Electron Acceleration And Magnetic Field Deflection: A Physics Deep Dive

Hey guys, let's dive into a classic physics problem that beautifully combines the concepts of electric potential, kinetic energy, and magnetic fields. We're going to explore what happens when an electron, initially at rest, gets a boost from a voltage difference, and then takes a wild ride through a uniform magnetic field. This is super interesting and relevant stuff, so buckle up! The core of our problem revolves around an electron that is accelerated from a standstill due to an 8 kV potential difference. Following this acceleration, the electron enters a homogenous magnetic field of 3 mT, entering perpendicular to it. The goal is to determine the electron's velocity upon entering the magnetic field, calculate the radius of its path within the magnetic field, and figure out the time it takes for a full rotation within the magnetic field. Throughout this exploration, we'll harness key physics principles, including energy conservation and the Lorentz force. Let's start with a breakdown of each part of the problem to get a good understanding.

(a) Electron Velocity Entering the Magnetic Field

Alright, first things first: let's figure out how fast this electron is moving before it hits the magnetic field. Because the electron starts from rest and is accelerated by the electric potential, we can use the principle of energy conservation. This essentially means the electric potential energy that the electron gains is converted into kinetic energy as it accelerates. The change in potential energy (ΔPE) of a charged particle in an electric field is given by qV, where 'q' is the charge of the electron and 'V' is the potential difference. The electron's kinetic energy (KE) is given by ½mv², where 'm' is the mass of the electron and 'v' is its velocity. So, the energy conservation equation becomes:

qV = ½mv²

Now, let's plug in the numbers. We know that the charge of an electron (q) is 1.6 x 10⁻¹⁹ C, the potential difference (V) is 8 kV, which is 8000 V, and the mass of the electron (m) is 9 x 10⁻³¹ kg. Solving for 'v' (velocity), we get:

v = √(2qV / m) v = √(2 * 1.6 x 10⁻¹⁹ C * 8000 V / 9 x 10⁻³¹ kg) v ≈ 5.33 x 10⁷ m/s

So, the electron is zooming into the magnetic field at a whopping 5.33 x 10⁷ meters per second! That's a serious speed, guys! This calculation really highlights how a relatively small potential difference can accelerate a tiny particle to extremely high speeds. Understanding how electric fields can accelerate charged particles is fundamental to many technologies, including particle accelerators and cathode ray tubes.

More on the initial velocity calculation

When calculating the velocity, we're assuming that all the electric potential energy is converted into kinetic energy. This is a reasonable assumption in this idealized scenario. However, in real-world situations, there might be factors like air resistance or other forms of energy loss that could affect the final velocity. Also, the energy conservation principle is a cornerstone of physics, demonstrating how energy transforms from one form to another, and how it must always be conserved within a closed system. It is a critical concept for understanding many physical phenomena. As you can see, the velocity calculation combines basic electrostatics with mechanics, underlining the interconnectedness of different physics concepts.

(b) Radius of the Electron's Circular Path in the Magnetic Field

Now, let's consider what happens inside the magnetic field. When a charged particle moves through a magnetic field, it experiences a force perpendicular to its velocity and the magnetic field direction. This force is known as the Lorentz force. The magnitude of the Lorentz force (F) on a charge 'q' moving with velocity 'v' in a magnetic field 'B' is given by:

F = qvBsinθ

Where 'θ' is the angle between the velocity and the magnetic field. In our case, the electron enters the field perpendicularly, so θ = 90 degrees, and sin(90°) = 1. Therefore, the force simplifies to F = qvB.

This force doesn't change the electron's speed; it only changes its direction. Consequently, the electron moves in a circular path. The magnetic force provides the centripetal force needed for this circular motion. The centripetal force is given by:

F_c = mv²/r

Where 'm' is the mass of the electron, 'v' is its velocity, and 'r' is the radius of the circular path. Since the magnetic force is the centripetal force, we can equate the two:

qvB = mv²/r

Solving for 'r', we get:

r = mv / qB

Plugging in the values we know (m = 9 x 10⁻³¹ kg, v ≈ 5.33 x 10⁷ m/s, q = 1.6 x 10⁻¹⁹ C, and B = 3 x 10⁻³ T), we get:

r = (9 x 10⁻³¹ kg * 5.33 x 10⁷ m/s) / (1.6 x 10⁻¹⁹ C * 3 x 10⁻³ T) r ≈ 0.1 m

So, the electron's path has a radius of approximately 0.1 meters, or 10 centimeters. This calculation shows how the magnetic field curves the path of the electron. The radius of the path depends on the electron's momentum (mv), the magnetic field strength (B), and the charge (q). A stronger magnetic field will result in a smaller radius, causing the electron to curve more sharply. The mass and charge of the electron also play a crucial role in determining the radius. A more massive particle or one with a larger charge would also have a different radius, assuming the other parameters remain the same.

The concept of Centripetal Force

This part is really a great illustration of centripetal force. The magnetic force acts as the centripetal force, continuously pulling the electron toward the center of the circle, changing its direction but not its speed. Without this centripetal force, the electron would simply move in a straight line, according to Newton's first law. The radius of the circular path is crucial in many applications, from mass spectrometers, which separate ions based on their mass-to-charge ratio, to the design of particle accelerators. Also, note that the magnetic field does no work on the electron because the force is always perpendicular to the direction of motion. This means the magnetic field can't change the kinetic energy or the speed of the electron. This is also why the magnetic field doesn't change the electron’s kinetic energy.

(c) Time for One Full Rotation in the Magnetic Field

Finally, let's find out how long it takes for the electron to complete one full orbit within the magnetic field. This is the period (T) of the circular motion. The period is the time it takes for one complete revolution. The electron travels the circumference of the circle (2πr) in one period. Because the electron's speed is constant, the period can be found using:

T = 2πr / v

We already know 'r' and 'v', so we can calculate:

T = 2π * 0.1 m / 5.33 x 10⁷ m/s T ≈ 1.18 x 10⁻⁸ s

So, the electron takes approximately 1.18 x 10⁻⁸ seconds, or 11.8 nanoseconds, to complete one full rotation. This extremely short time highlights how quickly the electron moves and how sensitive it is to magnetic fields. The electron's period is independent of the radius of the path. This means that the time for one rotation only depends on the magnetic field strength and the electron's charge and mass, but not the electron's initial velocity. Also, the frequency of the circular motion (f) is the reciprocal of the period:

f = 1 / T f ≈ 8.47 x 10⁷ Hz

This means the electron is orbiting the magnetic field at a frequency of approximately 84.7 MHz. This high frequency is used in various applications, including microwave ovens and medical imaging devices. It is very useful and shows the direct relationship between magnetic fields and the movement of charged particles.

Magnetic Field Applications and Impact

To sum it all up, this problem illustrates some of the fundamental relationships between electricity and magnetism. It is used in different technologies, from cathode-ray tubes (like old TVs) to mass spectrometers and even in the design of particle accelerators. Understanding how charged particles behave in electric and magnetic fields is key to developing many modern technologies. Furthermore, our results give us insight into how we can control and manipulate particles. The calculation of the electron's path radius and time for one full rotation depends entirely on the principles of electromagnetism, but also provides the basis for understanding more complex systems, such as the behavior of plasma in fusion reactors or the operation of MRI machines. These concepts show the power of physics to explain and predict real-world phenomena.

In conclusion, understanding how these concepts interact helps us better understand the world around us and the technologies that shape our lives. We have explored the electron's velocity, the radius of its path, and the time it takes to complete a full rotation. Each of these calculations is a testament to the fundamental principles of physics and highlights the beauty of how these principles come together to explain the behavior of charged particles in electric and magnetic fields. Keep exploring, guys, because there's so much more to discover!