Question

How far must two electrons be placed on the Earth's surface for there to be an electrostatic force between them equal to the weight of one of the electrons?

Short answer

Answer: The two electrons should be placed approximately \(5.47 * 10^{-11}\) meters apart.

Step-by-step solution

Write down the given information

The charge of an electron (e): \(1.6 * 10^{-19}\) C The mass of an electron (m): \(9.11 * 10^{-31}\) kg Coulomb's constant (k): \(8.988 * 10^{9} Nm^2/C^2\) Acceleration due to gravity (g): \(9.81 m/s^2\)

Compute the weight of one electron

To find the weight of one electron, we use the gravitational force formula: Weight (W) = mass (m) × acceleration due to gravity (g) \(W = m \times g\)

Write the formula for the electrostatic force between two electrons

The electrostatic force between two electrons with charges q1 and q2 at a distance r apart is given by the formula: \(F = k \frac{q1 * q2}{r^2}\) In our case, q1 = q2 = charge of an electron (e).

Set the weight equal to the electrostatic force and solve for r

We want to find the distance (r) between the electrons when the weight of one electron equals the electrostatic force between them: \(W = F\) \(m \times g = k \frac{e^2}{r^2}\) Now, let's solve for r: \(r^2 = k \frac{e^2}{m \times g}\) \(r = \sqrt{k \frac{e^2}{m \times g}}\) Now substitute the values of k, e, m, and g into the equation and calculate r: \(r = \sqrt{ (8.988 * 10^{9}) \frac{(1.6 * 10^{-19})^2}{(9.11 * 10^{-31}) \times (9.81)}}\) \(r \approx 5.47 * 10^{-11} m\) The two electrons must be placed approximately \(5.47 * 10^{-11}\) meters apart on the Earth's surface for the electrostatic force between them to be equal to the weight of one of the electrons.

Key concepts

Coulomb's law

Coulomb's law is a fundamental principle in electrostatics that describes how the electrostatic force acts between two charged particles. This force can either be attractive or repulsive, depending on the nature of the charges involved. According to Coulomb's law, the electrostatic force (\( F \) is calculated using the formula:

• \( F = k \frac{q1 * q2}{r^2} \)

Here, \( k \) is Coulomb's constant, which has a value of \( 8.988 \times 10^{9} \, \text{Nm}^2/\text{C}^2 \). \( q1 \) and \( q2 \) represent the magnitudes of the charges, while \( r \) is the distance separating the two charges.

This law is essential to understand how charged particles interact in various scenarios, whether it be two electrons as in the given exercise or other charged bodies in fields like electronics, chemistry, and even in biological systems.

Gravitational force

Gravitational force is a type of force that acts between two masses. It is always attractive and is described using Newton's law of universal gravitation. The gravitational force on an object at Earth's surface is commonly referred to as its weight, calculated as:

• Weight (W) = mass (m) × acceleration due to gravity (g)

Here, \( g \) is the acceleration due to gravity, which is approximately \( 9.81 \, \text{m/s}^2 \) on Earth.

For example, the weight of a particle like an electron can be determined using its mass and combining it with the gravitational acceleration to get the force it experiences due to gravity. Even though electrons are incredibly small and light (\( m = 9.11 \times 10^{-31} \, \text{kg} \)), they are still subject to gravitational force.

Electron charge

An electron possesses a fundamental negative charge, commonly expressed in units of Coulombs. This charge is critical in numerous physics problems involving electric forces, as it influences how electrons interact with other charged particles and fields. The charge of an electron is:

• \( e = 1.6 \times 10^{-19} \text{C} \)

This minuscule charge keeps every electron engaging with other particles, leading to various chemical and physical processes such as bonding in atoms or current flow in circuits.

In the exercise at hand, knowing the precise value of the electron charge allows for the accurate calculation of electrostatic forces using Coulomb's law, underlining the significance of the electron charge as a fundamental constant in physics.

Physics problem-solving

Solving physics problems often requires a systematic approach to identify and apply the correct principles. Here are some steps to incorporate into effective problem-solving:

• Identify and write down all given quantities, like mass, charge, and constants.
• Define what is being asked or what you need to find out.
• Choose the appropriate principles or laws. In this case, using both gravitational force formula and Coulomb's law.
• Set up equations based on these principles. For this exercise, equate weight to electrostatic force.
• Solve algebraically or computationally for the unknowns, substituting known values correctly.
• Check units for consistency and ensure that the solution makes physical sense.

By following these steps thoroughly and methodically, students can tackle complex physics problems with greater confidence and accuracy. The problem of finding the distance between two electrons where gravitational weight equals electrostatic force illustrates the application of these steps clearly.