Ch 25: The Electric Potential

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796당신이 사용하는 게 아니라요?교과서 변경

Knight Calc 5th Edition

Ch 25: The Electric Potential

문제 43

25장, 문제 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?

검증된 단계별 안내

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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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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교과서 질문

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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교과서 질문

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교과서 질문

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?

Living cells 'pump' singly ionized sodium ions, Na+, from the inside of the cell to the outside to maintain a membrane potential ΔV_membrane=V_in−V_out=−70 mV. It is called pumping because work must be done to move a positive ion from the negative inside of the cell to the positive outside, and it must go on continuously because sodium ions 'leak' back through the cell wall by diffusion. At rest, the human body uses energy at the rate of approximately 100 W to maintain basic metabolic functions. It has been estimated that 20% of this energy is used to operate the sodium pumps of the body. Estimate—to one significant figure—the number of sodium ions pumped per second.

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