Electron Revolution Frequency In A Transverse Magnetic Field Calculation And Discussion
Understanding the behavior of charged particles within magnetic fields is a cornerstone of physics, with profound implications across various fields, from particle physics to electronics. This article delves into a specific scenario: the influence of a transverse magnetic field on the revolution frequency of an electron. We'll explore the fundamental principles governing this interaction, perform the necessary calculations, and interpret the results within a broader context. This exploration not only enhances our theoretical understanding but also highlights the practical applications of these concepts in technologies like mass spectrometry and cyclotron particle accelerators. Transverse magnetic fields play a crucial role in confining and manipulating charged particles, making the study of their effects essential for advancing scientific knowledge and technological innovation. The revolution frequency, in particular, is a key parameter in characterizing the motion of charged particles in magnetic fields, and its accurate determination is vital for designing and optimizing various devices and experiments.
When an electron enters a transverse magnetic field, it experiences a force perpendicular to both its velocity and the magnetic field direction. This force, known as the Lorentz force, causes the electron to move in a circular path. The magnitude of this force (F) is given by the equation F = qvB, where q is the charge of the electron (approximately 1.602 x 10⁻¹⁹ Coulombs), v is the electron's velocity, and B is the magnetic field strength. This force acts as the centripetal force, which keeps the electron moving in a circle. The centripetal force (Fc) is given by Fc = mv²/r, where m is the mass of the electron (approximately 9.109 x 10⁻³¹ kilograms) and r is the radius of the circular path. By equating the Lorentz force and the centripetal force (qvB = mv²/r), we can derive the relationship between the electron's velocity, the magnetic field strength, and the radius of its orbit. This relationship is crucial for understanding how the magnetic field influences the electron's trajectory. The circular motion arises because the magnetic force continuously deflects the electron, preventing it from moving in a straight line. The balance between the magnetic force and the centripetal force determines the radius and the speed of the electron's circular path. Furthermore, the frequency of revolution, which is the number of complete circles the electron makes per unit time, is a critical parameter that depends on the electron's speed and the radius of its orbit. Understanding these fundamental principles allows us to predict and control the behavior of electrons in magnetic fields, which is essential for various applications.
The frequency of revolution (f) can be calculated using the formula f = v / (2πr), where v is the velocity and r is the radius of the circular path. Substituting the expression for velocity (v = qBr/m) derived from the force balance equation into the frequency formula, we obtain f = qB / (2πm). This equation reveals that the frequency of revolution is directly proportional to the magnetic field strength (B) and the charge of the electron (q), and inversely proportional to the mass of the electron (m). This is a significant result because it shows that the frequency of revolution is independent of the electron's velocity. This independence is a crucial factor in the operation of devices like cyclotrons, where particles are accelerated to high energies while maintaining a constant revolution frequency. The ratio of the charge to the mass of the electron, often denoted as e/m or q/m, is a fundamental constant that appears in this equation. The e/m ratio is a characteristic property of the electron and plays a vital role in many physical phenomena. Knowing the e/m ratio and the magnetic field strength, we can accurately determine the frequency of revolution of the electron. This understanding is not only essential for theoretical calculations but also for practical applications in various scientific and technological fields.
We are given a transverse magnetic field of 3.57 x 10⁻² T (Tesla). The charge-to-mass ratio (e/m) for the electron is provided as 1.76 x 10¹¹ C/kg (Coulombs per kilogram). The objective is to determine the frequency of revolution of the electron in this magnetic field. This problem is a classic example of applying the principles of electromagnetism to calculate the motion of a charged particle in a magnetic field. By understanding the relationship between the magnetic field, the charge-to-mass ratio, and the frequency of revolution, we can accurately predict the behavior of electrons in various electromagnetic environments. Solving this problem not only reinforces our understanding of the underlying physics but also provides valuable insights into the practical applications of these concepts. The given values of the magnetic field and the charge-to-mass ratio are typical for experiments involving electron motion in magnetic fields, making this problem highly relevant to real-world scenarios. The ability to calculate the frequency of revolution is crucial for designing and analyzing various devices that utilize electron beams, such as mass spectrometers and particle accelerators. Therefore, this problem serves as a fundamental exercise in applying electromagnetic theory to solve practical problems.
To calculate the frequency of revolution (f), we use the formula derived earlier: f = qB / (2πm). Since we are given the value of e/m, which is q/m, we can rewrite the formula as f = (e/m) * B / (2π). Substituting the given values, we have:
f = (1.76 x 10¹¹ C/kg) * (3.57 x 10⁻² T) / (2π)
Performing the calculation:
f ≈ (1.76 x 10¹¹ * 3.57 x 10⁻²) / (2 * 3.14159)
f ≈ 6.2832 x 10⁹ Hz
Converting this to MHz (Megahertz) by dividing by 10⁶:
f ≈ 62.8 MHz
This calculation demonstrates the direct relationship between the magnetic field strength and the frequency of revolution. A stronger magnetic field results in a higher frequency of revolution, as the electron is forced to move in a tighter circle due to the increased Lorentz force. The use of the e/m ratio simplifies the calculation, highlighting the importance of this fundamental constant in determining the behavior of electrons in electromagnetic fields. The result, approximately 62.8 MHz, is a typical frequency range for electron motion in magnetic fields of this magnitude. This frequency is readily measurable using standard electronic equipment, making it possible to experimentally verify the theoretical calculations. The accurate calculation of the frequency of revolution is crucial for various applications, such as tuning the resonance frequency in cyclotron particle accelerators or analyzing the mass-to-charge ratio of ions in mass spectrometers. Therefore, this calculation is a fundamental step in understanding and applying the principles of electromagnetism.
The calculated frequency of revolution of the electron in the given transverse magnetic field is approximately 62.8 MHz. This result corresponds to option (c) in the provided choices. The frequency falls within the radio frequency range, which is commonly used in various applications, including wireless communication and medical imaging. The significance of this frequency lies in its relationship to the electron's motion within the magnetic field. The electron completes approximately 62.8 million revolutions per second, a testament to the strong influence of the magnetic field on its trajectory. This high frequency is a direct consequence of the relatively large charge-to-mass ratio of the electron and the strength of the magnetic field. The result underscores the importance of understanding the interplay between magnetic fields and charged particles in various physical phenomena. The circular motion of the electron, characterized by this frequency, is a fundamental concept in electromagnetism and has far-reaching implications in various fields.
The magnitude of the calculated frequency highlights the precision required in experiments involving charged particles in magnetic fields. Small variations in the magnetic field strength or the electron's initial conditions can significantly alter the frequency of revolution and, consequently, its trajectory. This sensitivity is both a challenge and an opportunity. On one hand, it necessitates precise control and measurement of experimental parameters. On the other hand, it allows for highly sensitive measurements and manipulations of charged particles. For example, in mass spectrometry, the frequency of revolution (or the related cyclotron frequency) is used to determine the mass-to-charge ratio of ions with exceptional accuracy. Similarly, in cyclotron particle accelerators, the frequency of the accelerating electric field is synchronized with the revolution frequency of the particles to achieve high energies. The agreement between the calculated frequency and the experimental observations provides strong evidence for the validity of the theoretical framework used. The Lorentz force law and the concept of centripetal force are fundamental principles that accurately describe the behavior of charged particles in magnetic fields. This consistency between theory and experiment is a cornerstone of scientific progress, allowing us to build a robust understanding of the natural world.
In conclusion, the frequency of revolution of an electron in a transverse magnetic field of 3.57 x 10⁻² T, given the e/m value of 1.76 x 10¹¹ C/kg, is calculated to be approximately 62.8 MHz. This result is obtained by applying the fundamental principles of electromagnetism, specifically the Lorentz force law and the concept of centripetal force. The calculation demonstrates the direct relationship between the magnetic field strength, the charge-to-mass ratio of the electron, and the frequency of revolution. The frequency of revolution is a critical parameter in characterizing the motion of charged particles in magnetic fields and has significant implications in various scientific and technological applications. Understanding the behavior of electrons in magnetic fields is essential for a wide range of applications, from particle physics experiments to medical imaging techniques. The principles discussed in this article are fundamental to the design and operation of devices such as mass spectrometers, cyclotron particle accelerators, and magnetic resonance imaging (MRI) scanners. These technologies rely on the precise control and manipulation of charged particles using magnetic fields, and the frequency of revolution plays a central role in their functionality. Furthermore, the study of electron motion in magnetic fields provides valuable insights into the fundamental nature of electromagnetism and the interactions between charged particles and fields. The consistency between theoretical calculations and experimental observations reinforces the validity of the underlying physical laws and allows us to develop more advanced technologies and scientific tools. The ability to accurately calculate the frequency of revolution is a testament to our understanding of these principles and their practical applications. As technology continues to advance, the study of charged particles in magnetic fields will undoubtedly remain a crucial area of research and development.
