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Chapter 2: Problem 21

Two identical uniform line charges, with \(\rho_{l}=75 \mathrm{nC} / \mathrm{m}\), are located in free space at \(x=0, y=\pm 0.4 \mathrm{~m}\). What force per unit length does each line charge exert on the other?

Short Answer

Expert verified
Question: Determine the force per unit length exerted by one line charge on the other in a given configuration of two identical uniform line charges with linear charge density \(\rho_l = 75 \,\text{nC/m}\) located at \((0, -0.4 \,\text{m})\) and \((0, 0.4 \,\text{m})\). Answer: \(F_{\text{unit length}} = \frac{F}{0.8\,\text{m}}\), where \(F = \int_{-0.4\,\text{m}}^{0.4\,\text{m}} \frac{k \rho_l^2 dy^2}{(\sqrt{(0.4 \,\text{m} - (-0.4 \,\text{m}))^2 + y^2})^2}\)

Step by step solution

01

Understand the initial situation

We have two identical uniform line charges with linear charge density \(\rho_l = 75 \,\text{nC/m}\) located at \((0, -0.4 \,\text{m})\) and \((0, 0.4 \,\text{m})\). Let's name them line charge A and B respectively. To find the force per unit length exerted by each line charge on the other, we will first calculate the force exerted on a small segment of line charge A by a small segment of line charge B.
02

Define the infinitesimal charges

Now let's consider an infinitesimal segment of length \(dy\) at a distance \(y\) from the origin on both line charges. On line charge A, we can represent the infinitesimal charge \(dq_A = \rho_l dysin(\theta)\). Similarly, on line charge B, we can represent the infinitesimal charge \(dq_B = \rho_l dy\).
03

Coulomb's law

According to Coulomb's law, the force between two charges is given by \(F = \frac{kq_1q_2}{r^2}\), where \(k = \frac{1}{4\pi\epsilon_0}\) is the electrostatic constant and \(r\) is the distance between the charges. Therefore, the force between the infinitesimal charges \(dq_A\) and \(dq_B\) can be expressed as: \(dF = \frac{k dq_A dq_B}{r^2} = \frac{k \rho_l^2 dy^2}{r^2}\)
04

Determine the distance between infinitesimal charges

To find the force per unit length, we need to calculate the distance between the infinitesimal charges \(dq_A\) and \(dq_B\). We can use the Pythagorean theorem to find this distance: \(r = \sqrt{(0.4 \,\text{m} - (-0.4 \,\text{m}))^2 + y^2}\)
05

The force exerted on an infinitesimal charge of line charge A

Now we can integrate the force expression \(dF\) over the entire length of the line charge to find the total force acting on line charge A. To do this, we can simply integrate by substituting the values of \(r\) and the infinitesimal charges in our force formula. Then, we will integrate the force expression with respect to \(y\): \(F = \int_{-0.4\,\text{m}}^{0.4\,\text{m}} \frac{k \rho_l^2 dy^2}{(\sqrt{(0.4 \,\text{m} - (-0.4 \,\text{m}))^2 + y^2})^2}\)
06

Find the force per unit length

Since the line charges are identical, both will exert equal force on each other. Moreover, we are interested in finding the force per unit length, so we need to divide the result of the integral by the distance between the line charges which is \(0.8\,\text{m}\). \(F_{\text{unit length}} = \frac{F}{0.8\,\text{m}}\) Now we can simply solve the integral and find the force per unit length exerted by one line charge on the other.

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Most popular questions from this chapter

A charge \(Q_{0}\) located at the origin in free space produces a field for which \(E_{z}=1 \mathrm{kV} / \mathrm{m}\) at point \(P(-2,1,-1) .(a)\) Find \(Q_{0} .\) Find \(\mathbf{E}\) at \(M(1,6,5)\) in (b) rectangular coordinates; ( \(c\) ) cylindrical coordinates; \((d)\) spherical coordinates.

(a) Find \(\mathbf{E}\) in the plane \(z=0\) that is produced by a uniform line charge, \(\rho_{L}\), extending along the \(z\) axis over the range \(-L

A line charge of uniform charge density \(\rho_{0} \mathrm{C} / \mathrm{m}\) and of length \(\ell\) is oriented along the \(z\) axis at \(-\ell / 2

Eight identical point charges of \(Q \mathrm{C}\) each are located at the corners of a cube of side length \(a\), with one charge at the origin, and with the three nearest charges at \((a, 0,0),(0, a, 0)\), and \((0,0, a)\). Find an expression for the total vector force on the charge at \(P(a, a, a)\), assuming free space.

Within a region of free space, charge density is given as \(\rho_{v}=\frac{\rho_{v} r \cos \theta}{a} \mathrm{C} / \mathrm{m}^{3}\), where \(\rho_{0}\) and \(a\) are constants. Find the total charge lying within \((a)\) the sphere, \(r \leq a ;(b)\) the cone, \(r \leq a, 0 \leq \theta \leq 0.1 \pi ;(c)\) the region, \(r \leq a\) \(0 \leq \theta \leq 0.1 \pi, 0 \leq \phi \leq 0.2 \pi\)
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