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Ch 25: The Electric Potential
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoKnight Calc 5th EditionCh 25: The Electric PotentialProblema 43
Capitolo 25, Problema 43

A proton's speed as it passes point 1 is 50,000 m/s. It follows the trajectory shown in FIGURE P25.43. What is the proton's speed at point 2?
Diagram showing a proton's trajectory between two points, with electric potential values labeled at each point.

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1
Identify the principle of energy conservation: The total mechanical energy of the proton (kinetic energy + electric potential energy) remains constant if no non-conservative forces (like friction) are acting on it.
Write the expression for the total energy at point 1: \( E_1 = \frac{1}{2} m v_1^2 + q V_1 \), where \( m \) is the mass of the proton, \( v_1 \) is its speed at point 1, \( q \) is the charge of the proton, and \( V_1 \) is the electric potential at point 1.
Write the expression for the total energy at point 2: \( E_2 = \frac{1}{2} m v_2^2 + q V_2 \), where \( v_2 \) is the speed of the proton at point 2 and \( V_2 \) is the electric potential at point 2.
Set \( E_1 = E_2 \) because energy is conserved. This gives \( \frac{1}{2} m v_1^2 + q V_1 = \frac{1}{2} m v_2^2 + q V_2 \).
Rearrange the equation to solve for \( v_2 \): \( v_2 = \sqrt{v_1^2 + \frac{2q}{m} (V_1 - V_2)} \). Substitute the known values for \( v_1 \), \( q \), \( m \), \( V_1 \), and \( V_2 \) to calculate the final speed.

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Conservation of Energy

The principle of conservation of energy states that the total energy in a closed system remains constant over time. In the context of a proton moving through a field, its kinetic energy and potential energy can interchange, but their sum will remain the same. This concept is crucial for determining the speed of the proton at different points along its trajectory.
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Conservation Of Mechanical Energy

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion, calculated using the formula KE = 1/2 mv², where m is mass and v is velocity. For the proton, its initial speed at point 1 contributes to its kinetic energy, which can change as it moves through different potential energy regions. Understanding how kinetic energy varies helps in calculating the speed at point 2.
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Intro to Rotational Kinetic Energy

Electric Potential Energy

Electric potential energy is the energy a charged particle has due to its position in an electric field. As the proton moves through the field, its potential energy changes, affecting its speed. The relationship between potential energy and kinetic energy is essential for solving the problem, as it allows us to determine how the proton's speed changes from point 1 to point 2.
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Electric Potential Energy
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