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

- Consider two long, parallel, and oppositely charged wires of radius $d$ with their centers separated by a distance $D$ 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 $$\frac{C}{\ell}=\frac{\pi \epsilon_{0}}{\ln [(D-d) / d]}$$

- Consider two long, parallel, and oppositely charged wires of radius $d$ with their centers separated by a distance $D$ 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
$$\frac{C}{\ell}=\frac{\pi \epsilon_{0}}{\ln [(D-d) / d]}$$

Key Concepts

Capacitance per Unit Length

Capacitance is a measure of a system's ability to store charge per unit potential difference. When dealing with infinitely long conductors, the capacitance is defined per unit length. This concept simplifies the analysis by considering charge and potential differences along a uniform segment of an extended object, avoiding edge effects and focusing on the uniform distribution along its length.

Electric Potential in Cylindrical Geometries

In cylindrical systems, such as long wires or rods, the electric potential generated by a linear charge distribution exhibits a logarithmic dependence on distance from the axis. This arises from the integration of the electric field in a cylindrical coordinate system where the radial component of the field varies inversely with the distance, leading to a natural log in the resulting potential expression.

Logarithmic Dependence in Electrostatics

The logarithmic dependence observed in the potential difference between cylindrical conductors is a direct consequence of the nature of the electric field in two-dimensional geometries. When calculating the potential difference between two points in such systems, the integration of the 1/r field relationship results in a logarithm, which is central to deriving expressions for capacitance in these configurations.

Uniform Surface Charge Distribution

Assuming a uniform surface charge distribution simplifies the analysis of electric fields and potentials around conductors. In the context of cylindrical wires, this assumption allows for a straightforward application of Gauss’s law to determine the electric field, and subsequently, the potential, which is essential when calculating capacitance per unit length.

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