A solid conducting sphere with radius R that carries...

A solid conducting sphere with radius $R$ that carries positive charge $Q$ is concentric with a very thin insulating shell of radius $2 R$ that also carries charge $Q$. The charge $Q$ is distributed uniformly over the insulating shell. (a) Find the electric field (magnitude and direction) in each of the regions $0<r<R$, $R<r<2 R$, and $r>2 R$. (b) Graph the electric-field magnitude as a function of $r$.

Key Concepts

Gauss's Law

Gauss's law relates the electric flux through a closed surface to the total charge enclosed by that surface. It is a powerful tool for calculating electric fields when high symmetry is present, such as in spherical, cylindrical, or planar systems. By choosing a Gaussian surface that exploits the symmetry of the charge distribution, the computation of the electric field simplifies considerably.

Electric Field in Conductors

In a conductor at electrostatic equilibrium, the electric charge resides on the surface, and the electric field inside the conductor is zero. This is because any internal field would cause the mobile charges to rearrange until equilibrium is restored. Understanding this property is key when analyzing situations involving conducting materials.

Spherical Shell Charge Distribution

A uniformly charged thin spherical shell exhibits a unique behavior: the electric field outside the shell is as if the entire charge were concentrated at its center, yielding an inverse square law dependence, while the electric field inside the shell is zero. This property is a direct consequence of the spherical symmetry and can be derived using Gauss's law.

Superposition Principle

The superposition principle in electrostatics states that when multiple charge distributions are present, the total electric field at any point is the vector sum of the fields due to each individual charge distribution. This principle is essential for solving problems where different sources contribute to the overall electric field.

Graphing Electric Field Distributions

Graphing the electric field as a function of distance involves recognizing the different regimes dictated by the symmetry and enclosed charge. The graph typically shows regions where the field is zero (such as inside a conductor or within a spherical shell), regions where the field follows a 1/r² dependence outside of spherically symmetric charge distributions, and the transitions between these regions.

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