(a) A long metal pipe of square cross-section (side a ) is...

(a) A long metal pipe of square cross-section (side $a$ ) is grounded on three sides, while the fourth (which is insulated from the rest) is maintained at constant potential $V_{0} .$ Find the net charge per unit length on the side opposite to $V_{0}$. [Hint: Use your answer to Prob. 3.14 or Prob. $3.47 .]$ (b) A long metal pipe of circular cross-section (radius $R$ ) is divided (lengthwise) into four equal sections, three of them grounded and the fourth maintained at constant potential $V_{0}$ Find the net charge per unit length on the section opposite to $V_{0} .$ [Answer to both ( $a$ ) and ( $b$ ): $\left.\lambda=-\epsilon_{0} V_{0} \ln 2\right]^{13}.$

(a) A long metal pipe of square cross-section (side $a$ ) is grounded on three sides, while the fourth (which is insulated from the rest) is maintained at constant potential $V_{0} .$ Find the net charge per unit length on the side opposite to $V_{0}$. [Hint: Use your answer to Prob. 3.14 or Prob. $3.47 .]$
(b) A long metal pipe of circular cross-section (radius $R$ ) is divided (lengthwise) into four equal sections, three of them grounded and the fourth maintained at constant potential $V_{0}$ Find the net charge per unit length on the section opposite to $V_{0} .$ [Answer to both ( $a$ ) and ( $b$ ):
$\left.\lambda=-\epsilon_{0} V_{0} \ln 2\right]^{13}.$

Key Concepts

Electrostatic Boundary?Value Problems

These problems involve finding the electric potential in a region given specified conditions on the boundaries. In electrostatics, the potential must satisfy Laplace's equation in regions free of charge, and the boundary conditions (such as a conductor maintained at a fixed potential or at ground) play a crucial role. The uniqueness theorem guarantees that once the boundary conditions are set, the solution for the potential is unique.

Conductor Boundary Conditions

Conductors in electrostatic equilibrium have constant potential over their surfaces, and the electric field just outside a conductor is perpendicular to the surface. When parts of a conductor are grounded, their potential is fixed (typically at zero), which influences the distribution of surface charge. The insulated portions, when held at a different constant potential, result in induced charges that are determined by the surrounding boundary conditions.

Laplace’s Equation and the Uniqueness Theorem

In regions where there are no free charges, the electric potential must satisfy Laplace’s equation. This partial differential equation, combined with specified boundary conditions, yields a unique solution. This framework is fundamental in analyzing electrostatic systems, determining potential distributions, and subsequently calculating the induced surface charges.

Method of Images and Separation of Variables

These mathematical techniques are often used to solve Laplace’s equation in regions with complex geometries or mixed boundary conditions. The method of images replaces certain boundary conditions with equivalent charge distributions, while separation of variables breaks down Laplace’s equation into simpler ordinary differential equations, making it possible to solve problems with symmetry, even on non-standard geometries.

Charge Distribution and Capacitance

Once the potential distribution is determined, the surface charge density can be found by evaluating the discontinuity in the electric field across a conductor’s surface using Gauss’s law. The net charge per unit length or on a particular segment of a conductor is related to the potential difference between conductors and can be understood in the context of capacitance for the geometry in question.

Logarithmic Potential in Two-Dimensional Electrostatics

In two-dimensional electrostatic problems, such as those with translational symmetry (infinitely long conductors), the potential due to line charges typically varies logarithmically with distance. This logarithmic behavior is essential when deriving the relationships for the net charge per unit length in configurations where the potential is maintained constant on parts of the boundary.

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