A solid nonconducting sphere has a positive charge q spread...

A solid nonconducting sphere has a positive charge $q$ spread uniformly throughout its volume. The charge density or charge per unit volume, therefore, is $\frac{q}{\frac{4}{3} \pi R^{3}}$ . Use Gauss' law to show that the electric field at a point within the sphere at a radius $r$ has a magnitude of $\frac{q r}{4 \pi \epsilon_{0} R^{3}}$

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Symmetry Considerations

Spherical symmetry in the charged sphere implies that the electric field is radial and its magnitude depends only on the distance from the center. This symmetry simplifies the application of Gauss's Law since both the electric field and the area element of the Gaussian surface are aligned, allowing for easy computation of the electric flux.

Gaussian Surface

A Gaussian surface is an imagined closed surface used to apply Gauss's Law. For a uniformly charged sphere, a spherical Gaussian surface concentric with the charged sphere is typically chosen because it simplifies the integration, as the magnitude of the electric field remains constant over the surface.

Uniform Charge Distribution

A uniform charge distribution means that the charge density is constant throughout the volume of the sphere. This provides a straightforward way to calculate the charge enclosed by any spherical Gaussian surface, as it is directly proportional to the volume enclosed.

Gauss's Law

Gauss's Law states that the net electric flux through any closed surface is equal to the charge enclosed divided by the permittivity of free space (??). This principle allows one to relate the distribution of charge to the resulting electric field, especially in systems with high symmetry.

Transcript

Please give Ace some feedback