Problem 14 - Classical Electrodynamics

Question

A line charge of length \(2 d\) with a total charge \(Q\) has a linear charge density varying as \(\left(d^{2}-z^{2}\right)\), where \(z\) is the distance from the midpoint. A grounded, conducting. spherical shell of inner radius \(b>d\) is centered at the midpoint of the line charge. (a) Find the potential everywhere inside the spherical shell as an expansion in L.egendre polynomials. (b) Calculate the surface-charge density induced on the shell. (c) Discuss your answers to parts a and \(b\) in the limit that \(d \ll b\).

Step-by-Step Solution

Answer

Potential inside uses Legendre polynomials; surface charge density is calculated via induced charges; limit simplifies to a point-like charge behavior.

Step 1: Understand the Problem

We have a line charge with a non-uniform charge distribution along the length from \(-d\) to \(d\). This is surrounded by a grounded spherical shell, which will be influenced by the charge on the line.

Step 2: Express the Potential Using Line Charge Density

Identify the linear charge density \(\lambda(z) = \lambda_0 (d^2 - z^2)\). Integrate the contribution of elemental charges \(dq = \lambda(z)dz\) to the potential \(dV\) at a point inside the shell by considering contributions from each piece \(z\). This utilizes Green's function or direct integration methods depending on symmetry.

Step 3: Use Legendre Polynomial Expansion

Use coordinates suited for Legendre polynomials. The potential due to an axially symmetric line charge can often be expanded in terms of Legendre polynomials \(P_n(\cos \theta)\). Therefore: \[ V(r, \theta) = \sum_{n=0}^{\infty} A_n r^n P_n(\cos \theta) \] for the inside (\(r < b\)) of a spherical shell.

Step 4: Apply Boundary Conditions

Use boundary conditions to determine the coefficients \(A_n\). Since the shell is grounded, \(V(b, \theta) = 0\). Determine \(A_n\) such that the potential inside the sphere matches the conditions at the surface \(r=b\).

Step 5: Induced Surface Charge Density

Extract the induced surface charge density \(\sigma\) on the sphere using the relation:\(\sigma = -\varepsilon_0 \frac{\partial V}{\partial r}\big|_{r=b}\). Integrating over the surface confirms the presence of the image charge concept over conducting surfaces.

Step 6: Limit Analysis for \(d \ll b\)

For the limit \(d \ll b\), simplify the expansions and expressions to focus on terms that remain relevant at distances much larger than line charge separation. Evaluate how the line charge behaves similarly to a point charge centered around \(r=0\). The spherical shell remains effectively unaffected due to its large size relative to the line charge.

Key Concepts

Legendre polynomials

Legendre polynomials are a set of orthogonal polynomials that emerge in physics and engineering, especially in solving problems involving spherical coordinates. They are denoted by \(P_n(x)\) where \(n\) is a non-negative integer, and \(x\) is a variable typically representing \(\cos\theta\). These polynomials are useful when solving potential problems using separations of variables, like in the scenario of a line charge inside a spherical shell.

In the given exercise, the potential due to the line charge is expressed as a series of Legendre polynomials because the system exhibits axial symmetry around the line charge's axis. By expanding the potential \(V(r, \theta)\) in terms of \(P_n(\cos \theta)\), we can leverage the orthogonality and completeness of these polynomials to find a solution inside the grounded shell.

• The potential expansion is given by: \(V(r, \theta) = \sum_{n=0}^{\infty} A_n r^n P_n(\cos \theta)\).
• The coefficients \(A_n\) are determined using the boundary condition that the potential on the inner surface of the shell (\(r = b\)) is zero, due to it being grounded.

Legendre polynomials simplify expressing potentials in spherical environments, making them pivotal in this type of electrostatics problem.

Surface charge density

Surface charge density \(\sigma\) refers to the charge per unit area on a surface and plays an essential role in how conductors respond to external electric fields. In the exercise, the surface charge density induced on the grounded conducting shell needs to be calculated.

This induced surface charge density arises due to the presence of the line charge inside the spherical shell. Since the shell is grounded, it influences the electric potential such that the potential on its surface is zero. The redistributed charges on the shell's surface are directly related to maintaining this condition.

• The induced surface charge density \(\sigma\) can be found by determining the rate of change of the potential with respect to the radial coordinate at the shell's surface, using \(\sigma = -\varepsilon_0 \frac{\partial V}{\partial r}\big|_{r=b}\).
• The negative sign ensures the surface charge density opposes the internal potential variation to maintain a ground state \(V=0\) on the shell.

Understanding how the surface charge density on the shell responds provides insights into electrostatic shielding and the distribution's dynamics around conductors.

Line charge distribution

In electrostatics, line charge distribution is a model used to simplify the representation of charge spread continuously along a line, casting insight into complex systems like the one in this exercise.

The exercise specifies a line charge distribution with a varying linear charge density, \(\lambda(z) = \lambda_0 (d^2 - z^2)\), reflecting how the charge diminishes as we move away from the line's midpoint. This non-uniform distribution necessitates an integration method to quantify its electric potential contributions at different points in space.

• Each infinitesimal segment of the line contributes to the potential at a given point, described by \(dV\) from each charge element \(dq = \lambda(z)dz\).
• The symmetry of the problem allows us to use Green's functions or direct integration methods to calculate the resultant potential field.
• The variation \(\lambda(z)\) indicates a quadratic decrease in charge density, essential for correctly applying Legendre polynomial expansion in spherical coordinates as detailed in earlier steps.

Describing how this line charge distributes along its length is essential in analyzing potential fields and induced effects within symmetrical systems like a spherical conductor.