Problem 37 - Physics

Question

In a TV, electrons with a speed of \(8.4 \times 10^{7} \mathrm{~m} / \mathrm{s}\) strike the screen from behind, causing it to glow. The electrons come to a halt after striking the screen. Each electron has a mass of \(9.11 \times 10^{-31} \mathrm{~kg}\), and there are \(6.2 \times 10^{16}\) electrons per second hitting the screen over an area of \(\mathrm{m}^{2}\). What is the pressure that the electrons exert on the screen?

Step-by-Step Solution

Answer

The pressure exerted by the electrons on the screen is \(4.744 \times 10^{-6} \, \mathrm{Pa}\).

Step 1: Calculate the momentum of one electron

The momentum of an electron can be calculated using the formula \( p = mv \), where \( m \) is the mass and \( v \) is the velocity of the electron. Here, \( m = 9.11 \times 10^{-31} \, \mathrm{kg} \) and \( v = 8.4 \times 10^{7} \, \mathrm{m/s} \). Thus, the momentum is \[ p = 9.11 \times 10^{-31} \, \mathrm{kg} \times 8.4 \times 10^{7} \, \mathrm{m/s} \] \[ p = 7.6524 \times 10^{-23} \, \mathrm{kg} \, \mathrm{m/s} \].

Step 2: Calculate the change in momentum per second

The change in momentum per electron is equal to the momentum calculated since the electrons come to a halt. Since there are \( 6.2 \times 10^{16} \) electrons striking per second, the change in momentum per second is \( 6.2 \times 10^{16} \times 7.6524 \times 10^{-23} \, \mathrm{kg} \, \mathrm{m/s} \).

Step 3: Calculate the force exerted by the electrons

Force is the rate of change of momentum, thus \[ F = 6.2 \times 10^{16} \times 7.6524 \times 10^{-23} \] \[ F = 4.744 \times 10^{-6} \, \mathrm{N} \].

Step 4: Determine the pressure exerted on the screen

Pressure is defined as force per unit area. Assuming the area \( A \) is \( 1 \, \mathrm{m}^2 \), thus pressure \( P \) is \[ P = \frac{F}{A} = \frac{4.744 \times 10^{-6}}{1} \] \[ P = 4.744 \times 10^{-6} \, \mathrm{Pa} \].

Key Concepts

Momentum Calculation

To understand electron dynamics, it's essential to start with understanding momentum calculation. The momentum of an object is calculated by the formula \( p = mv \), where \( m \) represents mass and \( v \) represents velocity. For electrons in this context:

• Mass, \( m = 9.11 \times 10^{-31} \, \mathrm{kg} \)
• Velocity, \( v = 8.4 \times 10^{7} \, \mathrm{m/s} \)

So, the momentum (\( p \)) of a single electron is calculated as \( 7.6524 \times 10^{-23} \, \mathrm{kg} \, \mathrm{m/s} \). This value captures the motion's energy, which is brought to zero when the electron stops at the screen's surface.Calculating momentum helps in understanding how electrons transfer energy when they collide with surfaces.

Pressure on Surfaces

Pressure is the action of force applied perpendicularly to a surface's area. In this example, electrons striking the TV screen lead to the exertion of pressure. When calculating pressure, Newton's second law comes into play since it relates force, mass, and acceleration. Pressure calculation involves:

• Calculating force from momentum change.
• Determining pressure as force over area.

In this scenario, the electrons emit a pressure of \( 4.744 \times 10^{-6} \, \mathrm{Pa} \) on the screen,

Force and Motion

Electrons hitting the screen demonstrate core principles of force and motion. According to Newton’s laws, an object's velocity will not change unless acted upon by an external force. When a force is applied to the electrons, they accelerate, prompting a change in velocity until they stop. This encounter converts momentum into force.
The force exerted in this case is \(4.744 \times 10^{-6} \, \mathrm{N} \). Understanding how force acts upon motion helps in simply grasping electron dynamics in varied contexts like televisions or other electron-based devices.

Electron Mass

The electron mass plays a significant role in dictating the behavior of electrons.Mass remains a constant in physics problems, acting as a foundational parameter in calculations like momentum.For an electron, this is \(9.11 \times 10^{-31} \, \mathrm{kg} \). Given its small value, when multiplied by the electron's velocity, it yields a small momentum.Understanding mass's influence in electron dynamics ensures accurate computational insight, as it's vital in energy transfer and force-related calculations.

Kinematics

Kinematics explores the description of motion without considering the forces that cause it. For electrons moving towards a screen, speed and trajectory describe their kinematic journey.
Understanding electron dynamics in kinematic terms involves:

• Analyzing velocity: the speed at which electrons move.
• Understanding cessation: electrons halting at the screen.

Kinematics gives a clear picture of motion that helps contextualize force and momentum without tackling underlying forces directly.

Pressure Calculation

Pressure calculation bridges force and surface area in electron dynamics. Combining these quantities offers insights into the impact experience on a surface. The formula for pressure is: \( P = \frac{F}{A} \).
In our case:

• The force from the electrons is \(4.744 \times 10^{-6} \mathrm{N} \).
• The area is \(1 \, \mathrm{m}^2\).

Thus, the pressure is resolved as \(4.744 \times 10^{-6} \, \mathrm{Pa} \).This calculation highlights how distributed force affects surfaces, important for understanding mechanical impacts in devices.

Newton's Laws

Newton's laws serve as the fundamental principles underlying electron dynamics.

• Newton's First Law states an object remains in uniform motion or at rest unless acted upon by a force.
• Newton's Second Law relates force, mass, and acceleration by the formula \( F = ma \).
• Newton's Third Law focuses on action-reaction forces counterbalancing interactions.

These laws help illuminate how electrons initiate movement, transfer energy to surfaces, and come to a halt, providing essential theories behind pressure calculation and momentum shifts in electronics.