Using the results of Problem 66, we can find the electric...

Using the results of Problem $66,$ we can find the electric field at any radius for any spherically symmetrical charge distribution. A solid sphere of charge of radius $R$ has a total charge of $q$ uniformly spread throughout the sphere. (a) Find the magnitude of the electric field for $r \geq R .$ (b) Find the magnitude of the electric field for $r \leq R .$ (c) Sketch a graph of $E(r)$ for $0 \leq r \leq 3 R$

Key Concepts

Gauss's Law

Gauss's Law is a fundamental principle in electromagnetism which states that the total electric flux through a closed surface equals the net charge enclosed divided by the permittivity of free space. This concept is especially useful when dealing with problems that exhibit high degrees of symmetry, as it simplifies the calculation of electric fields by relating them directly to the charge distribution.

Uniform Charge Distribution

A uniform charge distribution means that the charge is spread evenly throughout the volume of the object. In the context of a solid sphere, this implies that every region within the sphere has the same charge density, which is a key factor in determining how the enclosed charge varies with the radius of an internal Gaussian surface.

Spherical Symmetry

Spherical symmetry indicates that the physical properties of a system depend only on the distance from the center, not on the direction. This symmetry makes it possible to assume that the electric field at a given radius is the same in all directions, greatly simplifying the computation using a spherical Gaussian surface.

Enclosed Charge Calculation

When applying Gauss's Law, calculating the charge enclosed by a chosen Gaussian surface is crucial. For a sphere with a uniform charge distribution, this involves determining the fraction of the total charge that lies within a radius smaller than the sphere’s full radius, often leading to expressions that vary as a function of radius.

Electric Field Behavior Inside and Outside the Sphere

The electric field exhibits distinct behavior depending on the region of interest. Outside the sphere (r ? R), the field behaves as though all the charge were concentrated at the center, following an inverse-square law. Inside the sphere (r ? R), the field increases linearly with the distance from the center due to the proportionally smaller amount of enclosed charge, highlighting the impact of the spherical charge distribution.

Graphical Representation of Electric Field

Graphing the electric field as a function of radius provides a visual representation of its behavior, showing a linear increase within the sphere and a transition to an inverse-square decay outside the sphere. Such a graph aids in understanding the continuity of the electric field at the boundary and the contrasting dependencies in different regions.

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