Problem 61 - Fundamentals of Physics

Question

A thin-walled metal spherical shell of radius \(a\) has a charge \(q_{a}\). Concentric with it is a thin-walled metal spherical shell of radius \(b>a\) and charge \(q_{b} .\) Find the electric field at points a distance \(r\) from the common center, where (a) \(rb\). (d) Discuss the criterion you would use to determine how the charges are distributed on the inner and outer surfaces of the shells.

Step-by-Step Solution

Answer

(a) E = 0 for r < a. (b) E = \(\frac{q_{a}}{4\pi\varepsilon_{0}r^2}\) for a < r < b. (c) E = \(\frac{q_{a} + q_{b}}{4\pi\varepsilon_{0}r^2}\) for r > b. (d) Charges reside on the outer surfaces of the shells.

Step 1: Electric Field Inside the Inner Shell (r < a)

For the region where \(r < a\), consider a Gaussian surface inside the inner shell. Due to symmetry, the electric field inside a charged spherical shell is zero. This is because any enclosed charge would be spherically symmetric, resulting in no net electric field within the hollow space. Thus, for \(r < a\), the electric field \(E = 0\).

Step 2: Electric Field Between the Shells (a < r < b)

For \(a < r < b\), consider a Gaussian surface between the two shells. The charge enclosed by this surface is the charge on the inner shell, \(q_{a}\). By Gauss's law, \(\oint \mathbf{E} \cdot d\mathbf{A} = \frac{q_{\text{enclosed}}}{\varepsilon_{0}}\), and noting that \(\mathbf{E}\) is radial and constant over the Gaussian surface, the electric field at distance \(r\) is given by \(E = \frac{q_{a}}{4\pi\varepsilon_{0}r^2}\).

Step 3: Electric Field Outside the Outer Shell (r > b)

For \(r > b\), consider a Gaussian sphere outside both shells. The total charge enclosed is the sum of charges on both shells, \(q_{a}+q_{b}\). Applying Gauss's law again, \(E = \frac{q_{a} + q_{b}}{4\pi\varepsilon_{0}r^2}\). The decreased dependency on \(r\) suggests the electric field behaves as if all charge were concentrated at the center.

Step 4: Charge Distribution on the Shells

Both shells are conductors, and in electrostatics, conductors in equilibrium have no electric field inside them. Therefore, all charge on a conductor resides on its surface. The inner shell's charge \(q_a\) resides entirely on its outer surface, while the outer shell's charge \(q_b\) also resides on its outer surface, contributing to the observed external field.

Key Concepts

Gaussian Surface

The concept of a Gaussian surface is pivotal in understanding electric fields, especially when dealing with symmetry. A Gaussian surface is an imaginary closed surface through which we calculate the flux of the electric field. The principle behind a Gaussian surface is utilized in Gauss's law, which states that the total electric flux through a closed surface is equal to the charge enclosed divided by the permittivity of space.
To apply this, imagine a spherical shell inside which another imaginary sphere (Gaussian surface) is constructed. Depending on where this surface is placed—inside a shell or between shells—it helps in simplifying calculations of electric fields due to the symmetry of charge distribution across spherical surfaces.

• For a point inside a spherical shell where no charge is enclosed, the electric field is zero.
• For points outside a shell, the charge enclosed is equivalent to considering it a point charge.

Conductors

Conductors are materials that allow the free movement of charge particles, specifically electrons. In electrostatics, which involves stationary charges, conductors display specific properties.
A conductor in electrostatic equilibrium has the electric field inside it equal to zero. This unique property arises because free electrons within the conductor rearrange themselves in response to any external electric fields so as to cancel it within the material.

• Because the electric field inside a conductor is zero, any excess charge will only reside on the surface.
• This movement of charge ceases when the system reaches equilibrium, necessitating the electric field inside to be zero.

Charge Distribution

Charge distribution refers to the way electric charge is spread over a space or an object. For a conductive spherical shell, the electric charge will always distribute itself on the surface.
Here’s why:

• A charged conductor in equilibrium has no electric field inside—any initial field quickly redistributes charge until the interior field cancels out.
• All excess charges reside on the outer surfaces of a conductor because like charges repel and move as far apart as possible.

These principles are particularly vital when analyzing the behavior of spherical shells of differing radii, such as in problems involving concentric shells.

Spherical Shells

Spherical shells are surfaces of a sphere defined by a specific radius. They are particularly interesting in electromagnetism because of their symmetrical properties.
A spherical shell can either serve as an insulator or a conductor.

• When considered in electrostatics, spherical shells demonstrate unique properties, especially considered as conductors where electric charges solely reside on the outer surface.
• This makes calculating the electric field inside and outside the shell simpler using Gaussian laws.

Understanding spherical shells ensures clearer insights into how charge behaves, especially in scenarios involving layers of conductors, such as multi-layered shells. These shells impact not just the distribution of the charges themselves, but also how the resultant electric fields manifest around them.