A transmission line consists of two parallel wires, of...

A transmission line consists of two parallel wires, of radius $a$ and separation $b,$ carrying uniform line charge densities $\pm \lambda,$ respectively. With $a \ll b$, their electric field is the superposition of the fields from two long straight lines of charge. Find the capacitance per unit length for this transmission line.

Key Concepts

Transmission Line Geometry

Understanding the geometry of a transmission line, which typically involves parallel conductors, is crucial. The spatial configuration determines how the electric fields from the different conductors interact, and influences the boundary conditions used in deriving the capacitance per unit length. Geometrical factors, such as the separation between the wires and their radii, play a significant role in the final capacitance expression.

Electric Potential and Its Relation to the Electric Field

The electric potential difference between two points in space is determined by the integral of the electric field along a path connecting the points. For fields that decay as 1/r, the integration yields a logarithmic function of the radial distance, which is a key mathematical concept when deriving expressions for capacitance in cylindrical geometries.

Electric Field of a Line Charge

The electric field produced by an infinitely long straight line of charge is derived using Gauss’s law, resulting in an expression where the field magnitude varies inversely with the distance from the line charge. This concept is critical for calculating the potential difference around cylindrical conductors and forms the basis for analyzing the fields in the transmission line scenario.

Capacitance per Unit Length

Capacitance per unit length is a measure of how much electric charge a transmission line can store per unit length for a given potential difference between its conductors. It is defined by the relationship C = Q/V, where Q is the charge per unit length and V is the voltage difference, and is essential in characterizing the transmission line’s behavior, especially at high frequencies or in microwave circuits.

Superposition Principle

The superposition principle states that the net electric field generated by multiple sources is the vector sum of the fields due to each individual source. This principle enables the analysis of complex configurations such as a two-wire transmission line by summing the contributions from each charged wire to determine the overall electric field and potential difference.

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