Question

Your sister wants to participate in the yearly science fair at her high school and asks you to suggest some exciting project. You suggest that she experiment with your recently created electron extractor to suspend her cat in the air. You tell her to buy a copper plate and bolt it to the ceiling in her room and then use your electron extractor to transfer electrons from the plate to the cat. If the cat weighs $7.00\mathrm{kg}$ and is suspended $2.00\mathrm{m}$ below the ceiling, how many electrons have to be extracted from the cat? Assume that the cat and the metal plate are point charges.

Short answer

Answer: Approximately 3.45 × 10^14 electrons need to be extracted from the cat to suspend it in the air.

Step-by-step solution

Calculate the gravitational force acting on the cat

The gravitational force (Fg) acting on the cat can be calculated using the formula: Fg = m * g Where m is the mass of the cat (7.00 kg) and g is the acceleration due to gravity (approximated to 9.81 m/s^2). Fg = 7.00 kg * 9.81 m/s^2 = 68.67 N

Determine the necessary electrostatic force to suspend the cat

We need the electrostatic force (Fe) to be equal to the gravitational force to keep the cat suspended in the air. So, Fe = Fg = 68.67 N

Calculate the charge on the cat

We can use Coulomb's Law to relate the electrostatic force with the charge on the cat (q). The formula for Coulomb's Law is: Fe = k * (q1 * q2) / r^2 Where k is the electrostatic constant (approximated to 8.99 × 10^9 N m^2/C^2), q1 and q2 are the charges of the cat and the copper plate (in Coulombs), and r is the distance between the cat and the plate (2.00 m). Since the cat and the copper plate have the same charge, we can rewrite the equation as: Fe = k * (q^2) / r^2 Now we can solve for q: q^2 = (Fe * r^2) / k q = sqrt((68.67 N * (2.00 m)^2) / (8.99 × 10^9 N m^2/C^2)) q = sqrt(274.68 / 8.99 × 10^9) q ≈ 5.52 × 10^{-5} C

Calculate the number of electrons extracted

The charge q represents the total charge transferred from the cat. Since we know that the charge of one electron is approximately 1.6 × 10^{-19} C, we can find the number of electrons (n) extracted by dividing the total charge by the charge of one electron: n = q / (charge of one electron) n = 5.52 × 10^{-5} C / 1.6 × 10^{-19} C n ≈ 3.45 × 10^{14} So approximately 3.45 × 10^{14} electrons must be extracted from the cat to suspend it in the air.

Key concepts

Coulomb's Law

Coulomb's Law is a fundamental principle in physics that describes how electrically charged objects interact with each other. It explains the electrostatic force between two charged objects. In its simplest form, the law states:$$F_e=k\frac{q_1{q}_2}{r^2}$$where:

• $F_e$ is the electrostatic force in newtons (N),
• $k$ is Coulomb's constant $(8.99\times {10}^9{\text{N m}}^2/{\text{C}}^2)$,
• $q_1$ and $q_2$ are the charges of the two objects in coulombs (C), and
• $r$ is the distance between the centers of the two charges in meters (m).

Coulomb's Law is similar to the gravitational force in that it acts over a distance and depends on a constant, which varies with the medium between the charges. However, unlike gravity, which is always attractive, the electrostatic force can be either attractive or repulsive depending on the nature of the charges.

Gravitational force

Gravitational force is the attractive force that exists between any two masses. It is one of the four fundamental forces of nature. The gravitational force acting on an object near the Earth's surface can be calculated using:$$F_g=m\cdot g$$where:

• $F_g$ is the gravitational force in newtons (N),
• $m$ is the mass of the object in kilograms (kg), and
• $g$ is the acceleration due to gravity, approximately $9.81{\text{m/s}}^2$ near Earth's surface.

In the original exercise, the gravitational force on a cat weighing 7 kg is 68.67 N, calculated using the formula. This force must be balanced by an equal electrostatic force to suspend the cat in air, demonstrating how gravitational and electrostatic forces can interact in a physics experiment.

Electron charge

The electron charge is a fundamental property of electrons, which are subatomic particles found in atoms. Each electron carries a negative charge denoted by $e$, approximately $-1.6\times {10}^{-19}\text{C}$. This charge is a key factor in electric phenomena such as conductivity and the electrostatic interactions described by Coulomb's Law.In any physics experiment involving charges, knowing the charge of an electron allows us to calculate the quantity of charge in terms of the number of electrons. For instance, in the exercise scenario, the charge needed to counteract the gravitational force and suspend the cat determines how many electrons are involved in creating the electrostatic force. By dividing the total charge by the charge per electron, we can find out the number of electrons transferred. This principle is crucial in demonstrating the scale and impact of electric charges in everyday and scientific contexts.

Physics experiment

Physics experiments are vital in illustrating abstract principles in understandable ways. They often involve setting up scenarios where variables can be controlled and outcomes measured accurately. In the exercise, a creative physics experiment proposes using a copper plate and an electron extractor to levitate a cat by transferring electrons and creating an electrostatic force to oppose gravity.

Key Steps in Designing a Physics Experiment:

• Start with a hypothesis: Define what you are trying to prove or demonstrate, such as using electrostatic force to balance gravitational force.
• Outline the procedure: Detail every step, including materials and methods. Here, it involves using a copper plate and determining electron transfer.
• Data collection: Record measurements with precision, as calculating forces and charges requires accuracy.
• Analysis: Use mathematical formulas to interpret results, like using Coulomb's Law to calculate necessary electronic charges.

Thus, physics experiments not only bring theoretical principles to life but also enhance our understanding of the natural world through practical demonstration and analysis.