Ch 25: The Electric Potential

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 25: The Electric Potential

Problem 43

Chapter 25, Problem 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?

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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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Key Concepts

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

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.

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.

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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Textbook Question

The electron gun in an old TV picture tube accelerates electrons between two parallel plates 1.2 cm apart with a 25 kV potential difference between them. The electrons enter through a small hole in the negative plate, accelerate, then exit through a small hole in the positive plate. Assume that the holes are small enough not to affect the electric field or potential. With what speed does an electron exit the electron gun if its entry speed is close to zero?

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Textbook Question

An arrangement of source charges produces the electric potential V=5000x² along the x-axis, where V is in volts and x is in meters. What is the maximum speed of a 1.0 g, 10 nC charged particle that moves in this potential with turning points at ±8.0 cm?

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Textbook Question

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Textbook Question

Two small metal cubes with masses 2.0 g and 4.0 g are tied together by a 5.0-cm-long massless string and are at rest on a frictionless surface. Each is charged to +2.0 μC. What is the tension in the string?

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Textbook Question

The four 1.0 g spheres shown in FIGURE P25.42 are released simultaneously and allowed to move away from each other. What is the speed of each sphere when they are very far apart?

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