Problem 9 - 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 \(4.89 \times 10^{14}\) electrons to each sphere; they will accelerate apart.

Step 1: Understanding the given values

First, understand the given quantities. We have two spheres, each with a mass of \(8.55\text{ g}\), separated by \(15.0\text{ cm}\). They will accelerate at \(25.0\, g\) when charged and released, where \(g\) is the acceleration due to gravity, approximately \(9.8\, \text{m/s}^2\). We need to know how many electrons are added to each sphere.

Step 2: Calculating the force required for acceleration

The required acceleration \(a\) is \(25.0 \times 9.8 = 245\, \text{m/s}^2\). The force \(F\) required for this acceleration using \(F = ma\) is \[F = 8.55\, \text{g} \times 245\, \text{m/s}^2 = 2.09675\, \text{N}\text{ (converted from grams to kilograms: } 8.55 \times 10^{-3}\, \text{kg).}\]

Step 3: Understanding the force between charged spheres

The force between two charged objects is given by Coulomb's law: \[F = k \frac{|q_1 q_2|}{r^2}\] where \(k\) is Coulomb's constant, \(8.99 \times 10^9\, \text{N}\cdot\text{m}^2/\text{C}^2\), and \(r = 0.15\, \text{m}\). Since the charges \(q_1\) and \(q_2\) are equal, \(q^2 = q_1 q_2\).

Step 4: Setting up the equation with charges

From Step 3, set \(q_1 = q_2 = q\) and rearrange Coulomb’s law to find \(q\): \[245 = 8.99 \times 10^9 \frac{q^2}{(0.15)^2}\] solving for \(q^2\), \[q^2 = \frac{245 \times (0.15)^2}{8.99 \times 10^9}\].

Step 5: Solving for the charge on each sphere

Calculate \(q^2\): \[q^2 = \frac{245 \times 0.0225}{8.99 \times 10^9} = 6.137 \times 10^{-9}\] thus, \(q = \sqrt{6.137 \times 10^{-9}} = 7.835 \times 10^{-5}\, \text{C}\).

Step 6: Determining the number of electrons

Find the charge of one electron \(e = 1.602 \times 10^{-19}\, \text{C}\). The number of electrons \(n\) needed for this charge is \[n = \frac{q}{e} = \frac{7.835 \times 10^{-5}}{1.602 \times 10^{-19}} = 4.89 \times 10^{14}\] electrons.

Step 7: Determining the direction of acceleration

Since the forces are due to electron charge, and like charges repel, the spheres will accelerate away from each other, along the line joining them.

Key Concepts

Charged Spheres

In physics, charged spheres are objects that are either positively or negatively charged. Here, our task involves two spheres, each weighing 8.55 grams, which are initially neutrally charged. When electrons are added to the spheres, they become negatively charged.
By adding electrons to both spheres equally, you create like charges. According to the basic principles of electromagnetism, like charges repel each other. This repulsion force results from the charges and is observed as the interaction between two electric fields.
In this exercise, the spheres are brought 15.0 cm apart, allowing the charged spheres to exert an electric force on one another. This force can lead to noteworthy acceleration of the spheres when they are released.

Coulomb's Constant

Coulomb's constant, also known as the electrostatic constant, plays a vital role in calculating the electrostatic force between two charges. It is represented as \( k \) and holds a constant value of approximately \(8.99 \times 10^9 \, \text{N} \cdot \text{m}^2 / \text{C}^2\). Coulomb's constant originates from Coulomb's Law, formulated by Charles-Augustin de Coulomb.
Using this constant, you can determine the force between two charged objects by inserting it into the formula:

• \(F = k \frac{|q_1 q_2|}{r^2}\)

In this problem, the constant helps us calculate the force needed to achieve the acceleration of the spheres. By setting up the equation with the charges and rearranging, we find the necessary charge on each sphere to cause the desired separation force.

Electron Charge

The electron charge is a fundamental property of electrons, which are subatomic particles carrying a negative charge. This charge is denoted by \( e \), and its known value is \( 1.602 \times 10^{-19} \, \text{C} \). Electrons are responsible for creating negative charges when added to particles like our charged spheres.
In this exercise, to achieve the required charge of \( 7.835 \times 10^{-5} \, \text{C} \) per sphere, you will determine how many electrons must be added. The process involves calculating the electron count using the formula:

• \( n = \frac{q}{e} \)

Here, \( n \) is the number of electrons. With the calculated charge \( q \), we find that approximately \( 4.89 \times 10^{14} \) electrons are needed. This allows each sphere to have an equal negative charge to cause the electrostatic repulsion.

Acceleration Due to Forces

Acceleration is a change in velocity caused by forces acting on an object. For this exercise, when the charged spheres are released, they accelerate at 25.0 times the standard gravitational acceleration \( g \). Standard \( g \) is roughly \( 9.8 \, \text{m/s}^2 \). Thus, the spheres' acceleration \( a \) is \( 25.0 \times g \), resulting in \( 245 \, \text{m/s}^2 \).
Forces between charged spheres arise due to electron repulsion. According to Newton's second law, \( F = ma \), the force \( F \) required can be calculated by multiplying mass \( m \) and acceleration \( a \). In the solution, this force equals approximately \( 2.09675 \, \text{N} \), setting the stage for the spheres' separation.
When released, the spheres will move away from each other because the repulsive force from like charges causes them to accelerate in opposite directions.