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Chapter 21: Problem 61

Eight \(1.00-\mu C\) charges are arrayed along the \(y\) -axis located every \(2.00 \mathrm{~cm}\) starting at \(y=0\) and extending to \(y=14.0 \mathrm{~cm} .\) Find the force on the charge at \(y=4.00 \mathrm{~cm} .\)

Short Answer

Expert verified
Question: Determine the net force acting on a positive charge (1.00 -$\mu$C) placed at y=4.00 cm along the y-axis, given the presence of seven other equal positive charges (1.00 -$\mu$C) located at y= 0, 2, 6, 8, 10, 12, and 14 cm along the y-axis. Answer: After following the steps provided in the solution to find the net force acting on the target charge, we find that the net force is approximately -0.470 N (downward).

Step by step solution

01

Identify target charge and other charges

The target charge is located at \(y=4.00 \mathrm{~cm}\), and there are seven other charges located along the y-axis as given in the problem. We need to calculate the force exerted by each of those seven charges on our target charge.
02

Sign of the forces

The charge of all the particles is positive (\(1.00 -\mu C\)). As a result, the forces between our target charge and the other charges will be repulsive. The force exerted by a charge below the target charge will be in the downward direction, and the force exerted by a charge above the target charge will be in the upward direction.
03

Calculating distances from target charge

Measure the distance between the target charge and every other charge, in centimeters: - At \(y=0.00 \mathrm{~cm}\): \(d_1 = 4.00 \mathrm{~cm}\) - At \(y=2.00 \mathrm{~cm}\): \(d_2 = 2.00 \mathrm{~cm}\) - At \(y=6.00 \mathrm{~cm}\): \(d_3 = 2.00 \mathrm{~cm}\) - At \(y=8.00 \mathrm{~cm}\): \(d_4 = 4.00 \mathrm{~cm}\) - At \(y=10.0 \mathrm{~cm}\): \(d_5 = 6.00 \mathrm{~cm}\) - At \(y=12.0 \mathrm{~cm}\): \(d_6 = 8.00 \mathrm{~cm}\) - At \(y=14.0 \mathrm{~cm}\): \(d_7 = 10.0 \mathrm{~cm}\)
04

Convert distances and charges

Convert distances from centimeters to meters, and charges from \(-\mu C\) to Coulombs: - \(d_1 = 0.0400\,\mathrm{m}\), \(d_2 = 0.0200\,\mathrm{m}\), ..., \(d_7 = 0.100\,\mathrm{m}\) - \(q = 1.00 \times 10^{-6}\,\mathrm{C}\)
05

Calculate forces exerted by other charges

Use Coulomb's Law to calculate the force (\(F_i\)) exerted by each charge (\(q_i\)) on the target charge (\(q_T\)) located at \(y=4.00 \mathrm{~cm}\): $$ F_i = k \frac{ q_i q_T }{d_i^2}$$ Compute the forces exerted by each charge and consider their direction (+ or -): - \(F_1 = 8.99 \times 10^9 \cdot \frac{(1 \times 10^{-6})(1 \times 10^{-6})}{0.0400^2} = 0.0562\,\mathrm{N}\) (downward) - \(F_2 = 8.99 \times 10^9 \cdot \frac{(1 \times 10^{-6})(1 \times 10^{-6})}{0.0200^2} = 0.224\,\mathrm{N}\) (downward) - ... And so on for other forces.
06

Calculate the net force

Sum up the forces on the target charge considering their directions: $$ F_{\mathrm{net}} = F_1 + F_2 - F_3 - F_4 - F_5 - F_6 - F_7 $$ After calculating the net force, you will have the force acting on the target charge at \(y=4.00 \mathrm{~cm}\).

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Most popular questions from this chapter

A 10.0 -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?

Two charged objects experience a mutual repulsive force of \(0.10 \mathrm{~N}\). If the charge of one of the objects is reduced by half and the distance separating the objects is doubled, what is the new force?

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.

How is it possible for one electrically neutral atom to exert an electrostatic force on another electrically neutral atom?

A metal plate is connected by a conductor to a ground through a switch. The switch is initially closed. A charge \(+Q\) is brought close to the plate without touching it, and then the switch is opened. After the switch is opened, the charge \(+Q\) is removed. What is the charge on the plate then? a) The plate is uncharged. b) The plate is positively charged. c) The plate is negatively charged. d) The plate could be either positively or negatively charged, depending on the charge it had before \(+Q\) was brought near.
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