Problem 60 - Physics

Question

In a vacuum, two particles have charges of \(q_{1}\) and \(q_{2}\), where \(q_{1}=+3.5 \mu \mathrm{C}\). They are separated by a distance of \(0.26 \mathrm{~m}\), and particle 1 experiences an attractive force of \(3.4 \mathrm{~N}\). What is \(q_{2}\) (magnitude and sign)?

Step-by-Step Solution

Answer

\(q_2 = -2.88 \, \mu\text{C}\), negative sign.

Step 1: Identify Given and Required Information

We are given the charge of the first particle \(q_1 = +3.5 \, \mu\text{C}\), the distance between the particles \(r = 0.26 \, \text{m}\), and the force acting on particle 1 is \(F = 3.4 \, \text{N}\). We need to determine the charge \(q_2\).

Step 2: Recall Coulomb's Law

Coulomb's Law states that the force \(F\) between two charges is given by the formula: \[ F = k \frac{|q_1 q_2|}{r^2} \]where \(k = 8.99 \times 10^9 \, \text{N m}^2/\text{C}^2\) is Coulomb's constant.

Step 3: Rearrange Coulomb's Law to Solve for \(q_2\)

Rearrange the formula to solve for \(q_2\):\[ |q_2| = \frac{F \cdot r^2}{k \cdot |q_1|} \]

Step 4: Substitute Known Values

Substitute \(F = 3.4 \, \text{N}\), \(r = 0.26 \, \text{m}\), and \(|q_1| = 3.5 \times 10^{-6} \, \text{C}\) into the formula:\[ |q_2| = \frac{3.4 \cdot (0.26)^2}{8.99 \times 10^9 \cdot 3.5 \times 10^{-6}} \]

Step 5: Calculate \(q_2\) and Determine its Sign

Calculate the value:\[ |q_2| \approx \frac{3.4 \cdot 0.0676}{8.99 \times 10^9 \cdot 3.5 \times 10^{-6}} \approx 2.88 \times 10^{-6} \, \text{C} \]Since the force is attractive, the charges must be opposite in sign. Therefore, \(q_2\) is negative, \(q_2 = -2.88 \, \mu\text{C}\).

Key Concepts

Electric Force

The electric force is a fundamental interaction between charged objects. It plays a critical role in our everyday lives, though often operating at scales unseen by the naked eye. According to Coulomb's Law, the magnitude of this force depends on:

• The amount of charge on each object.
• The distance between the objects.
• A constant known as Coulomb’s constant, denoted as \( k \).

This law is described mathematically as follows: \[F = k \frac{|q_1 q_2|}{r^2} \]Where:

• \( F \) is the magnitude of the force.
• \( q_1 \) and \( q_2 \) are the magnitudes of the charges.
• \( r \) is the separation distance between the charges.
• \( k \) is a constant (\( 8.99 \times 10^9 \, \text{N m}^2/\text{C}^2\)).

The electric force can be attractive or repulsive, depending on the signs of the interacting charges. Understanding electric force is crucial for explaining how atoms stay together, how electricity flows, and many other phenomena.

Charge Interactions

Charge interactions are key to understanding the behavior of electric forces between particles. Simply put:

• Like charges repel each other.
• Unlike charges attract each other.

This concept is essential when predicting the motion and behavior of charged particles in various environments. In the specific scenario from the exercise, the interaction leads to an attractive force. What does this indicate? It implies that the two charges involved are opposite in nature, one positive and one negative. This attraction results in the electric force calculated in the problem. The interaction can be visualized as a tug-of-war, where opposite charges pull on each other, trying to get closer. Understanding these interactions helps clarify how electric circuits function and how charge distribution in substances can lead to static electricity.

Electrostatics

Electrostatics is the study of electric charges at rest. Within this realm, forces emerge due to the presence of charged bodies rather than their motion. As such, it encompasses phenomena such as:

• Charging by friction, where electrons are transferred by rubbing two different materials.
• Induction, where charges are rearranged in a material without direct contact.
• Polarization, where the positive and negative charges within an object are aligned, usually in the presence of an external charged object.

By examining electrostatics, we understand how charges behave in isolation or when influenced by other charges. The scenario in the exercise presents static charges that exert a force on each other but aren't moving relative to each other. The principles of electrostatics are instrumental not just in simple experiments but also in technologies like capacitors, which store energy in electronic devices.

Electric Charges

Electric charges are the building blocks of electromagnetism. They come in two types: positive and negative. Typically, protons carry a positive charge, while electrons carry a negative charge. Neutrons are neutral and don't affect the charge balance.The exercise deals with charges measured in microcoulombs (\( \mu\text{C}\)), a convenient unit for small quantities of charge often encountered in problems.Key properties of electric charges include:

• Charge Conservation: Total charge remains constant in an isolated system.
• Quantization: Charge exists in discrete amounts, typically multiples of the elementary charge \( e \).
• Interaction: Opposite charges attract, while like charges repel.

These attributes are fundamental to numerous applications, from designing atomic models to creating sophisticated sensors and devices. In our exercise, knowing the sign and magnitude of each charge helps unravel the nature of their interaction and the resulting force dynamics.