Problem 6 - University Physics with Modern Physics

Question

Two small spheres spaced 20.0 \(\mathrm{cm}\) 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 \(4.57 \times 10^{-21} \mathrm{N} ?\)

Step-by-Step Solution

Answer

Each sphere must have 893 excess electrons.

Step 1: Understand Coulomb’s Law

Coulomb's Law calculates the force between two charged objects. The formula is \[ F = k \frac{|q_1 q_2|}{r^2} \]where:- \( F \) is the force between the charges,- \( q_1 \) and \( q_2 \) are the charges,- \( r \) is the distance between the centers of the two spheres, - \( k \) is Coulomb's constant, \( 8.99 \times 10^9 \, \mathrm{Nm^2/C^2} \). Here, \( F = 4.57 \times 10^{-21} \, \mathrm{N} \) and \( r = 0.2 \, \mathrm{m} \). Since the spheres have equal charge, let \( q_1 = q_2 = q \).

Step 2: Solve for Charge q

Substitute the known values into Coulomb's Law:\[4.57 \times 10^{-21} = 8.99 \times 10^9 \frac{q^2}{(0.2)^2}\]Simplify and solve for \( q^2 \):\[q^2 = \left(4.57 \times 10^{-21}\right) \cdot (0.2)^2 / 8.99 \times 10^9 \]Calculate:\[q^2 \approx 2.03 \times 10^{-31}\]Take the square root:\[q \approx \sqrt{2.03 \times 10^{-31}} \approx 1.43 \times 10^{-16} \, \mathrm{C}\]

Step 3: Calculate Excess Electrons

The charge of one electron is \( e = 1.602 \times 10^{-19} \, \mathrm{C} \). To find the number of excess electrons \( n \), use:\[n = \frac{q}{e} = \frac{1.43 \times 10^{-16}}{1.602 \times 10^{-19}}\]Calculate:\[n \approx 8.93 \times 10^2\]Since the number of electrons must be a whole number, round \( 8.93 \times 10^2 \) to the nearest whole number: \( n = 893 \).

Key Concepts

Electric Charge

Electric charge is a fundamental property of matter that enables particles to experience a force when placed in an electromagnetic field. Every charged particle carries a certain quantity of charge, measured in Coulombs (C). There are two types of electric charges: positive and negative.

Protons possess a positive charge, while electrons carry a negative charge. The interaction of these charges is what leads to electric forces acting between objects. The unit of electric charge is derived from the charge of an electron, with one electron having a charge of approximately \( 1.602 \times 10^{-19} \text{ C} \).

In scenarios where two objects have the same type of charge, either both positive or both negative, they repel each other. Conversely, opposite charges attract. This property is essential in understanding phenomena like static electricity and electric forces.

Force of Repulsion

The force of repulsion is the push that occurs when two like-charged objects encounter each other. In this context, when two small spheres with like charges are located near each other, they tend to push away from each other due to the electric force.

Coulomb's Law is used to calculate the magnitude of this force. According to Coulomb's Law, the repulsive force \( F \) 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 expression is given as:

• \( F = k \frac{|q_1 q_2|}{r^2} \)

Here, \( k \) is the Coulomb's constant, \( q_1 \) and \( q_2 \) are the charges, and \( r \) is the distance between them.

Excess Electrons

Excess electrons on an object indicate that an object has gained additional electrons making it negatively charged. In the original exercise, we determined the number of excess electrons needed to result in a known level of repulsive force between two charged spheres.

The process involves first calculating the total charge \( q \) on each sphere using Coulomb's Law. Once \( q \) is determined, the number of excess electrons can be found by dividing the total charge by the charge of a single electron:

• \( n = \frac{q}{e} \)
• where \( e \) is the charge of an electron, approximately \( 1.602 \times 10^{-19} \text{ C} \).

This calculation reveals how many extra electrons are present to create the specified repulsive force.

Coulomb's Constant

Coulomb's constant \( (k) \) is a key figure used in Coulomb's Law and reflects how electric forces scale in the formula. Its value is approximately \( 8.99 \times 10^9 \text{ Nm}^2/ ext{C}^2 \).

This constant helps quantify the force between two point charges in a vacuum. It is a reflection of the interaction strength of electric charges at a specific distance in an unimpeded medium, and it plays a critical role in predicting how strongly charged objects will attract or repel each other in space.

By applying Coulomb's constant, we can determine how much repulsion or attraction force will be experienced by objects depending on their charge and distance. Understanding this allows for precise calculations in both academic and practical applications.