Problem 1 - Chemistry The Central Science In Si Units

Question

Two positively charged spheres, each with a charge of \(2.0 \times\) \(10^{-5} \mathrm{C},\) a mass of \(1.0 \mathrm{~kg},\) and separated by a distance of \(1.0 \mathrm{~cm},\) are held in place on a frictionless track. (a) What is the electrostatic potential energy of this system? (b) If the spheres are released, will they move toward or away from each other? (c) What speed will each sphere attain as the distance between the spheres approaches infinity?

Step-by-Step Solution

Answer

(a) The electrostatic potential energy is 35.96 J. (b) The spheres will move away from each other. (c) Each sphere will have a speed of 8.48 m/s as they separate to infinity.

Step 1: Identify the formula for electrostatic potential energy

The electrostatic potential energy between two point charges is given by the formula \( U = \frac{k \cdot q_1 \cdot q_2}{r} \), where \( k \) is Coulomb's constant \( 8.99 \times 10^9 \text{ N m}^2\text{/C}^2 \), \( q_1 \) and \( q_2 \) are the charges, and \( r \) is the separation distance between the charges.

Step 2: Apply values to the potential energy formula

Substitute the given values into the formula: \( q_1 = q_2 = 2.0 \times 10^{-5} \) C and \( r = 1.0 \times 10^{-2} \) m (converted from cm to meters). The potential energy \( U = \frac{8.99 \times 10^9 \cdot (2.0 \times 10^{-5})^2}{1.0 \times 10^{-2}} = 35.96 \) J.

Step 3: Determine the motion of charged spheres

Since both charges are positive, they repel each other. Therefore, when released on a frictionless track, the spheres will move away from each other.

Step 4: Apply conservation of energy to find the speed

Initially, all energy is potential energy. As they move away, potential energy converts to kinetic energy. The potential energy will be zero when spheres are infinitely apart. Apply energy conservation: \( 2 \cdot \frac{1}{2} m v^2 = U_i \) (both spheres have the same speed). Solving for \( v \) gives: \( v = \sqrt{\frac{2U_i}{m}} = \sqrt{\frac{2 \times 35.96}{1.0}} = 8.48 \text{ m/s} \) for each sphere.

Key Concepts

Coulomb's Law

Coulomb's Law is one of the fundamental principles in electrostatics. It quantifies the electrostatic force between two charges. Imagine you have two small charged spheres. The force between them depends on their charges and the distance separating them.
Coulomb's Law formula is:

• \( F = \frac{k \cdot q_1 \cdot q_2}{r^2} \)

- \( F \) is the force between the charges.- \( k \) is Coulomb’s constant, approximately \(8.99 \times 10^9 \text{ N m}^2\text{/C}^2\). - \( q_1 \) and \( q_2 \) are individual charges.- \( r \) is the distance between the charges.
Coulomb's Law tells us whether charged particles will attract or repel each other. Like charges repel, while opposite charges attract. This reflection in potential energy forms the basis for calculating the electrostatic potential energy using the formula \( U = \frac{k \cdot q_1 \cdot q_2}{r} \).
Understanding this principle helps to determine how charged spheres interact and respond when released on a frictionless track.

Conservation of Energy

The principle of Conservation of Energy states that energy cannot be created or destroyed. Rather, it can only change from one form to another. When two charged spheres are released, the energy transformation is evident. Initially, all energy present is electrostatic potential energy. As the spheres move apart, this energy shifts into kinetic energy.
Here's how it works for our charged spheres:

• Initially, potential energy \( U_i = 35.96 \) J.
• As the spheres move infinitely apart, \( U_f = 0 \) J.

The initial potential energy converts entirely to kinetic energy. Thus:

• \( 2 \cdot \frac{1}{2} m v^2 = U_i \)

Each sphere acquires speed from the conversion of potential to kinetic energy. By applying the Conservation of Energy, students can understand the dynamics of motion for charged particles set in such a configuration.
Recognizing energy's unchanging nature in totality simplifies such calculations.

Motion of Charged Particles

The Motion of Charged Particles is influenced by their interactions under electrostatic forces. With two positive charges, like in our example, the spheres repel each other. This repulsion results in both spheres moving away when released.
The principles guiding their motion include:

• Electrostatic repulsion due to like charges.
• Energy transformations, where potential energy transforms into kinetic energy.
• Frictionless surface allowing perfect energy conversion.

As charged particles accelerate apart, their velocity increases due to energy conservation. Both particles attain the same final speed of \( 8.48 \text{ m/s} \), calculated by:

• \( v = \sqrt{\frac{2 \times 35.96}{1.0}} \)

Through observing set examples, students gain insights into how charged particles behave under electrostatic conditions. The exercise highlights the predictable nature of electrostatic interactions and their resulting motion.