Textbook Question
Two point charges 2.0 cm apart have an electric potential energy −180 μJ. The total charge is 30 nC. What are the two charges?
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25. Electric Potential
Electric Potential Energy
4:51 minutesProblem 59b
Textbook Question
An electron starting from rest acquires 4.8 keV of kinetic energy in moving from point A to point B. Determine the ratio of their speeds at the end of their respective trajectories.
Verified step by step guidance1
Convert the given kinetic energy of the electron from kiloelectronvolts (keV) to joules (J). Use the conversion factor: 1 eV = 1.602 × 10⁻¹⁹ J. Therefore, 4.8 keV = 4.8 × 10³ × 1.602 × 10⁻¹⁹ J.
Recall the relationship between kinetic energy and velocity for a particle: \( KE = \frac{1}{2}mv^2 \), where \( KE \) is the kinetic energy, \( m \) is the mass of the electron (\( 9.11 × 10^{-31} \) kg), and \( v \) is the velocity. Rearrange this equation to solve for \( v \): \( v = \sqrt{\frac{2KE}{m}} \).
Substitute the converted kinetic energy (in joules) and the mass of the electron into the velocity equation to calculate the speed of the electron at the end of its trajectory.
To determine the ratio of their speeds, note that the problem does not specify a second particle or trajectory. If this is a continuation of a multi-part problem, clarify the second particle or trajectory to compute the ratio. For now, the ratio would depend on the specific details of the second case.
If the second trajectory involves another particle or energy, repeat the process for that particle (convert energy to joules, use \( v = \sqrt{\frac{2KE}{m}} \)), and then compute the ratio \( \frac{v_1}{v_2} \), where \( v_1 \) and \( v_2 \) are the speeds of the two particles.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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. In this context, the electron acquires 4.8 keV (kilo-electronvolts) of kinetic energy, which can be converted to joules for calculations. Understanding how kinetic energy relates to speed is crucial for determining the ratio of speeds at points A and B.
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Conservation of Energy
The principle of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. In this scenario, the potential energy associated with the electron's position is converted into kinetic energy as it moves from point A to point B. This concept is essential for analyzing the energy changes that occur during the electron's motion.
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Velocity Ratio
The velocity ratio refers to the comparison of the speeds of two objects, which can be derived from their kinetic energies. For two objects with kinetic energies KE1 and KE2, the ratio of their speeds can be expressed as v1/v2 = sqrt(KE1/KE2). In this problem, calculating the ratio of the electron's speed at points A and B involves applying this relationship to the kinetic energy acquired.
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