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Chapter 11: Problem 1580

Two equal charges \(q\) are placed at a distance of \(2 \mathrm{a}\) and a third charge \(-2 q\) is placed at the midpoint. The potential energy of the system is ....... (A) $\left[\left(9 \mathrm{q}^{2}\right) /\left(8 \pi \epsilon_{0} \mathrm{a}\right)\right]$ (B) \(\left[q^{2} /\left(8 \pi \epsilon_{0} a\right)\right]\) (C) \(-\left[\left(7 q^{2}\right) /\left(8 \pi \epsilon_{0} a\right)\right]\) (D) \(\left[\left(6 q^{2}\right) /\left(8 \pi \in_{0} a\right)\right]\)

Short Answer

Expert verified
The short answer to the question is: (C) \(-\left[\left(7 q^{2}\right) /\left(8 \pi \epsilon_{0} a\right)\right]\)

Step by step solution

01

Identify charges and their positions

Denote the equal charges as \(q_1 = q_2 = q\), and the third charge as \(q_3 = -2q\). Their positions can be represented as follows: - Charge \(q_1\) at position \((-a,0)\) - Charge \(q_2\) at position \((a,0)\) - Charge \(q_3\) at position \((0,0)\)
02

Calculate distances between charges

We will find the distances between each pair of charges: - Distance between \(q_1\) and \(q_3\): \(r_{13} = a\) - Distance between \(q_2\) and \(q_3\): \(r_{23} = a\) - Distance between \(q_1\) and \(q_2\): \(r_{12} = 2a\)
03

Calculate potential energy between each pair of charges

Using the formula for potential energy between two point charges, \(U = \frac{k*q_1*q_2}{r}\) (with \(k = \frac{1}{4 \pi \epsilon_0}\)), calculate the potential energy between each pair of charges: 1. Potential energy between \(q_1\) and \(q_3\): \( U_{13} = \frac{k*q_1*q_3}{r_{13}}= \frac{k*q*(-2q)}{a} =-\frac{2kq^2}{a}\) 2. Potential energy between \(q_2\) and \(q_3\): \( U_{23} = \frac{k*q_2*q_3}{r_{23}} = \frac{k*q*(-2q)}{a} =-\frac{2kq^2}{a}\) 3. Potential energy between \(q_1\) and \(q_2\): \(U_{12} = \frac{k*q_1*q_2}{r_{12}} = \frac{k*q*q}{2a} = \frac{kq^2}{2a}\)
04

Calculate total potential energy of the system

Now, we will sum up the potential energies of each pair of charges to find the total potential energy of the system: \(U_{total} = U_{13} + U_{23} + U_{12} = -\frac{2kq^2}{a} -\frac{2kq^2}{a} + \frac{kq^2}{2a}\) \(U_{total} = -\frac{8kq^2}{2a} + \frac{kq^2}{2a} = -\frac{7kq^2}{2a}\) Recall that \(k = \frac{1}{4 \pi \epsilon_0}\), substituting this value into the expression: \(U_{total} = -\frac{7\left[\frac{1}{4 \pi \epsilon_0}\right]q^2}{2a} = -\frac{7q^2}{8 \pi \epsilon_0 a}\)
05

Match the answer to the options given

Comparing our result to the given options, we find that the total potential energy of the system matches option (C): \(-\left[\left(7 q^{2}\right) /\left(8 \pi \epsilon_{0} a\right)\right]\)

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Most popular questions from this chapter

A sphere of radius \(1 \mathrm{~cm}\) has potential of \(8000 \mathrm{v}\), then energy density near its surface will be ...... (A) \(64 \times 10^{5}\left(\mathrm{~J} / \mathrm{m}^{3}\right)\) (B) \(2.83\left(\mathrm{~J} / \mathrm{m}^{3}\right)\) (C) \(8 \times 10^{3}\left(\mathrm{~J} / \mathrm{m}^{3}\right)\) (D) \(32\left(\mathrm{~J} / \mathrm{m}^{3}\right)\)

Charges of \(+(10 / 3) \times 10^{-9} \mathrm{C}\) are placed at each of the four corners of a square of side \(8 \mathrm{~cm}\). The potential at the intersection of the diagonals is ...... (A) \(150 \sqrt{2}\) Volt (B) \(900 \sqrt{2}\) Volt (C) \(1500 \sqrt{2}\) Volt (D) \(900 \sqrt{2} \cdot \sqrt{2}\) Volt

Two point charges \(100 \mu \mathrm{c}\) and \(5 \mu \mathrm{c}\) are placed at points \(\mathrm{A}\) and \(B\) respectively with \(A B=40 \mathrm{~cm}\). The work done by external force in displacing the charge \(5 \mu \mathrm{c}\) from \(\mathrm{B}\) to \(\mathrm{C}\) where \(\mathrm{BC}=30 \mathrm{~cm}\), angle \(\mathrm{ABC}=(\pi / 2)\) and \(\left[1 /\left(4 \pi \epsilon_{0}\right)\right]\) \(=9 \times 10^{9} \mathrm{Nm}^{2} / \mathrm{c}^{2}\). (A) \(9 \mathrm{~J}\) (B) \((9 / 25) \mathrm{J}\) (C) \((81 / 20) \mathrm{J}\) (D) \(-(9 / 4) \mathrm{J}\)

The displacement of a charge \(Q\) in the electric field $E^{-}=e_{1} i \wedge+e_{2} j \wedge+e_{3} k \wedge\( is \)r^{-}=a i \wedge+b j \wedge$ The work done is \(\ldots \ldots\) (A) \(Q\left(e_{1}+e_{2}\right) \sqrt{\left(a^{2}+b^{2}\right)}\) (B) \(Q\left[\sqrt{ \left.\left(e_{1}^{2}+e_{2}^{2}\right)\right](a+b)}\right.\) (C) \(Q\left(a e_{1}+b e_{2}\right)\) (D) \(\left.Q \sqrt{[}\left(a e_{1}\right)^{2}+\left(b e_{2}\right)^{2}\right]\)

Two identical metal plates are given positive charges \(\mathrm{Q}_{1}\) and \(\mathrm{Q}_{2}\left(<\mathrm{Q}_{1}\right)\) respectively. If they are now brought close to gather to form a parallel plate capacitor with capacitance \(\mathrm{c}\), the potential difference between them is (A) \(\left[\left(Q_{1}+Q_{2}\right) /(2 c)\right]\) (B) \(\left[\left(\mathrm{Q}_{1}+\mathrm{Q}_{2}\right) / \mathrm{c}\right]\) (C) \(\left[\left(\mathrm{Q}_{1}-\mathrm{Q}_{2}\right) /(2 \mathrm{c})\right]\) (D) \(\left[\left(\mathrm{Q}_{1}-\mathrm{Q}_{2}\right) / \mathrm{c}\right]\)
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