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Chapter 21: Problem 36

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

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

Step by step solution

01

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): \(9.81 m/s^2\)
02

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) \(W = m \times g\)
03

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).
04

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.

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Most popular questions from this chapter

If two charged particles (the charge on each is \(Q\) ) are separated by a distance \(d\), there is a force \(F\) between them. What is the force if the magnitude of each charge is doubled and the distance between them changes to \(2 d ?\)

A point charge \(q_{1}=100 . \mathrm{nC}\) is at the origin of an \(x y\) -coordinate system, a point charge \(q_{2}=-80.0 \mathrm{nC}\) is on the \(x\) -axis at \(x=2.00 \mathrm{~m},\) and a point charge \(q_{3}=-60.0 \mathrm{nC}\) is on the \(y\) -axis at \(y=-2.00 \mathrm{~m} .\) Determine the net force (magnitude and direction) on \(q_{1}\).

Suppose the Earth and the Moon carried positive charges of equal magnitude. How large would the charge need to be to produce an electrostatic repulsion equal to \(1.00 \%\) of the gravitational attraction between the two bodies?

How many electrons does \(1.00 \mathrm{~kg}\) of water contain?

In the Bohr model of the hydrogen atom, the electron moves around the one- proton nucleus on circular orbits of well-determined radii, given by \(r_{n}=n^{2} a_{\mathrm{B}}\), where \(n=1,2,3, \ldots\) is an integer that defines the orbit and \(a_{\mathrm{B}}=5.29 \cdot 10^{-11} \mathrm{~m}\) is the radius of the first (minimum) orbit, called the Bohr radius. Calculate the force of electrostatic interaction between the electron and the proton in the hydrogen atom for the first four orbits. Compare the strength of this interaction to the gravitational interaction between the proton and the electron.
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