An infinitely long line of charge has linear charge density 5.00×10−125.00\$$times10^{-12}5.00×10$$−12 C/m. A proton (mass 1.67×10−271.67\$$times10^{-27}1.67×10$$−27 kg, charge +1.60×10−19+1.60\$$times10^{-19}+1.60×10$$−19 C) is 18.018.018.0 cm from the line and moving directly toward the line at 3.50×1033.50\$$times10^33.50×103 m/s. $$How close does the proton get to the line of charge?

Ch 23: Electric Potential

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 23: Electric Potential

Problema 30b

Capitolo 23, Problema 30b

An infinitely long line of charge has linear charge density $5.00$\times$10^{-12}$ C/m. A proton (mass $1.67$\times$10^{-27}$ kg, charge $+1.60$\times$10^{-19}$ C) is $18.0$ cm from the line and moving directly toward the line at $3.50$\times$10^3$ m/s. How close does the proton get to the line of charge?

Guida verificata passo dopo passo

First, understand that the electric field (E) due to an infinitely long line of charge with linear charge density (λ) is given by the formula: E = (2 * k * λ) / r, where k is Coulomb's constant (8.99 x 10^9 N m²/C²) and r is the distance from the line of charge.

Next, calculate the initial electric potential energy (U_i) of the proton when it is 18.0 cm (0.18 m) from the line of charge. The potential energy is given by U = q * V, where V is the electric potential due to the line of charge. The potential V at a distance r from the line is V = (2 * k * λ) * ln(r).

Determine the initial kinetic energy (K_i) of the proton using the formula K = (1/2) * m * v², where m is the mass of the proton and v is its initial velocity.

Apply the conservation of energy principle, which states that the total mechanical energy (sum of kinetic and potential energy) remains constant. Set up the equation: K_i + U_i = K_f + U_f, where K_f and U_f are the final kinetic and potential energies when the proton is at its closest distance to the line.

Solve for the closest distance (r_f) by setting the final kinetic energy (K_f) to zero (since the proton momentarily stops at the closest point) and rearranging the conservation of energy equation to find r_f. This involves solving the equation: (1/2) * m * v² + q * (2 * k * λ) * ln(0.18) = q * (2 * k * λ) * ln(r_f).

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Electric Field of a Line Charge

An infinitely long line of charge creates an electric field that decreases with distance from the line. The electric field (E) at a distance (r) from the line is given by E = λ/(2πε₀r), where λ is the linear charge density and ε₀ is the permittivity of free space. This field influences the motion of charged particles nearby.

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Electric Field Lines

Kinetic and Potential Energy in Electric Fields

A charged particle in an electric field experiences changes in kinetic and potential energy. As the proton moves toward the line of charge, its kinetic energy decreases while its electric potential energy increases. The conservation of energy principle allows us to calculate the closest approach by equating initial kinetic energy with the change in potential energy.

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Electric Potential Energy

Conservation of Energy

The conservation of energy states that the total energy of an isolated system remains constant. For the proton, the sum of its kinetic and potential energy at any point in its motion is constant. By applying this principle, we can determine the point where the proton's kinetic energy is zero, indicating its closest approach to the line of charge.

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Conservation Of Mechanical Energy

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