Question

An alpha particle \(\left(m=6.6 \cdot 10^{-27} \mathrm{~kg}, q=+2 e\right)\) is accelerated by a potential difference of \(2700 \mathrm{~V}\) and moves in a plane perpendicular to a constant magnetic field of magnitude \(0.340 \mathrm{~T}\), which curves the trajectory of the alpha particle. Determine the radius of curvature and the period of revolution.

Short answer

Answer: The radius of curvature of the alpha particle's trajectory is approximately \(1.20 \cdot 10^{-2} \mathrm{~m}\), and the period of revolution is approximately \(3.10 \cdot 10^{-8} \mathrm{~s}\).

Step-by-step solution

Find the speed of the alpha particle

To find the speed (\(v\)) of the alpha particle after being accelerated by the potential difference, we can use the relationship between kinetic energy (K) and the potential difference (V): K = \(\frac{1}{2} m v^2 = eV\) where \(m\) is the mass of the particle, \(e\) is the charge of an electron, and \(q = 2e\) is the charge of the alpha particle. We can solve for \(v\): \(v = \sqrt{\frac{2eV}{m}}\) Plug in the given values for \(m\), \(e\), and \(V\): \(v = \sqrt{\frac{2(2 \cdot 1.6 \cdot 10^{-19} C)(2700V)}{6.6 \cdot 10^{-27} kg}}\) \(v \approx 1.94 \cdot 10^7 \mathrm{~m/s}\)

Find the radius of curvature

To find the radius of curvature (r) of the alpha particle's trajectory, we can use the formula: \(r = \frac{mv}{|q|B}\) where \(B\) is the magnitude of the magnetic field. Plug in our values for \(m\), \(v\), \(q\), and \(B\): \(r = \frac{(6.6 \cdot 10^{-27} kg)(1.94 \cdot 10^7 \mathrm{~m/s})}{|2(1.6 \cdot 10^{-19} C)|(0.340 T)}\) \(r \approx 1.20 \cdot 10^{-2} \mathrm{~m}\)

Find the period of revolution

To find the period of revolution (T), we will use the formula: \(T = \frac{2 \pi m}{|q|B}\) Plugging in the values for \(m\), \(q\), and \(B\): \(T = \frac{2 \pi (6.6 \cdot 10^{-27} kg)}{|2(1.6 \cdot 10^{-19} C)|(0.340 T)}\) \(T \approx 3.10 \cdot 10^{-8} \mathrm{~s}\) The radius of curvature of the alpha particle's trajectory is \(1.20 \cdot 10^{-2} \mathrm{~m}\), and the period of revolution is \(3.10 \cdot 10^{-8} \mathrm{~s}\).

Key concepts

Alpha Particle Trajectory

Understanding the trajectory of an alpha particle in a magnetic field opens a window into the realm of particle physics and electromagnetic force interactions. An alpha particle, which consists of two protons and two neutrons, carries a positive charge due to its protons. When this particle is introduced to a magnetic field at a right angle, it doesn't proceed in a straight line. Instead, it follows a circular path, a direct result of the Lorentz force acting upon it. This force, which is perpendicular to both the velocity of the charged particle and the magnetic field, bends the path of the particle into a curve.

The textbook example considers an alpha particle accelerated through a voltage that gives it kinetic energy. Once in the magnetic field, its trajectory curves due to the interaction between the magnetic field and the particle's charge, causing it to spiral along a fixed radius. This behavior illustrates the classic principles of electromagnetism as described by the Lorentz force equation. This fundamental concept is crucial not only in particle physics but also in applications such as cyclotrons and mass spectrometers, where the curvature of charged particles' paths is exploited for analysis and experimentation.

When discussing improvements to the exercise, it's essential to stress that the curvature occurs because of the particle's charge and the magnetic field's influence. Through practical examples, like the path of electrons in old-fashioned television tubes or the operation of cyclotrons, students can better visualize the trajectory's circular nature and connect the abstract equations to real-world phenomena.

Radius of Curvature

The radius of curvature can seem like an abstract concept, but it is, essentially, the radius of the circular path taken by a charged particle, such as an alpha particle, when it moves through a magnetic field. This radius can be determined by the balance between the magnetic force and the particle's inertia—in a way, it's the 'tightness' of the particle's path. In our exercise, the radius depends on the particle's mass, its velocity after being accelerated by the potential difference, its charge, and the magnetic field's strength.

Students often struggle with the equation for the radius of curvature, which might seem non-intuitive at first glance. It's worthwhile to reinforce that the velocity of the alpha particle is a result of the energy gain from the voltage applied, represented by the kinetic energy formula. Further, recognizing that the charge of the particle and the strength of the magnetic field inversely affect the radius helps students understand why a larger charge or a stronger magnetic field would result in a tighter curve, and vice versa.

For better clarity during exercise improvements, one could utilize visual aids such as diagrams showing how variations in charge and magnetic field affect the curvature. Simplified explanations alongside the mathematical formulas would also help in demystifying the path's 'tightness,' making the concept more approachable and comprehensible.

Period of Revolution

The period of revolution is the time it takes for a charged particle to complete one full orbit along its curved path in a magnetic field. For our alpha particle, it represents the duration of one full sweep of its circular trajectory. This period is crucial for understanding the particle's motion dynamics and frequency of revolution. It’s a central concept not only in physics but also significantly in fields like astronomy and engineering, where orbital periods are fundamental.

As with the radius, the period of revolution depends on the particle's properties, such as its mass and charge, as well as the magnetic field's strength. The insightful aspect of this relationship is that the velocity cancels out when we apply the period of revolution formula, showing how the period is independent of the speed and only depends on the ratio of the particle’s charge to its mass and the magnetic field's intensity.

When improving the exercise presentation, making the independence from velocity clear is useful. It may seem counterintuitive—not many students would guess that a particle could circle faster or slower without changing the time it takes to complete each orbit. Including a comparison to familiar situations, such as the Earth’s orbit around the Sun taking one year regardless of minor speed changes, can help solidify this concept. The beauty of physics often lies in such counterintuitive truths, and bringing them to light can be a fascinating revelation for students.