An infinitely long line of charge has linear charge density 5.00×10−125.00\$$times10^{-12} C/m. A $$proton (mass 1.67×10−271.67\$$times10^{-27} kg$$, charge +1.60×10−19+1.60\$$times10^{-19} C) is 18.018.0 cm $$from the line and moving directly toward the line at 3.50×1033.50\$$times10^3 m/s. $$Calculate the proton's initial kinetic energy.

First, understand that kinetic energy (KE) is given by the formula: KE=12mv2, where m is the mass of the proton and v is its velocity. Identify the given values: the mass of the proton m=1.67×10-27 kg and its velocity v=3.50×103 m/s. Substitute these values into the kinetic energy formula: KE=12×1.67×10-27 kg × 3.50×1032 m/s. Calculate the square of the velocity: 3.50×1032 m/s. Multiply the mass by the squared velocity and then by 12 to find the initial kinetic energy of the proton.

Ch 23: Electric Potential

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 23: Electric Potential

Problem 30a

Chapter 23, Problem 30a

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 $3.50 times 10 cubed$ m/s. Calculate the proton's initial kinetic energy.

Verified step by step guidance

First, understand that kinetic energy (KE) is given by the formula:

K E = \frac{1}{2} m v^{2}

, where

m

is the mass of the proton and

v

is its velocity.

Identify the given values: the mass of the proton

m = 1.67 \times 10^{- 27}

kg and its velocity

v = 3.50 \times 10^{3}

m/s.

Substitute these values into the kinetic energy formula:

K E = \frac{1}{2} \times 1.67 \times 10^{- 27}

kg × 3.50×1032 m/s.

Calculate the square of the velocity:

{3.50 \times 10^{3}}^{2}

m/s.

Multiply the mass by the squared velocity and then by

\frac{1}{2}

to find the initial kinetic energy of the proton.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Linear Charge Density

Linear charge density is a measure of electric charge per unit length along a line. It is denoted by λ and is expressed in coulombs per meter (C/m). In this problem, the linear charge density helps determine the electric field generated by the infinitely long line of charge, which influences the motion of the proton.

Kinetic Energy

Kinetic energy is the energy possessed by an object due to its motion, calculated using the formula KE = 0.5 * m * v^2, where m is the mass and v is the velocity of the object. For the proton in this problem, its initial kinetic energy is determined by its mass and initial velocity as it moves toward the line of charge.

Electric Field of an Infinite Line of Charge

The electric field generated by an infinitely long line of charge is uniform and radial, decreasing with distance from the line. It is calculated using the formula E = (λ / (2πε₀r)), where λ is the linear charge density, ε₀ is the permittivity of free space, and r is the distance from the line. This field affects the proton's motion and energy as it approaches the line.

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Textbook Question

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Textbook Question

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?

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