Figure 23-52 gives the magni- tude of the electric ficld...

Figure $23-52$ gives the magni-
tude of the electric ficld inside and
outside a sphere with a positive charge
distributed uniformly throughout its
volume. The scale of the vertical axis is
set by $E_{g}=5.0 \times 10^{7} \mathrm{N} / \mathrm{C}$ . What is the
charge on 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 used to derive the electric field for symmetric charge distributions, making it especially useful for calculating the field inside and outside uniformly charged objects.

Spherical Symmetry

Spherical symmetry implies that the properties of the system are the same in every direction from the center of the sphere. This symmetry simplifies the analysis because the electric field at any point depends only on the radial distance from the center, allowing the use of Gauss’s Law with spherical Gaussian surfaces.

Uniform Charge Distribution

A uniform charge distribution means that the charge is spread evenly throughout the volume of the sphere. This uniformity leads to different expressions for the electric field inside and outside the sphere—inside, the field varies linearly with distance from the center, while outside, it behaves as if all the charge were concentrated at the center.

Electric Field of a Charged Sphere

The electric field of a charged sphere is derived by applying Gauss’s Law to the sphere. For points outside the sphere, the field is given by an inverse-square law, similar to that of a point charge. Inside the sphere, the electric field increases linearly with distance from the center because only the charge enclosed by the Gaussian surface contributes to the field at that point.

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