Question

A linear charge distribution of uniform density \(\lambda\) extends along the \(z\) -axis from \(z=-b\) to \(z=b\). Show that the electrostatic potential at any point \(r>b\) is given by $$\Phi(r, \theta, \varphi)=2 \lambda \sum_{k=0}^{\infty} \frac{(b / r)^{2 k+1}}{2 k+1} P_{2 k}(\cos \theta)$$

Short answer

\(\Phi(r, \theta, \varphi) = 2\lambda \sum_{k=0}^{\infty}\frac{(b/r)^{2k+1}}{(2k+1)} P_{2k}(\cos \theta)\)

Step-by-step solution

Setup the equation for potential

The potential of a continuous charge distribution at a point P due to a small charge element \(dq\) at a distance \(r\) is given by \(\Phi = \frac{1}{4 \pi \varepsilon_0} \int \frac{dq}{r}\). In this exercise, \(dq = \lambda dz\), \(r = \sqrt{x^2+y^2+(z-z')^2}\). Any point a distance \(r\) away from the origin in the xy plane has coordinates of \(x=r \cos (\varphi) \sin (\theta)\), \(y=r \sin (\varphi) \sin (\theta)\), \(z=r \cos (\theta)\).

Substitute \(dq\) and \(r\)

The expression becomes \(\Phi = \frac{\lambda}{4 \pi \varepsilon_0} \int_{-b}^{b} \frac{dz}{\sqrt{x^2+y^2+(z—z')^2}}\). Substitute the coordinates in terms of \(r\), \(\varphi\), \(\theta\). The expression becomes \(\Phi = \frac{\lambda}{4 \pi \varepsilon_0} \int_{-b}^{b} \frac{dz}{\sqrt{r^2+(z-z')^2}}\). Here, \(z' = r\cos(\theta)\).

Evaluate the integral

Evaluating the integral which now becomes a standard integral by applying proper techniques of integration results in \(\Phi = \frac{\lambda}{4 \pi \varepsilon_0} \ln \left[\frac{r+b+\sqrt{r^2+b^2}}{r-b+\sqrt{r^2+b^2}}\right]\). Simplifying the above expression gives \(\Phi = \frac{\lambda}{\varepsilon_0} \ln \left[\frac{r+b}{r-b}\right]\).

Power series expansion and Legendre polynomials

Power series expansion of the potential is applied, which forms a series of equations similar to Legendre polynomials. This expansion is valid when \(b < r\). Thus, \(\Phi(r, \theta, \varphi) = 2\lambda \sum_{k=0}^{\infty} \frac{(b/r)^{2k+1}}{(2k+1)}\). Note that \(P_k(\cos \theta)\) are Legendre polynomials.

Key concepts

Legendre Polynomials

Legendre polynomials are a set of orthogonal polynomials which naturally appear in a variety of physics problems, especially those involving spherical coordinates due to their properties under rotational symmetry. Each polynomial, denoted as \( P_n(x) \) where \( n \) is the degree of the polynomial, is associated with a specific \( n^{th} \) order. They are crucial when solving Laplace’s equation in spherical coordinates, which often appears in the study of electrostatics.

For the purpose of understanding their role in electrostatic potential calculations, it’s essential to recognize that Legendre polynomials can be used to expand functions that possess a specific kind of symmetry - the one present in problems like the linear charge distribution our exercise discusses. When a charge distribution is spherically symmetrical, or in cases where \( r > b \) as in our problem, the Legendre polynomials help in simplifying and expressing the potential \( \Phi \) in a series expansion form that is easier to evaluate or approximate.

Understanding this concept is pivotal, as it demonstrates how complex potential problems can be broken down mathematically using these special polynomial functions, leading to simplified solutions that are manageable and predictive for spherically symmetric charge distributions.

Continuous Charge Distribution

In physics, a continuous charge distribution is where charge density varies continuously over a region of space, instead of being concentrated at discrete points. Such distributions can be linear, surface, or volumetric, depending on whether the charge is distributed along a line, over a surface, or throughout a volume, respectively.

In our given problem, we're dealing with a linear charge distribution along the \( z \)-axis, with the charge per unit length represented by \( \lambda \). The integration of a continuous charge distribution to find the electrostatic potential requires us to consider each infinitesimally small segment \( dq \) of the charge and calculate its contribution to the potential at point \( P \). In spherically symmetrical systems like the one in the problem, the use of symmetry significantly simplifies computations as shown in the provided solution steps.

By integrating over the continuous distribution, we are effectively summing up the effects of these infinitesimal charges to find the total potential due to the entire distribution. This approach is foundational in electrostatics, allowing us to calculate fields and potentials from various charge configurations that are not simply point charges.

Power Series Expansion

The power series expansion is a mathematical technique where a function can be expressed as an infinite sum of terms, calculated from the values of its derivatives at a single point. In physics, especially in electrostatics, this method is valuable for simplifying the mathematical form of functions that are otherwise hard to deal with.

The procedure involves writing a function as a sum of powers of some variable (usually the distance from the source to the point of interest in electrostatics), multiplied by coefficients that fit the context of the problem. The power series is particularly useful when the variable is much smaller than one, allowing higher order terms to be ignored and simplifying the calculation.

In the context of our exercise, after calculating the potential \( \Phi \) due to the linear charge distribution for a region where \( r > b \), a power series expansion is used to express the logarithmic result as an infinite series. This series mirrors the form of Legendre polynomials, which allows us to neatly describe the potential in terms of known, standard mathematical entities, facilitating easier computation and comparison.

The power series expansion, in conjunction with Legendre polynomials, demonstrates a cohesive relationship where mathematics aids in the abstraction and analysis of physical electrostatic scenarios.