Question

A \(10.0-\mathrm{g}\) mass is suspended \(5.00 \mathrm{~cm}\) above a nonconducting flat plate, directly above an embedded charge of \(q\) (in coulombs). If the mass has the same charge, \(q\), how much must \(q\) be so that the mass levitates (just floats, neither rising nor falling)? If the charge \(q\) is produced by adding electrons to the mass, by how much will the mass be changed?

Short answer

Answer: The charge required for the mass to levitate is 1.042 × 10⁻⁵ C. The change in mass due to the addition of electrons is approximately 5.929 × 10⁻¹⁸ kg.

Step-by-step solution

Identify the forces acting on the mass

There are two forces acting on the mass: The gravitational force (weight), which is pulling the mass downward, and the electrostatic force due to the charge interaction between the mass and the embedded charge on the plate, which is pushing the mass upward. We will denote the gravitational force as F_g and the electrostatic force as F_e.

Calculate the gravitational force (F_g)

In order to find the gravitational force acting on the mass, we can use the formula: F_g = m × g where m is the mass (10.0 g) and g is the acceleration due to gravity (approximately 9.81 m/s²). First, we need to convert the mass from grams to kilograms: m = 10.0 g × (1 kg / 1000 g) = 0.01 kg Now, we can calculate F_g: F_g = 0.01 kg × 9.81 m/s² = 0.0981 N

Write the expression for the electrostatic force (F_e)

The electrostatic force between two charged objects can be calculated using Coulomb's law: F_e = \frac{k \cdot q^2}{r^2} where k is the electrostatic constant (approximately 8.99 × 10⁹ N·m²/C²), q is the charge on the mass and the embedded charge, and r is the distance between the mass and the embedded charge (5.00 cm). First, we need to convert the distance from centimeters to meters: r = 5.00 cm × (1 m / 100 cm) = 0.05 m

Set F_g equal to F_e and solve for q

Since the mass is levitating, the gravitational and electrostatic forces must be equal. Thus, we can write: F_g = F_e Substituting the expressions for F_g and F_e, we get: 0.0981 N = \frac{8.99 × 10⁹ N·m²/C² × q^2}{(0.05 m)^2} Now we can solve for q: q^2 = \frac{0.0981 N × (0.05 m)^2}{8.99 × 10⁹ N·m²/C²} = 1.086 \cdot 10^{-11} C² Taking the square root of both sides, we find: q = \sqrt{1.086 \cdot 10^{-11} C²} = 1.042 \cdot 10^{-5} C

Calculate the change in mass due to the addition of electrons

The charge q is produced by adding electrons to the mass. The charge of an electron is approximately -1.602 × 10⁻¹⁹ C. To find the number of electrons added, we can divide the charge q by the charge of an electron: n = \frac{1.042 \cdot 10^{-5} C}{-1.602 \cdot 10^{-19} C/electron} ≈ 6.506 \cdot 10¹³ electrons Each electron has a mass of approximately 9.109 × 10⁻³¹ kg. To find the change in mass, we can multiply the number of electrons by the mass of an electron: Δm = 6.506 \cdot 10¹³ electrons × 9.109 \cdot 10⁻³¹ kg/electron ≈ 5.929 \cdot 10^{-18} kg The mass will be changed by approximately 5.929 × 10⁻¹⁸ kg due to the addition of electrons.

Key concepts

Coulomb’s Law

At the heart of electrostatic levitation is Coulomb’s Law, a fundamental principle in physics. It describes the electrostatic force between two charged particles: the force is proportional to the product of the charges and inversely proportional to the square of the distance between them.

Mathematically, it is expressed as:
\[ F_e = \frac{k \cdot q^2}{r^2} \] where:\

• \( F_e \) is the electrostatic force,
• \( k \) is Coulomb's constant (about \( 8.99 \times 10^9 \) Nm²/C²),
• \( q \) represents the charges involved, and
• \( r \) is the distance separating them.

In the context of our exercise, this law explains how two objects with the same charge can repel each other. By finding the precise charge needed, we can balance this repulsive force with the force of gravity, allowing levitation to occur.

Gravitational Force

Whereas Coulomb's Law gives us insight into the workings of electric charges, gravitational force is a measure of attraction between two masses. The universally experienced force is governed by the formula:
\[ F_g = m \times g \] where:

• \( F_g \) is the gravitational force,
• \( m \) is the mass of the object, and
• \( g \) is the gravitational acceleration (approximately \( 9.81 \) m/s² on Earth).

In our levitation scenario, the gravitational downward pull must be precisely offset by the upward electrostatic force. This balance is what allows the object to 'float' mid-air. Understanding and computing this delicate equilibrium is key to solving problems involving gravitational and electrostatic competences.

Charge Quantization

Charge quantization is a less widely known concept but equally fundamental. It refers to the fact that charge comes in discrete 'packets' - the smallest being the charge of a single electron, approximately \( -1.602 \times 10^{-19} \) coulombs.

In practice, this means that any charge is an integer multiple of this elementary charge. For our levitating mass, the required charge to achieve equilibrium is achieved by adding or removing electrons.

When we calculate how many electrons need to be added to a particle to achieve a specific charge, we are applying the concept of charge quantization. In our example, introducing electrons changes the mass negligibly (by about \( 5.929 \times 10^{-18} \) kg), illustrating how the quantization of charge is essential to understanding physical changes in electrostatic experiments.