A long, straight coaxial cable consists of two thin, tubular conCH ductors, the inner of radius

step 1 p at the distance r from axis is we have p is equals to u i divided by 2 n pi for area of unit l

A long, straight coaxial cable consists of two thin, tubular...

A long, straight coaxial cable consists of two thin, tubular con$\mathrm{CH}$ ductors, the inner of radius $a$ and the outer of radius $b$. Current $I$ flows out along one conductor and back along the other. Show that the self-inductance per unit length of the cable is $\frac{\mu_{0}}{2 \pi} \ln (b / a)$.

Key Concepts

Logarithmic Dependence in Inductance

The logarithmic relationship in the self-inductance per unit length of a coaxial cable arises from the integration of the magnetic field over the radial distance between the inner and outer conductors. This reflects how the field strength decreases with distance and how the geometry influences the total flux linkage.

Self-Inductance

Self-inductance is a property of a conductor whereby a changing current induces an electromotive force (emf) in the same conductor. It is defined as the ratio of the magnetic flux linked with the circuit to the current producing it, and it plays a central role in understanding energy storage in magnetic fields.

Magnetic Flux Linkage

Magnetic flux linkage refers to the total magnetic flux passing through a given area that is linked with a current-carrying circuit. In the context of inductance, it involves integrating the magnetic field over the area enclosed by the circuit, which then establishes a relationship between the flux and the current.

Ampère's Law

Ampère's Law is a fundamental principle in electromagnetism that relates the integrated magnetic field around a closed loop to the electric current passing through the loop. It is essential for calculating the magnetic field in symmetric current configurations, such as those found in coaxial cables.

Coaxial Cable Geometry

The coaxial cable configuration involves two concentric cylindrical conductors, where the inner conductor carries current in one direction and the outer conductor carries current in the opposite direction. This symmetry simplifies the magnetic field calculations and leads to a logarithmic dependence of inductance on the radii of the two conductors.

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