Problem 19 - Physics

Question

A charge of \(-3.00 \mu \mathrm{C}\) is fixed in place. From a horizontal distance of \(0.0450 \mathrm{~m}, \mathrm{a}\) particle of mass \(7.20 \times 10^{-3} \mathrm{~kg}\) and charge \(-8.00 \mu \mathrm{C}\) is fired with an initial speed of \(65.0 \mathrm{~m} / \mathrm{s}\) directly toward the fixed charge. How far does the particle travel before its speed is zero?

Step-by-Step Solution

Answer

The particle travels a distance of approximately 0.234 m before stopping.

Step 1: Identify the Forces

Two charges of the same sign will exert a repulsive force on each other. Calculate the electrostatic force using Coulomb's Law, where \( F = k \frac{|q_1 q_2|}{r^2} \). Here, \( q_1 = -3.00 \, \mu \mathrm{C} \) and \( q_2 = -8.00 \, \mu \mathrm{C} \), converted to Coulombs as \( q_1 = -3.00 \times 10^{-6} \, \mathrm{C} \) and \( q_2 = -8.00 \times 10^{-6} \, \mathrm{C} \). The constant \( k \) is \( 8.99 \times 10^9 \, \mathrm{N} \cdot \mathrm{m}^2/\mathrm{C}^2 \). The initial distance \( r = 0.0450 \, \mathrm{m} \). Substitute these values into Coulomb's Law to find the initial force.

Step 2: Calculate Initial Kinetic Energy

The initial kinetic energy \( KE_i \) of the moving charge is given by \( KE_i = \frac{1}{2}mv^2 \), where \( m \) is the mass \( 7.20 \times 10^{-3} \, \mathrm{kg} \) and \( v \) is the initial velocity \( 65.0 \, \mathrm{m/s} \). Calculate \( KE_i \) using these values.

Step 3: Determine Electric Potential Energy Change

The work done by the electrostatic force as the charges repel is equal to the change in electric potential energy. The potential energy \( U \) between two point charges is given by \( U = k \frac{|q_1q_2|}{r} \). Calculate the difference in potential energy as the distance changes from 0.0450 m to the final stopping point.

Step 4: Use Energy Conservation

Apply the principle of conservation of energy: the total mechanical energy initially is equal to the total mechanical energy at the stopping point. The initial kinetic energy is converted into electric potential energy at the stopping point, where kinetic energy becomes zero. Set \( KE_i = \Delta U \), where \( \Delta U = U_f - U_i \). Solve for the final distance \( r_f \).

Step 5: Solve for Stopping Distance

Using the conservation equation, substitute the known values and solve for the final distance \( r_f \) where the kinetic energy is zero. Rearrange the expression to find \( r_f = \frac{k |q_1 q_2|}{KE_i} + r_i \). Calculate \( r_f \) with the calculated values from previous steps.

Key Concepts

Coulomb's Law

Coulomb's Law is fundamental in understanding how charged particles interact. It describes the force between two point charges and can be used to determine the magnitude of the electrostatic force. The formula is \[ F = k \frac{|q_1 q_2|}{r^2} \]where:

• \( F \) is the force in Newtons (N),
• \( q_1 \) and \( q_2 \) are the magnitudes of the charges in Coulombs (C),
• \( r \) is the distance between the charges in meters (m), and
• \( k \) is the electrostatic constant, approximately \( 8.99 \times 10^9 \ \mathrm{N} \cdot \mathrm{m}^2/\mathrm{C}^2 \).

For two like charges, such as the ones in the problem, the force is repulsive. The closer they are, the stronger the force. This formula helps determine the initial forces that act on charged particles as they move closer or further apart. Understanding this concept is essential for solving problems involving electric charges and can help predict how charged objects will behave.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. It's a critical concept in physics and helps us understand how this energy can transform in a system. The formula for kinetic energy is \[ KE = \frac{1}{2}mv^2 \]where:

• \( KE \) is the kinetic energy in Joules (J),
• \( m \) is the mass of the object in kilograms (kg), and
• \( v \) is the velocity of the object in meters per second (m/s).

In our exercise, the particle has initial kinetic energy because it is moving initially. The importance of kinetic energy lies in its ability to change as it transforms into other energy forms like potential energy. As the particle travels toward the fixed charge, this kinetic energy is gradually converted into potential energy until the particle comes to a stop.

Potential Energy

Potential energy in the context of electrostatics refers to the stored energy due to the position of charged particles relative to each other. The potential energy between two point charges can be calculated using the formula \[ U = k \frac{|q_1 q_2|}{r} \]where:

• \( U \) is the electric potential energy in Joules (J),
• \( q_1 \) and \( q_2 \) are the charges in Coulombs (C),
• \( r \) is the distance between the charges in meters (m), and
• \( k \) is the electrostatic constant.

As the particle in our scenario moves toward the fixed charge, the potential energy increases. Eventually, the particle uses up all its kinetic energy, and its speed becomes zero. The total mechanical energy is conserved, which includes both kinetic and potential energy.

Conservation of Energy

The principle of conservation of energy is a powerful tool in physics, stating that in a closed system, the total energy remains constant over time. In other words, energy cannot be created or destroyed, only transformed.In our problem, this means that the initial kinetic energy of the moving charge and the electric potential energy between the charges equals the final electric potential energy since the kinetic energy becomes zero at the stopping point. The conservation of energy equation can be written as\[ KE_i = \Delta U \]where:

• \( KE_i \) is the initial kinetic energy, and
• \( \Delta U \) is the change in potential energy as the charged particle moves.

This relation allows us to determine the point at which the moving charge stops. Solving this equation gives us the distance traveled by the particle. This reflects the pivotal role of energy conservation in analyzing and solving physics problems.