Problem 6 - University Physics with Modern Physics

Question

Two small spheres spaced 20.0 cm apart have equal charge. How many excess electrons must be present on each sphere if the magnitude of the force of repulsion between them is 3.33 \(\times\) \(10^{-21}\) N?

Step-by-Step Solution

Answer

Each sphere must have about 537 excess electrons.

Step 1: Understand Coulomb's Law

Coulomb's Law explains the force between two charged objects. It is given by:\[ F = k \frac{|q_1 q_2|}{r^2} \]where \( F \) is the force, \( k \) is the Coulomb's constant \( 8.988 \times 10^9 \text{ N m}^2/\text{C}^2 \), \( q_1 \) and \( q_2 \) are the charges, and \( r \) is the distance between the charges. Here, \( q_1 = q_2 = q \) and \( r = 0.2 \text{ meters} \).

Step 2: Solve for Charge

Rearrange Coulomb's Law to solve for the charge \( q \):\[ q = \sqrt{\frac{F r^2}{k}} \]Substitute the given \( F = 3.33 \times 10^{-21} \text{ N} \), \( r = 0.2 \text{ m} \), and \( k = 8.988 \times 10^9 \text{ N m}^2/\text{C}^2 \):\[ q = \sqrt{\frac{3.33 \times 10^{-21} \times (0.2)^2}{8.988 \times 10^9}} \approx \sqrt{7.396 \times 10^{-33}} \approx 8.6 \times 10^{-17} \text{ C} \]

Step 3: Determine Number of Electrons

The charge of one electron is approximately \( 1.602 \times 10^{-19} \text{ C} \). To find the number of excess electrons \( n \), use:\[ n = \frac{q}{e} \]where \( e \) is the elementary charge. Therefore,\[ n = \frac{8.6 \times 10^{-17}}{1.602 \times 10^{-19}} \approx 537 \]

Step 4: Interpret the Result

Thus, each sphere must have approximately 537 excess electrons to produce the given force of repulsion.

Key Concepts

Electric Charge

Electric charge is a fundamental property of matter that causes it to experience a force when placed in an electromagnetic field. In simple terms, it's what makes particles attract or repel each other. There are two types of electric charges: positive and negative. These are often carried by protons and electrons, respectively. Charges of the same type repel, while opposite charges attract. This is easy to see in daily life, such as when tiny pieces of paper cling to a charged balloon.
Understanding electric charge is crucial in the context of Coulomb's Law, as it determines how charged objects interact. Coulomb's Law quantifies the force between two charges, allowing us to calculate how much they attract or repel.
In our example, each sphere has an equal and like charge, meaning they will repel each other. The amount of excess charge, in terms of electrons, helps us understand the magnitude of this repulsive force.

Force of Repulsion

The force of repulsion occurs when two objects with similar charges push away from each other. This is described by Coulomb's Law, which states that the force between two charges is directly proportional to the product of their magnitudes and inversely proportional to the square of the distance between them. Simply put, if you increase the charge of either object or decrease the distance between them, the force of repulsion becomes stronger.

• The equation for this is: \[ F = k \frac{|q_1 q_2|}{r^2} \] where \( F \) is the force, \( q_1 \) and \( q_2 \) are the charges, \( r \) is the distance, and \( k \) is Coulomb's constant.
• In this exercise, we calculated the force of repulsion to be \( 3.33 \times 10^{-21} \text{ N} \).
• This force pushes the two spheres apart because they both carry the same type of charges (either positive or negative).

Understanding the force of repulsion is crucial for interpreting interactions between charged particles, especially in fields like physics and chemistry.

Elementary Charge

An elementary charge is the smallest unit of electric charge that is considered indivisible in classical physics. It is carried by a single proton or electron, where protons are positive and electrons are negative. The value of an elementary charge is exactly \( 1.602 \times 10^{-19} \text{ C} \).This tiny yet incredibly fundamental constant forms the basis for understanding various physical phenomena, such as how much charge is needed to balance forces in an atom or a molecule.In our example, we calculated how many electrons, which each have one negative elementary charge, would be required to account for the total charge causing the force of repulsion.

• Given a total charge of \( 8.6 \times 10^{-17} \text{ C} \) per sphere, dividing by the elementary charge of an electron shows how many electrons are needed.
• The result was approximately 537 excess electrons per sphere, which gives insight into the detailed workings of charge and force in this problem.

Understanding the concept of an elementary charge is essential in fields like chemistry and physics, where the behavior of charged particles is a fundamental aspect.