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 that at a particular moment it carries charge per length $1.40 \mu \mathrm{C} / \mathrm{m}$ and is at potential $345 \mathrm{kV}$. Find the potential $12.0 \mathrm{m}$ below the wire. Ignore the other conductors of the transmission line and assume the electric field is everywhere purely radial.

Key Concepts

Integration of Electric Field to Determine Potential

The relationship between the electric field and the electric potential is defined by the negative gradient of the potential. For radially symmetric fields, the potential difference between two points is found by integrating the radial component of the electric field along a path between those points. This fundamental concept allows one to compute the potential at a specific location when the electric field is known as a function of the radial distance.

Cylindrical Symmetry in Electrostatics

Cylindrical symmetry arises when the distribution of charge and resulting fields are invariant under rotations about and translations along an axis. This symmetry is key for utilizing Gauss’s law effectively, as it leads to an electric field that depends solely on the radial distance from the axis of symmetry, simplifying both field and potential calculations.

Electric Potential

Electric potential represents the work done per unit charge in bringing a test charge from a reference point (often infinity) to a point in space within an electric field. In the context of continuous charge distributions, such as a long charged wire, the potential often varies as a logarithmic function of the radial distance from the distribution, reflecting the cumulative contributions from all elements of the charge distribution.

Linear Charge Density

Linear charge density is the amount of electric charge per unit length along a conductor. It is crucial in problems involving extended one-dimensional charge distributions because it simplifies the calculation of the electric field and potential by allowing the use of symmetry and integration techniques specific to line charges.

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