One conductor of an overhead electric transmission line is a...

One conductor of an overhead electric transmission line is a long aluminum wire 2.40 $\mathrm{cm}$ in radius. Suppose it carries a charge per length of 1.40$\mu \mathrm{C} / \mathrm{m}$ at a particular moment and its potential is 345 $\mathrm{kV}$ . Find the potential 12.0 $\mathrm{m}$ below the wire. Ignore the other conductors of the transmission line and assume the electric field is radial everywhere.

Key Concepts

Electric Potential and Its Relation to the Electric Field

The electric potential difference between two points is defined as the negative line integral of the electric field between those points. For radially symmetric fields, this integration typically involves the inverse relationship of the field with distance (as in 1/r), leading to logarithmic expressions for the potential difference. This concept is crucial for calculating how potential changes with distance from a charged wire, such as finding the potential at a point below an overhead conductor.

Electric Field of an Infinite Line Charge

For an infinite line of charge with uniform linear charge density, the electric field at a distance r from the wire is given by a simple expression derived using Gauss's Law: E = ?/(2???r), where ? is the linear charge density and ?? is the permittivity of free space. This field is directed radially outward (or inward, depending on the sign of the charge), and its dependence on 1/r is key for understanding the potential differences around the wire.

Gauss's Law

Gauss's Law is a fundamental principle in electromagnetism that relates the electric flux flowing out of a closed surface to the charge enclosed by that surface. For symmetrical charge distributions, such as an infinite line of charge, Gauss’s Law makes it possible to derive analytical expressions for the electric field, simplifying the analysis of problems in electrostatics.

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