Consider two long, parallel, and oppositely charged wires of...

Consider two long, parallel, and oppositely charged wires of radius $r$ with their centers separated by a distance $D$ that is much larger than $r$ . Assuming the charge is distributed uniformly on the surface of each wire, show that the capacitance per unit length of this pair of wires is $$\frac{C}{\ell}=\frac{\pi \epsilon_{0}}{\ln (D / r)}$$

Consider two long, parallel, and oppositely charged wires of radius $r$ with their centers separated by a distance $D$ that is much larger than $r$ . Assuming the charge is distributed uniformly on the surface of each wire, show that the capacitance per unit length of this pair of wires is
$$\frac{C}{\ell}=\frac{\pi \epsilon_{0}}{\ln (D / r)}$$

Key Concepts

Uniform Surface Charge Distribution

Assuming a uniform surface charge distribution simplifies the electrostatic analysis by exploiting symmetry. In problems involving cylindrical conductors, this assumption allows the use of Gauss's law to derive the electric field and potential, leading to tractable analytical expressions. It is a common and valid approximation in steady-state configurations where external influences on the charge distribution are negligible.

Capacitance per Unit Length

This concept refers to the ability of a pair of conductors to store electric charge per unit length along their extended geometry. It is especially useful in systems such as long wires or transmission lines, where analyzing the entire three-dimensional structure can be simplified by considering the capacity for storing charge along a one-dimensional length. The calculation involves determining the charge per unit length and the potential difference between the conductors.

Electric Potential of Cylindrical Conductors

For cylindrical conductors with a uniform surface charge, the electric potential at a given radial distance can be derived using Gauss's law. Owing to the cylindrical symmetry, the resulting potential function typically has a logarithmic dependence on the radial distance. This characteristic is essential when evaluating the potential difference between conductors in configurations such as parallel wires.

Logarithmic Dependence in Potential and Capacitance

The logarithmic function appears in the analysis of cylindrical charge distributions because, when integrating the electric field in cylindrical coordinates, the resulting expression for the electric potential involves the natural logarithm of the ratio between distances. This logarithmic behavior is central to understanding and calculating parameters like the capacitance per unit length in systems with cylindrical symmetry, such as parallel wires.

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