylindrical symmetry. In problems involving spherical charge distributions, Gauss's Law can simplify the calculation of the electric field both inside (where the field varies with the radial distance) and outside the sphere (where the field behaves as if all the charge were concentrated at the center).

Uniform Charge Distribution

A uniform charge distribution implies that the electric charge is spread evenly throughout a volume, resulting in constant charge density. This uniformity allows for straightforward application of symmetry arguments and simplifies the integration process when applying Gauss's Law. The concept is critical for determining how the enclosed charge varies as a function of radius, especially when calculating the electric field at different distances from the center of a sphere.

Electric Field Outside a Spherically Symmetric Charge Distribution

For a spherically symmetric charge distribution, the electric field calculated at a point outside the distribution acts as if all the charge were concentrated at the center of the sphere. This permits the use of the simple inverse-square law, where the electric field decreases with the square of the distance from the center. This principle is essential in analyzing how the electric field changes when moving from points within the charged sphere to points in the surrounding space.

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