An infinitely long line of charge has linear charge density 5.00 ×10^-12 C / m . A proton (mass

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An infinitely long line of charge has linear charge density...

An infinitely long line of charge has linear charge density $5.00 \times 10^{-12} \mathrm{C} / \mathrm{m} .$ A proton (mass $1.67 \times 10^{-27} \mathrm{kg}$ . charge $+1.60 \times 10^{-19} \mathrm{C} )$ is 18.0 $\mathrm{cm}$ from the line and moving directly toward the line at $1.50 \times 10^{3} \mathrm{m} / \mathrm{s}$ (a) Calculate the proton's initial kinetic energy. (b) How close does the proton get to the line of charge? (Hints See Example $23.10 . )$

Key Concepts

Electric Field of an Infinite Line of Charge

The electric field due to an infinite line of charge is characterized by its radial symmetry and is given by the expression E = ?/(2???r), where ? is the linear charge density, ?? is the permittivity of free space, and r is the distance from the line. This field governs the force experienced by other charges in its vicinity, which in turn affects their motion and energy.

Electric Potential Energy in a Radial Electric Field

Electric potential energy represents the work needed to move a charged particle within an electric field. In situations involving an infinite line charge, the potential energy associated with a point charge depends logarithmically on the distance from the line. This concept is essential for determining how a particle’s kinetic energy is converted to potential energy as it approaches or recedes from the line charge.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. It is calculated using the equation KE = ½ m v², where m is the mass of the object and v is its velocity. This concept is fundamental in mechanics and is the starting point for analyzing energy changes in dynamic systems.

Conservation of Energy in Electrostatics

Conservation of energy is a foundational principle stating that the total energy in an isolated system remains constant, though it can transform from one form to another. In electrostatic problems, this means that the sum of a charged particle's kinetic energy and its electric potential energy is conserved as it moves within an electric field.

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