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A magnetic field is a region around a magnetic material or a moving electric charge within which the force of magnetism acts. It is measured in teslas (T), and in this exercise, we're focusing on the horizontal component of the Earth's magnetic field. This component is given as a tiny force, specifically \(1.6 \times 10^{-5}\, \text{T}\), yet it can exert significant forces on moving charged particles like electrons.
The force experienced by a charged particle in a magnetic field depends on several factors: the strength of the magnetic field, the charge of the particle, its velocity, and the angle at which it meets the field. Importantly, the force is maximized when the motion of the particle is perpendicular to the magnetic field, as is the case in this exercise.
Understanding how magnetic fields interact with charged particles is crucial in fields ranging from physics to engineering. It explains phenomena seen in technologies such as cathode ray tubes and the behavior of particles in space.

The velocity of an electron determines how it interacts with a magnetic field. In this scenario, the electron has an initial velocity of \(2.1 \times 10^6 \text{ m/s}\), directed vertically upwards. This high speed is crucial because the magnetic force exerted on the electron is directly proportional to its velocity.
When a charged particle moves through a magnetic field, its path can curve depending on the field's direction and the speed of the particle. This happens because the magnetic force acts perpendicular to both the velocity and the magnetic field.
The high velocity in this exercise means that the interaction between the electron and the magnetic field is robust. Understanding velocity's role in magnetic forces brings insight into designing technologies like particle accelerators and understanding how solar winds interact with the Earth's magnetic field.

Calculating the acceleration of an electron under a magnetic force involves applying known physics principles and equations. The initial step is to determine the magnetic force acting on the electron. This force \( F \) is calculated using the formula: \[ F = qvB \sin \theta \] where:

• \( q \) is the charge of the electron (\(-1.6 \times 10^{-19} \text{ C}\))
• \( v \) is the velocity of the electron (\(2.1 \times 10^6 \text{ m/s}\))
• \( B \) is the magnetic field (\(1.6 \times 10^{-5} \text{ T}\))
• \( \theta \) is the angle between the velocity and the magnetic field, which is \(90^\circ\), making \(\sin\theta = 1\)

This results in the necessary force to further calculate acceleration.
Next, using Newton's second law, the acceleration \( a \) can be determined by rearranging the formula \( F = ma \) to \( a = \frac{F}{m} \). Substituting the mass of the electron \(9.11 \times 10^{-31} \text{ kg}\), we find that the acceleration is approximately \(5.9 \times 10^{10} \text{ m/s}^2\). This precise calculation reveals the remarkable effects even a tiny magnetic field can have on subatomic particles.

Newton's second law of motion is fundamental in physics, stating that the force acting on an object is equal to the mass of that object multiplied by its acceleration: \[ F = ma \] This principle acts as a cornerstone for understanding how forces affect motion. In this exercise, it's essential for transforming a force value into something more meaningful: the electron's acceleration.
Once the magnetic force on the electron is calculated, applying Newton's second law allows us to find the acceleration by dividing the force by the electron's mass. The concept highlights the proportionality between force and acceleration and inversely with mass, explaining why less massive objects like electrons can have significant acceleration under relatively small forces.
This principle not only assists in academic exercises but also provides insight into dynamics in everyday life, from the motion of planets to the functioning of engines.