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Chapter 11: Problem 1598

If 3 charges are placed at the vertices of equilateral triangle of charge ' \(q\) ' each. What is the net potential energy, if the side of equilateral triangle is \(\ell \mathrm{cm}\). (A) $\left[1 /\left(4 \pi \epsilon_{0}\right)\right]\left(3 q^{2} / \ell\right)$ (B) $\left[1 /\left(4 \pi \epsilon_{0}\right)\right]\left(2 q^{2} / \ell\right)$ (C) \(\left[1 /\left(4 \pi \epsilon_{0}\right)\right]\left(q^{2} / \ell\right)\) (D) $\left[1 /\left(4 \pi \epsilon_{0}\right)\right]\left(4 q^{2} / \ell\right)$

Short Answer

Expert verified
The net potential energy of three charges placed at the vertices of an equilateral triangle with side length \(\ell\) and charge \(q\) each is given by \(\frac{3 q^{2}}{(4\pi\epsilon_0\ell)}\), and the correct answer is option (A) \(\left[1/\(4 \pi \epsilon_{0}\)\right]\left(3 q^{2}/\ell\right)\).

Step by step solution

01

Understand the system of charges

There are 3 charges of magnitude 'q' placed at the vertices of an equilateral triangle. This means there are three pairs of charges whose interactions contribute to the total potential energy. Remember the formula for the potential energy between two charges: \(U = \frac{(k*q1*q2)}{r}\).
02

Calculate the potential energy between each pair of charges

The potential energy for each pair of charges is the same since all charges have the same magnitude and the distances between them are equal (since it's an equilateral triangle). The potential energy between any two charges is \(U = \frac{(k*q*q)}{\ell}\), where the constant \(k = 1/\(4 \pi \epsilon_{0}\) is Coulomb's constant.
03

Sum up the potential energy from all pairs

Since there are three pairs of charges, simply multiply the result from the previous step by 3. Therefore, the total potential energy is \(3*U = 3*\frac{(k*q*q)}{\ell} = \frac{(3 * k* q^{2})}{\ell}\).
04

Replace and solve

Substituting the value of \(k = 1/\(4 \pi \epsilon_{0}\), the total energy becomes \(\frac{3 q^{2}}{(4\pi\epsilon_0\ell)}\). Therefore, the correct answer is option (A) \(\left[1/\(4 \pi \epsilon_{0}\)\right]\left(3 q^{2}/\ell\right)\).

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Most popular questions from this chapter

Iwo small charged spheres repel each other with a force $2 \times 10^{-3} \mathrm{~N}$. The charge on one sphere is twice that of the other. When these two spheres displaced \(10 \mathrm{~cm}\) further apart the force is $5 \times 10^{-4} \mathrm{~N}$, then the charges on both the spheres are....... (A) \(1.6 \times 10^{-9} \mathrm{C}, 3.2 \times 10^{-9} \mathrm{C}\) (B) \(3.4 \times 10^{-9} \mathrm{C}, 11.56 \times 10^{-9} \mathrm{C}\) (C) \(33.33 \times 10^{9} \mathrm{C}, 66.66 \times 10^{-9} \mathrm{C}\) (D) \(2.1 \times 10^{-9} \mathrm{C}, 4.41 \times 10^{-9} \mathrm{C}\)

Large number of capacitors of rating $10 \mu \mathrm{F} / 200 \mathrm{~V} \mathrm{~V}$ are available. The minimum number of capacitors required to design a \(10 \mu \mathrm{F} / 700 \mathrm{~V}\) capacitor is (A) 16 (B) 8 (C) 4 (D) 7

The plates of a parallel capacitor are charged up to \(100 \mathrm{~V}\). If $2 \mathrm{~mm}$ thick plate is inserted between the plates, then to maintain the same potential difference, the distance between the capacitor plates is increased by \(1.6 \mathrm{~mm}\) the dielectric constant of the plate is (A) 5 (B) 4 (C) \(1.25\) (D) \(2.5\)

A simple pendulum consists of a small sphere of mass \(\mathrm{m}\) suspended by a thread of length \(\ell\). The sphere carries a positive charge q. The pendulum is placed in a uniform electric field of strength \(\mathrm{E}\) directed Vertically upwards. If the electrostatic force acting on the sphere is less than gravitational force the period of pendulum is (A) $\mathrm{T}=2 \pi[\ell /\\{\mathrm{g}-(\mathrm{q} \mathrm{E} / \mathrm{m})\\}]^{(1 / 2)}$ (B) \(\mathrm{T}=2 \pi(\ell / \mathrm{g})^{(1 / 2)}\) \(\left.\left.\left.\mathrm{m}_{\mathrm{}}\right\\}\right\\}\right]^{(1 / 2)}\) (D) \(\mathrm{T}=2 \pi[(\mathrm{m} \ell / \mathrm{qE})]^{(1 / 2)}\) (C) \(\mathrm{T}=2 \pi[\ell /\\{\mathrm{g}+(\mathrm{qE} / \mathrm{t}\)

Two Points \(P\) and \(Q\) are maintained at the Potentials of \(10 \mathrm{v}\) and \(-4 \mathrm{v}\), respectively. The work done in moving 100 electrons from \(\mathrm{P}\) to \(\mathrm{Q}\) is \(\ldots \ldots \ldots\) (A) \(2.24 \times 10^{-16} \mathrm{~J}\) (B) \(-9.60 \times 10^{-17} \mathrm{~J}\) (C) \(-2.24 \times 10^{-16} \mathrm{~J}\) (D) \(9.60 \times 10^{-17} \mathrm{~J}\)
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