Problem 11 - University Physics with Modern Physics

Question

Two very small \(8.55-\) g spheres, 15.0 \(\mathrm{cm}\) apart from center to center, are charged by adding equal numbers of electrons to each of them. Disregarding all other forces, how many electrons would you have to add to each sphere so that the two spheres will accelerate at 25.0 \(\mathrm{g}\) when released? Which way will they accelerate?

Step-by-Step Solution

Answer

Add \(4.53 \times 10^{13}\) electrons to each sphere; they accelerate towards each other.

Step 1: Understand the Forces Involved

The spheres are influenced by electric and gravitational forces. However, the problem states to disregard all other forces except the electric force, which is responsible for the acceleration. So, we should focus on the electric force, given by Coulomb's law: \( F = \frac{k \cdot |q_1 \cdot q_2|}{r^2} \), where \( k \) is Coulomb's constant \(8.99 \times 10^9 \mathrm{N} \cdot \mathrm{m}^2/\mathrm{C}^2\).

Step 2: Relate Electric Force to Mass and Acceleration

We need the electric force to provide the same acceleration as gravity at 25 times the strength of Earth's gravity. Therefore, the equation can be expressed as \( F = m \cdot a = m \cdot 25g \), where \( g = 9.81 \mathrm{m/s^2} \).

Step 3: Calculate the Required Force

The mass of each sphere is 8.55 g or 0.00855 kg. The required acceleration is \( a = 25 \times 9.81 \mathrm{m/s^2} = 245.25 \mathrm{m/s^2} \). Therefore, the force on each sphere is: \( F = 0.00855 \mathrm{kg} \times 245.25 \mathrm{m/s^2} = 2.09665375 \mathrm{N} \).

Step 4: Apply Coulomb's Law

Using Coulomb's law, substitute the force from Step 3: \( F = \frac{k \cdot |q_1 \cdot q_2|}{r^2} \). Since the charge on each sphere is equal, we substitute \( q_1 = q_2 = q \). Thus, \( F = \frac{k \cdot q^2}{r^2} \). Given \( r = 0.15 \mathrm{m} \), solve for \( q \): \(\frac{k \cdot q^2}{(0.15)^2} = 2.09665375 \),\( q^2 = \frac{2.09665375 \times (0.15)^2}{k} \).

Step 5: Solve for Total Charges

Substituting \( k = 8.99 \times 10^9 \mathrm{N} \cdot \mathrm{m}^2/\mathrm{C}^2\), find \( q \): \( q^2 = \frac{2.09665375 \times 0.0225}{8.99 \times 10^9} \), \( q^2 = \frac{0.047175 \cdot 10^{-9}}{8.99} \), \( q = \sqrt{5.246 \times 10^{-12}} = 7.24 \times 10^{-6} \mathrm{C} \).

Step 6: Calculate the Number of Electrons

The charge of a single electron is \( 1.6 \times 10^{-19} \mathrm{C} \). The number of electrons needed is: \( \frac{7.24 \times 10^{-6} \mathrm{C}}{1.6 \times 10^{-19} \mathrm{C/electron}} = 4.53 \times 10^{13} \mathrm{electrons} \).

Step 7: Direction of Acceleration

Both spheres will accelerate towards each other because they have opposite charges induced by adding electrons. The electric force is attractive since like charges attract each other under the given conditions.

Key Concepts

Coulomb's Law

Coulomb's Law is a fundamental principle in physics, describing how charged particles interact. When two charged objects are placed in proximity, they exert a force on each other. This force can be either attractive or repulsive, depending on the nature of the charges. The mathematical expression for Coulomb's Law is:
\[ F = \frac{k \cdot |q_1 \cdot q_2|}{r^2} \]
where:

• \( F \) is the magnitude of the force between the charges.
• \( k \) is Coulomb's constant, approximately \(8.99 \times 10^9 \mathrm{N \cdot m^2/C^2}\).
• \( q_1 \) and \( q_2 \) are the values of the charges.
• \( r \) is the distance between the centers of the two charges.

Coulomb's Law highlights how the force is inversely proportional to the square of the distance between the charges. This means if the distance between the charges increases, the force decreases non-linearly. Understanding this principle helps in solving problems involving electric forces, like calculating the force between two spheres in the exercise.

Acceleration

Acceleration refers to the rate of change of velocity of an object. It is a vector quantity, meaning it has both magnitude and direction. In the exercise, the spheres need to accelerate towards each other with a specific acceleration, which is 25 times the acceleration due to Earth's gravity.
Mathematically, acceleration \( a \) is connected to force \( F \) and mass \( m \) by the equation:
\[ a = \frac{F}{m} \]
This means that for a constant mass, the acceleration of an object is directly proportional to the force applied to it. In the context of the spheres, the electric force calculated using Coulomb’s law must provide the needed acceleration to match \( 25g \), where \( g = 9.81 \mathrm{m/s^2} \). Therefore, understanding acceleration is key to ensuring the right amount of force is achieved in the situation depicted by the exercise.

Charge of Electrons

The charge of an electron is an essential constant in physics, with a value of \( -1.6 \times 10^{-19} \mathrm{C} \). It is a fundamental property that makes up the negative charge of electrons. In scenarios involving charged particles, knowing the charge of an electron helps in understanding how charges add or subtract to produce a net charge.
Adding electrons to the spheres in the exercise results in each acquiring a negative charge due to the electron's charge. This situation involves calculating the number of electrons needed to achieve a total charge that produces the required force for the calculated acceleration.

• When electrons are added, they increase the negative charge of the spheres.
• The total charge can be calculated by multiplying the number of electrons by the charge per electron.

As in the exercise, knowing the precise charge helps determine how many electrons need to be added to achieve a specific force that meets the given conditions.

Newton's Second Law

Newton's Second Law of Motion is a pivotal concept in dynamics, relating force, mass, and acceleration. It states that the force acting on an object is equal to the mass of that object multiplied by its acceleration:
\[ F = m \cdot a \]
This law is key to understanding how forces affect motion. It means that for a given mass, the greater the force applied, the more the object will accelerate, and vice versa. In the context of the exercise, once the electric force is determined, Newton's Second Law can be applied to figure out the resulting acceleration of the spheres.
Here is how it becomes crucial:

• It helps relate the electric force calculated from Coulomb’s Law to the acceleration required (\( 25g \)).
• Knowing the mass of the spheres allows calculation of this force precisely.
• Thus, enabling prediction of the motion behavior, as outlined in the problem.

Newton's Second Law offers the framework to systematically analyze and solve for the kinematic properties, ensuring a comprehensive understanding of motion dynamics in this context.