What Is The Final Speed Of A Helium Ion With Mass 4m And Charge 2e Accelerated From Rest Through A Potential Difference V In Vacuum?

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In the realm of physics, understanding the behavior of charged particles within electric fields is paramount. This article delves into the scenario of a helium ion, possessing a mass of 4m and a charge of 2e, being accelerated from a state of rest through a potential difference V in a vacuum. We aim to elucidate the process of determining the ion's final speed, a concept rooted in the principles of energy conservation and electromagnetism. By exploring this problem, we not only reinforce fundamental physics knowledge but also demonstrate the practical application of these principles in analyzing charged particle motion.

Our core objective is to ascertain the final speed attained by a helium ion. This ion, characterized by its mass 4m and charge 2e, embarks on its journey from a standstill, propelled by a potential difference V within the confines of a vacuum. The absence of air resistance in a vacuum simplifies our calculations, allowing us to focus solely on the electrical forces at play. To unravel this problem, we will meticulously dissect the energy transformations that occur as the ion accelerates, ultimately arriving at a precise expression for its final velocity. The question presented to us is a classic example of how electrical potential energy converts into kinetic energy, a cornerstone concept in electrodynamics.

At the heart of our analysis lies the principle of energy conservation. As the helium ion traverses the potential difference, it undergoes a transformation of energy. Initially, the ion possesses electrical potential energy, a consequence of its charge and the electric potential it experiences. As it accelerates, this potential energy is converted into kinetic energy, the energy of motion. The work-energy theorem elegantly quantifies this transformation, stating that the work done on the ion by the electric field is precisely equal to the change in its kinetic energy. Mathematically, this can be expressed as:

Work Done = Change in Kinetic Energy

The work done on the ion can be further expressed as the product of the charge (2e) and the potential difference (V). The change in kinetic energy, on the other hand, is the difference between the final kinetic energy (1/2 * (4m) * v^2) and the initial kinetic energy (which is zero since the ion starts from rest). Equating these two expressions allows us to establish a direct relationship between the potential difference, the ion's charge and mass, and its final velocity. This relationship forms the bedrock of our solution, enabling us to calculate the ion's speed with precision.

To determine the final speed of the helium ion, we begin by equating the electrical potential energy gained by the ion to its kinetic energy. The electrical potential energy (U) is given by:

U = qV

where q is the charge of the ion and V is the potential difference. In this case, the charge q is 2e, so the electrical potential energy is:

U = 2eV

This potential energy is converted into kinetic energy (K) as the ion accelerates. The kinetic energy is given by:

K = (1/2)mv^2

where m is the mass of the ion and v is its final speed. Here, the mass m is 4m, so the kinetic energy becomes:

K = (1/2)(4m)v^2 = 2mv^2

By the principle of energy conservation, the electrical potential energy gained equals the kinetic energy gained:

2eV = 2mv^2

Now, we solve for v:

v^2 = (2eV) / (2m) = eV/m

Taking the square root of both sides, we find the final speed:

v = √(eV/m)

Thus, the final speed of the helium ion is √(eV/m). This result showcases the interplay between electrical potential energy and kinetic energy, a fundamental concept in physics. The derived equation not only provides the answer to our specific problem but also serves as a template for analyzing the motion of charged particles in electric fields.

Based on our calculations, the final speed of the helium ion is:

B. √(eV/m)

This answer aligns perfectly with our theoretical framework and demonstrates the effective application of energy conservation principles in solving physics problems. The speed is directly proportional to the square root of the potential difference and inversely proportional to the square root of the mass, which makes intuitive sense: a larger potential difference imparts more energy, leading to a higher speed, while a larger mass requires more energy to achieve the same speed.

It is crucial not only to arrive at the correct answer but also to understand why the other options are incorrect. This process, known as distractor analysis, enhances our comprehension of the underlying concepts and prevents us from making similar mistakes in the future. Let's examine why options A, C, and D are not the correct answers:

  • A. √(2eV/m): This option might arise from a miscalculation in the energy conservation equation. Perhaps the factor of 2 in the charge (2e) was not properly accounted for when equating potential and kinetic energies. This highlights the importance of meticulously tracking all numerical factors throughout the calculation process.
  • C. eV/2m: This option is dimensionally incorrect, as it represents energy divided by mass, resulting in units of velocity squared rather than velocity. This underscores the significance of dimensional analysis as a tool for verifying the correctness of physical equations.
  • D. eV/m: Similar to option C, this answer is also dimensionally incorrect, representing velocity squared rather than velocity. This could stem from an error in taking the square root during the final step of the calculation.

By dissecting these distractors, we reinforce our understanding of the correct solution and develop a more robust approach to problem-solving in physics. Recognizing common errors and pitfalls is an integral part of mastering the subject.

In summary, we have successfully determined the final speed of a helium ion accelerated through a potential difference in a vacuum. By applying the principle of energy conservation and carefully accounting for the ion's charge and mass, we arrived at the solution v = √(eV/m). This exercise exemplifies the power of fundamental physics principles in elucidating the behavior of charged particles within electric fields. Furthermore, our distractor analysis has provided valuable insights into potential errors and pitfalls, reinforcing our understanding of the underlying concepts. This problem serves as a testament to the beauty and elegance of physics in describing the natural world.

Helium ion, final speed, potential difference, energy conservation, kinetic energy, electrical potential energy, vacuum, charge, mass, electromagnetism, physics, √(eV/m), work-energy theorem, electrodynamics.