Problem 77 - College Physics

Question

\bullet A 150 g ball containing \(4.00 \times 10^{8}\) excess electrons is dropped into a 125 vertical shaft. At the bottom of the shaft, the ball suddenly enters a uniform horizontal 0.250 T magnetic field directed from east to west. If air resistance is negligibly small, find the magnitude and direction of the force that this magnetic field exerts on the ball just as it enters the field.

Step-by-Step Solution

Answer

The magnetic force is \(7.92 \times 10^{-11}\text{ N}\) directed south.

Step 1: Understand the Problem

The ball with an excess of electrons enters a magnetic field. We need to calculate the force exerted by the field using the charge of the electrons and the strength and direction of the magnetic field. We must determine both the magnitude and direction of this force.

Step 2: Calculate the Charge on the Ball

Each electron has a fundamental charge of approximately \(1.6 \times 10^{-19} \text{ C}\). Given \(4.00 \times 10^{8}\) excess electrons, the total charge \(q\) on the ball is calculated as:\[ q = n \times e = 4.00 \times 10^{8} \times 1.6 \times 10^{-19} = 6.4 \times 10^{-11} \text{ C} \]

Step 3: Find the Velocity of the Ball

The ball falls freely from a height of 125 m, thus the velocity just before it hits the field can be found using the equation for free fall: \[ v = \sqrt{2gh} = \sqrt{2 \times 9.81 \text{ m/s}^2 \times 125 \text{ m}} \approx 49.5 \text{ m/s} \] where \(g = 9.81 \text{ m/s}^2\) is the acceleration due to gravity and \(h = 125 \text{ m}\) is the height.

Step 4: Determine the Direction of the Velocity

Since the ball falls straight down, the velocity vector is directed vertically downwards.

Step 5: Use the Right-Hand Rule

To find the direction of the magnetic force, apply the right-hand rule. Point your fingers in the direction of the velocity (downward), curl them toward the direction of the magnetic field (east to west), and your thumb will point north, indicating the direction of the force on a positive charge. For electrons, the force direction is opposite, so it is south.

Step 6: Calculate the Magnitude of the Magnetic Force

The formula for the magnetic force is given by:\[ F = qvB \]Substituting the known values:\[ F = (6.4 \times 10^{-11} \, \text{C})(49.5 \, \text{m/s})(0.250 \, \text{T}) \]Calculating, we obtain:\[ F \approx 7.92 \times 10^{-11} \, \text{N} \]

Step 7: State the Final Result

The magnitude of the force is \(7.92 \times 10^{-11} \text{ N}\), and its direction is toward the south.

Key Concepts

Electromagnetism

Electromagnetism is a fundamental force of nature that describes the interaction between charged particles. It combines electricity and magnetism into a single theory. In this exercise, electromagnetism comes into play when the ball enters a magnetic field. The force experienced by the ball is due to the electromagnetic interaction between its charge and the magnetic field. A charged particle moving through a magnetic field experiences a magnetic force, perpendicular to both the velocity of the particle and the direction of the magnetic field. This force is what redirects the ball's path, as seen in our scenario.

Projectile Motion

Projectile motion refers to the motion of an object thrown or projected into the air, subject only to acceleration due to gravity. Although the ball is initially in free fall, it transitions into projectile motion when it interacts with the magnetic field. The force exerted by the magnetic field alters its trajectory. Normally, projectile motion is purely influenced by gravitational pull, but in this context, the magnetic force adds an additional component influencing motion. Non-gravitational forces, like this magnetic force, complicate the path of the projectile, and such interactions are a significant study area in physics.

Charge of Electrons

The concept of the charge of electrons is crucial in understanding how the ball interacts with the magnetic field. Electrons have a negative fundamental charge of approximately \(1.6 \times 10^{-19}\) coulombs. In this problem, the ball contains an excess of \( 4.00 \times 10^{8}\) electrons giving it a total charge calculated by multiplying the number of electrons by the charge of a single electron. This charge allows the magnetic field to exert a force on the ball, affecting its motion. Charge is a foundational concept in electromagnetism, engaging in forces that can be attractively or repulsively mediated by electric and magnetic fields.

Right-Hand Rule

The right-hand rule is a helpful mnemonic that allows us to determine the direction of the magnetic force on a moving charge. To apply the right-hand rule correctly, use your right hand: point your fingers in the direction of the particle's velocity (for the ball, this is downward as it falls). Then, rotate your fingers to follow the magnetic field lines (from east to west in this case). Your thumb will point towards the north, indicating the force's direction for positive charges. However, since electrons have a negative charge, the actual force direction is opposite—north becomes south. Understanding this concept is vital for predicting the behavior of charged particles in magnetic fields.