Problem 63 - Fundamentals of Physics

Question

Two point charges of \(30 \mathrm{nC}\) and \(-40 \mathrm{nC}\) are held fixed on an \(x\) axis, at the origin and at \(x=72 \mathrm{~cm},\) respectively. A particle with a charge of \(42 \mu \mathrm{C}\) is released from rest at \(x=28 \mathrm{~cm} .\) If the initial acceleration of the particle has a magnitude of \(100 \mathrm{~km} / \mathrm{s}^{2},\) what is the particle's mass?

Step-by-Step Solution

Answer

The particle's mass is approximately 0.048 kg.

Step 1: Understanding Coulomb's Law

Coulomb's law gives the force between two point charges:\[F = k \frac{|q_1 q_2|}{r^2}\]where \(k\) is the Coulomb's constant \(8.99 \times 10^9 \text{ Nm}^2/\text{C}^2\), \(q_1\) and \(q_2\) are the charges, and \(r\) is the distance between them. We will apply this formula to calculate the forces exerted on the particle by each fixed charge.

Step 2: Calculate the Force from Each Charge

The particle at \(x = 28 \text{ cm} = 0.28 \text{ m}\) feels forces from the charges at the origin (\(q_1 = 30 \text{ nC} = 30 \times 10^{-9} \text{ C}\)) and at \(x = 72 \text{ cm} = 0.72 \text{ m}\) (\(q_2 = -40 \text{ nC}\)). Calculate the force:**Force due to 30 nC at the origin:**\[ r_1 = 0.28 \text{ m}, \, q_1 = 30 \times 10^{-9} \text{ C} \]\[ F_1 = k \frac{|42 \times 10^{-6} \times 30 \times 10^{-9}|}{0.28^2} \]**Force due to -40 nC at 0.72 m:**\[ r_2 = 0.72 - 0.28 = 0.44 \text{ m}, \, q_2 = 40 \times 10^{-9} \text{ C} \]\[ F_2 = k \frac{|42 \times 10^{-6} \times 40 \times 10^{-9}|}{0.44^2} \]

Step 3: Determine the Net Force

Since the particle is between the positive charge at the origin and the negative charge at \(x = 0.72 \text{ m}\), the forces are in opposite directions. Calculate the net force:\[ F_\text{net} = F_1 - F_2 \] (assuming the force due to the positive charge is in the positive direction)

Step 4: Calculate the Mass from Acceleration

We know from Newton's Second Law:\[ F = ma \]Given that the initial acceleration \(a = 100,000 \text{ m/s}^2\), we can solve for the mass \(m\):\[ m = \frac{F_\text{net}}{a} \]

Step 5: Final Calculation

Substitute the values from previous steps into the formula to find \(m\). Simplify the expressions to obtain the numerical value for the mass of the particle.

Key Concepts

Electrostatics

Electrostatics is a branch of physics that focuses on the study of electric charges at rest. It deals with the forces, fields, and potentials associated with charges that do not move. In electrostatics, the fundamental concept is the electric force, which is the force exerted by a charge on another charge. This force can be either repulsive or attractive depending on whether the charges are like or opposite. Understanding electrostatics is crucial when calculating forces between stationary charges.

Coulomb's law is a principal law used in electrostatics. It states that the magnitude of the electric force between two point charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between them. The formula for Coulomb's law is:

• \[ F = k \frac{|q_1 q_2|}{r^2} \]

Here,

• \( F \) is the force between the charges,
• \( k \) is Coulomb's constant, approximately \( 8.99 \times 10^9 \, \text{Nm}^2/\text{C}^2 \),
• \( q_1 \) and \( q_2 \) are the charges,
• \( r \) is the distance between the charges.

This law allows us to calculate the forces acting on a charge when it is located near other charges. It is essential to consider that this force can impact the position, motion, and acceleration of the charge.

Point Charge

The concept of a point charge is fundamental in electrostatics. A point charge is an idealized model of a particle that carries a charge concentrated at a single point in space. This simplification is vital in theoretical physics because it allows for easier mathematical dealing with electrostatic interactions.

Point charges are often used in calculations to predict the behavior of actual charges in various physical situations. For instance, in the given exercise, both the stationary and the moving charges are considered as point charges, which allows for simple application of Coulomb's Law to analyze their interactions. These point charges create electric fields in space, which influence the forces acting on other charges within these fields.

The superposition principle is another key concept when dealing with point charges. It states that the total force exerted by multiple point charges on a test charge is the vector sum of individual forces exerted by each charge. This principle is critical as it simplifies the analysis of systems with multiple charges by allowing us to focus on one pair of interactions at a time.

Newton's Second Law

Newton's Second Law of Motion is a cornerstone in classical mechanics that relates the motion of an object to the forces acting upon it. It states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This principle provides the link between force and motion for dynamics analysis.

The equation form of Newton's Second Law is:

• \[ F = ma \]

Where:

• \( F \) is the net force acting on an object,
• \( m \) is the mass of the object,
• \( a \) is the acceleration of the object.

In the exercise, by applying Newton's Second Law, we can relate the net force determined from the electrostatic forces acting on the particle to find its mass, given the known initial acceleration. This method brilliantly simplifies solving the problem by bridging electrostatics with dynamics, helping students intuitively understand how forces translate into motion in physical systems.