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Chapter 11: Problem 7

A right circular cylinder has an electrostatic potential of \(\psi(\rho, \varphi)\) on both ends. The potential on the curved cylindrical surface is zero. Find the potential at all interior points. Hint. Choose your coordinate system and adjust your \(z\) dependence to exploit the symmetry of your potential.

Short Answer

Expert verified
The potential inside the cylinder satisfies Laplace's equation using Bessel and exponential functions, with a general solution built from boundary conditions and Fourier series.

Step by step solution

01

Understanding the Problem

We need to find the electrostatic potential \( \psi(\rho, \varphi, z) \) within a right circular cylinder. It's important to note that the potential on the curved surface is zero, and the potential \( \psi(\rho, \varphi) \) is given on the two flat ends. The problem hints at exploiting symmetry in the \( z \) direction to simplify the solution.
02

Establishing the Coordinate System

Use cylindrical coordinates \( (\rho, \varphi, z) \) because of the symmetry of the cylinder. The problem has azimuthal symmetry, meaning the potential does not depend on \( \varphi \), and since the side surface has zero potential, the problem simplifies.
03

Laplace's Equation

Inside the cylinder, the potential satisfies Laplace's equation \( abla^2 \psi = 0 \). In cylindrical coordinates, this becomes:\[\frac{1}{\rho} \frac{\partial}{\partial \rho} \left( \rho \frac{\partial \psi}{\partial \rho} \right) + \frac{1}{\rho^2} \frac{\partial^2 \psi}{\partial \varphi^2} + \frac{\partial^2 \psi}{\partial z^2} = 0.\]Exploiting the azimuthal symmetry, the \( \varphi \) dependence disappears.
04

Separation of Variables

Use the method of separation of variables by assuming a solution of the form \( \psi(\rho, z) = R(\rho) Z(z) \). Substitute into the Laplace equation to achieve:\[\frac{1}{R} \frac{d}{d\rho} \left( \rho \frac{dR}{d\rho} \right) = -\frac{1}{Z} \frac{d^2 Z}{dz^2} = -k^2,\]where \( k^2 \) is a separation constant.
05

Solving Radial Part

The radial part becomes:\[\frac{d}{d\rho} \left( \rho \frac{dR}{d\rho} \right) + k^2 \rho R = 0.\]This is a Bessel's equation, whose solution is given by \( R(\rho) = A J_0(k\rho) + B Y_0(k\rho) \). Due to boundary conditions \( R(\rho) \) must be finite at \( \rho = 0 \), thus \( B = 0 \).
06

Solving Axial Part

The axial part becomes:\[\frac{d^2 Z}{dz^2} = k^2 Z,\]which is solved by \( Z(z) = C e^{kz} + D e^{-kz} \). Utilize boundary conditions \( Z(z) = \psi(\text{given}) \) at the ends, leading to a discrete set of \( k \) values derived from boundary conditions, typically leading to a Fourier series.
07

Constructing the General Solution

Combine solutions:\[\psi(\rho, z) = \sum_{n=1}^{\infty} A_n J_0(k_n \rho) (C_n e^{k_n z} + D_n e^{-k_n z}).\]Use the boundary conditions \( \psi(\rho, 0) \) and \( \psi(\rho, L) \) to determine coefficients \( A_n, C_n, \) and \( D_n \) through Fourier analysis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Electrostatic Potential
Electrostatic potential is a fundamental concept in electromagnetism, representing the potential energy per unit charge at a point in a static electric field. In simpler terms, it is the measure of the work done in bringing a unit positive charge from infinity to that point without any acceleration. For a physical system like a cylinder, this potential can be defined over surfaces and volumes to understand how electric fields interact with matter.
In the context of this exercise, the cylinder has a given electrostatic potential on its ends, while its curved surface is held at zero potential, known as a boundary condition. These conditions help determine how the potential behaves inside the cylinder.
  • Boundary conditions: Essential to solving problems with Laplace's equation. Indicate known values on the system's boundary.
  • Symmetry: Exploiting the symmetry of the system, such as the homogeneity in the azimuthal direction, simplifies the mathematical solution.
Ultimately, solving for the electrostatic potential allows us to understand how electric fields are distributed within the cylinder.
Cylindrical Coordinates
Cylindrical coordinates are particularly useful for problems with cylindrical symmetry, like this one. These coordinates are represented by three variables: radial distance \( \rho \), azimuthal angle \( \varphi \), and height \( z \). Each point in space is described in terms of its distance from a central axis, its rotation around that axis, and its height along the axis.
Cylindrical coordinates are beneficial here because they simplify the equations involved due to the symmetry of the cylinder.
  • \( \rho \): Measures the radial distance from the axis of the cylinder.
  • \( \varphi \): Represents the angle around the central axis.
  • \( z \): Denotes the height along the cylinder's axis.
This coordinate system helps in reducing the complexity of Laplace's equation, making it manageable by eliminating unnecessary variables.
Laplace's Equation
Laplace's equation is a second-order partial differential equation frequently appearing in physics and engineering. It is written as \( abla^2 \psi = 0 \). The equation arises in electrostatics and fluid dynamics scenarios where the potential or field is static over time.
For cylindrical coordinates, Laplace's equation adapts to the symmetry of the problem, allowing for the separation of variables method, which simplifies solving procedures.
  • Basic form: \( abla^2 \psi = 0 \), indicating no charge present in the region.
  • Cylindrical adaptation: In cylindrical systems, it expands to include radial and axial terms, which can be separated due to symmetry.
This form is crucial, because solving it will provide the potential distribution inside the cylinder under given boundary conditions.
Bessel's Equation
Bessel's equation is a type of differential equation that often appears in problems with cylindrical symmetry, such as this cylinder problem. It takes the form \( \rho^2 \frac{d^2 R}{d\rho^2} + \rho \frac{dR}{d\rho} + (\rho^2 - n^2)R = 0 \). This equation arises when solving the radial part of the separated Laplace's equation.
The solutions to Bessel's equation are known as Bessel functions, which come in several forms, with \( J_n \) being the most common. For the problem at hand, due to zero boundary conditions, the suitable radial solution involves the first kind of Bessel function \( J_0 \) for the zero-order case.
  • Finite at origin: Bessel functions ensure the solution remains bounded at the cylinder's center.
  • Application: Used to describe oscillations and waveforms in cylindrical systems.
These functions are integral in combining the radial solution with the other components to form the complete potential expression.

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Most popular questions from this chapter

A set of functions \(C_{n}(x)\) satisfies the recurrence relations $$ \begin{array}{l} C_{n-1}(x)-C_{n+1}(x)=\frac{2 n}{x} C_{n}(x) \\ C_{n-1}(x)+C_{n+1}(x)=2 C_{n}^{\prime}(x) \end{array} $$ (a) What linear second-order ODE does the \(C_{n}(x)\) satisfy? (b) By a change of variable transform your ODE into Bessel's equation. This suggests that \(C_{n}(x)\) may be expressed in terms of Bessel functions of transformed argument.

The amplitude \(U(\rho, \varphi, t)\) of a vibrating circular membrane of radius \(a\) satisfies the wave equation $$ \nabla^{2} U-\frac{1}{v^{2}} \frac{\partial^{2} U}{\partial t^{2}}=0 $$ Here, \(v\) is the phase velocity of the wave fixed by the elastic constants and whatever damping is imposed. (a) Show that a solution is $$ U(\rho, \varphi, t)=J_{m}(k \rho)\left(a_{1} e^{i m \varphi}+a_{2} e^{-i m \varphi}\right)\left(b_{1} e^{i \omega t}+b_{2} e^{-i \omega t}\right) . $$ (b) From the Dirichlet boundary condition \(J_{m}(k a)=0\) find the allowable values of the wavelength \(\lambda(k=2 \pi / \lambda)\). Note. There are other Bessel functions besides \(J_{m}\) but they all diverge at \(\rho=0 .\) This is shown explicitly in Section 11.3. The divergent behavior is actually implicit in Eq. (11.6).

Defining the spherical modified Bessel functions (Fig. \(11.16)\) by $$ i_{n}(x)=\sqrt{\frac{\pi}{2 x}} I_{n+1 / 2}(x), \quad k_{n}(x)=\sqrt{\frac{2}{\pi x}} K_{n+1 / 2}(x) $$ show that $$ i_{0}(x)=\frac{\sinh x}{x}, \quad k_{0}(x)=\frac{e^{-x}}{x} \text { . } $$ Note that the numerical factors in the definitions of \(i_{n}\) and \(k_{n}\) are not identical.

Using only the generating function $$ e^{(x / 2)(t-1 / t)}=\sum_{n=-\infty}^{\infty} J_{n}(x) t^{n} $$ and not the explicit series form of \(J_{n}(x)\), show that \(J_{n}(x)\) has odd or even parity according to whether \(n\) is odd or even, that is, \({ }^{12}\) $$ J_{n}(x)=(-1)^{n} J_{n}(-x) . $$

The circular aperture diffraction amplitude \(\Phi\) of Eq. \((17.35)\) is proportional to \(f(z)=J_{1}(z) / z .\) The corresponding single slit diffraction amplitude is proportional to \(g(z)=\sin z / z\) (a) Calculate and plot \(f(z)\) and \(g(z)\) for \(z=0.0(0.2) 12.0 .\) (b) Locate the two lowest values of \(z(z>0)\) for which \(f(z)\) takes on an extreme value. Calculate the corresponding values of \(f(z)\). (c) Locate the two lowest values of \(z(z>0)\) for which \(g(z)\) takes on an extreme value. Calculate the corresponding values of \(g(z)\).
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