Horizontal electric power lines supported by vertical poles...

Horizontal electric power lines supported by vertical poles can carry large currents. Assume that Earth's magnetic field runs parallel to the surface of the ground from south to north with a magnitude of $0.50 \times 10-4 \mathrm{~T}$ $0.50 \times 10^{-4} \mathrm{~T}$ and that the supporting poles are $32 \mathrm{~m}$ apart. Find the magnitude and direction of the force that Earth's magnetic field exerts on a 32-m segment of power line (a wire) carrying $95 \mathrm{~A}$ if the current runs (a) from north to south, (b) from east to west, or (c) toward the northeast making an angle of $30.0^{\circ}$ north of east. (d) Are any of the above forces large enough to have an appreciable effect on the power lines? Example $19-3$

Key Concepts

Magnetic Force on a Current-Carrying Conductor

This concept explains that when an electric current flows through a conductor placed in a magnetic field, it experiences a force given by the equation F = I (L x B), where I is the current, L is the length vector of the conductor, and B is the magnetic field. The force's magnitude and direction are determined using the cross product of these vectors, which is foundational in understanding electromagnetic interactions in a variety of physical systems.

Cross Product and the Right-Hand Rule

The cross product is a vector operation used to calculate the magnetic force on a current-carrying wire. Its result is a vector that is perpendicular to both original vectors. The right-hand rule is a mnemonic that helps in determining the direction of the force: by pointing the fingers in the direction of the current and curling them toward the magnetic field, the thumb points in the direction of the force.

Vector Representation of Magnetic Fields and Currents

Both the current in a wire and the magnetic field are vector quantities, meaning they have both magnitude and direction. Properly representing these quantities in a coordinate system allows one to use vector operations, such as the cross product, to accurately determine the resultant forces in problems involving electromagnetic interactions.

Earth's Magnetic Field

Earth's magnetic field, though relatively weak compared to many man-made fields, is a natural magnetic field that affects current-carrying conductors on or near the Earth's surface. Understanding its magnitude and orientation is crucial when evaluating the force it exerts, especially in scenarios involving long conductors such as power lines.

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