Problem 22 - Physics

Question

A positively charged particle of mass \(7.2 \times 10^{-8} \mathrm{~kg}\) is traveling due east with a speed of \(85 \mathrm{~m} / \mathrm{s}\) and enters a 0.31 -T uniform magnetic field. The particle moves through onequarter of a circle in a time of \(2.2 \times 10^{-3} \mathrm{~s}\), at which time it leaves the field heading due south. All during the motion the particle moves perpendicular to the magnetic field. (a) What is the magnitude of the magnetic force acting on the particle? (b) Determine the magnitude of its charge.

Step-by-Step Solution

Answer

(a) Magnetic force: \( F = 1.6 \times 10^{-4} \mathrm{~N} \). (b) Charge: \( q = 6.6 \times 10^{-6} \mathrm{~C} \).

Step 1: Identify the Formula for Magnetic Force

The magnetic force on a moving charge in a magnetic field is given by the formula: \( F = qvB \sin \theta \), where \( q \) is the charge of the particle, \( v \) is its velocity, \( B \) is the magnetic field strength and \( \theta \) is the angle between the velocity and the magnetic field. Since the particle moves perpendicular to the field, \( \theta = 90^{\circ} \) and \( \sin \theta = 1 \), simplifying the formula to \( F = qvB \).

Step 2: Use Centripetal Force to Find Magnetic Force

While the particle is moving in a circular path, the magnetic force acts as the centripetal force. Thus, we have: \( F = \frac{mv^2}{r} \). This force equates to the magnetic force calculated previously: \( qvB = \frac{mv^2}{r} \).

Step 3: Determine the Radius of the Path

The fact that the particle moves in a circle indicates that it completes a quarter circle trajectory in the given time. The time for one full revolution is 4 times the quarter circle time, i.e., \( T = 4 \times 2.2 \times 10^{-3} \mathrm{~s} \). The formula for the period of circular motion is \( T = \frac{2\pi r}{v} \). Rearrange to solve for \( r \), the radius: \( r = \frac{Tv}{2\pi} \).

Step 4: Calculate the Magnetic Force

We calculate the radius using \( r = \frac{(4 \times 2.2 \times 10^{-3}) \times 85}{2\pi} \). Then, substitute the calculated \( r \), \( v = 85 \mathrm{~m/s} \), and \( m = 7.2 \times 10^{-8} \mathrm{~kg} \) into \( F = \frac{mv^2}{r} \), resulting in the magnetic force \( F \).

Step 5: Determine Charge Using Magnetic Force Formula

From the relation \( qvB = F \), solve for \( q \): \( q = \frac{F}{vB} \). Use the known values for \( v \), \( B = 0.31 \mathrm{~T} \), and the magnetic force \( F \) calculated previously to find the magnitude of the charge \( q \).

Key Concepts

Moving Charge

A moving charge, such as our positively charged particle, acts as the starting point for understanding magnetic fields and forces. When a charged particle moves, it inherently carries an electric potential with it. This is crucial when interacting with magnetic fields. Such interactions produce various electromagnetic phenomena, including magnetic forces. Here, our moving charge is defined by its velocity, represented as 85 m/s eastward. This motion indicates not just the presence of kinetic energy but also the potential for interaction with magnetic fields when present.
In this scenario, the charge moves perpendicular to the magnetic field, maximizing the magnetic force effect experienced by the charge. The velocity, movement direction, and perpendicular alignment all ensure that the magnetic force becomes effective, simplifying calculations to determine forces acting on the particle.
This mechanism highlights the importance of understanding how motion affects charged particles and their interactions within magnetic fields, forming the foundation for more complex electromagnetic concepts.

Centripetal Force

Centripetal force plays a critical role when our moving charge follows a circular path. In circular motion scenarios, any object moving along a curved path requires a continuous inward force to keep it on this path. This inward force is called the centripetal force. Its purpose is to counteract the inertial tendency of the object to move in a straight line, thereby maintaining circular motion.
For the particle in our exercise, the magnetic force serves as this centripetal force. Mathematically, it's expressed as \( F = \frac{mv^2}{r} \), where \( m \) is the mass of the particle, \( v \) is its speed, and \( r \) is the radius of the path. The balance between the magnetic force and required centripetal force determines the curvature of the path taken by the charge. Hence, the relationship between force, velocity, and radius is key to analyzing motion in circular pathways under electromagnetic influence.

Circular Motion

Circular motion emerges from the interaction of forces that act perpendicular to the velocity of a moving charge. In this exercise, the charge moves in a circular trajectory due to the magnetic field's influence. The particle initially travels east, enters the magnetic field, and completes a quarter circle before heading south.
This trajectory signifies a consistent circular motion, compelling the object to maintain a perpendicular velocity to its direction of rotation due to the magnetic force. The time taken for this path, specifically the quarter circle, helps to determine the frequency and radius of the complete motion. By understanding circular motion principles, such as its dependency on radius and speed, we grasp how external forces like magnetic fields orchestrate the motion characteristics and trajectory of particles.

Magnetic Field

A magnetic field is an invisible force field that influences moving charges. It surrounds magnetic materials and currents, exerting significant forces on charged particles. In our scenario, a uniform 0.31 T magnetic field enables the exertion of consistent force across the particle's path.
This uniformity simplifies calculations, indicating the magnitude of the field remains constant through the motion. The magnetic field determines the velocity trajectory by causing perpendicular force, culminating in circular motion. Thus, understanding the characteristics of magnetic fields, such as their direction (indicated by field lines) and strength (measured in Teslas), is essential to predicting and calculating how moving charges will behave within these fields.