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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

The capacities of three capacitors are in the ratio \(1: 2: 3\). Their equivalent capacity when connected in parallel is \((60 / 11) \mathrm{F}\) more then that when they are connected in series. The individual capacitors are of capacities in \(\mu \mathrm{F}\) (A) \(4,6,7\) (B) \(1,2,3\) (C) \(1,3,6\) (D) \(2,3,4\)

There are 10 condensers each of capacity \(5 \mu \mathrm{F}\). The ratio between maximum and minimum capacities obtained from these condensers will be (A) \(40: 1\) (B) \(25: 5\) (C) \(60: 3\) (D) \(100: 1\)

Two thin wire rings each having a radius \(R\) are placed at a distance \(d\) apart with their axes coinciding. The charges on the two rings are \(+q\) and \(-q\). The potential difference between the centers of the two rings is $\ldots .$ (A) 0 (B) $\left.\left[\mathrm{q} /\left(2 \pi \epsilon_{0}\right)\right]\left[(1 / \mathrm{R})-\left\\{1 / \sqrt{(}^{2}+\mathrm{d}^{2}\right)\right\\}\right]$ (C) $\left[\mathrm{q} /\left(4 \pi \epsilon_{0}\right)\right]\left[(1 / \mathrm{R})-\left\\{1 / \sqrt{\left. \left.\left(\mathrm{R}^{2}+\mathrm{d}^{2}\right)\right\\}\right]}\right.\right.$ (D) \(\left[(q R) /\left(4 \pi \epsilon_{0} d^{2}\right)\right]\)

Two air capacitors \(A=1 \mu F, B=4 \mu F\) are connected in series with $35 \mathrm{~V}\( source. When a medium of dielectric constant \)\mathrm{K}=3$ is introduced between the plates of \(\mathrm{A}\), change on the capacitor changes by (A) \(16 \mu \mathrm{c}\) (B) \(32 \mu \mathrm{c}\) (C) \(28 \mu \mathrm{c}\) (D) \(60 \mu \mathrm{c}\)

Let $\mathrm{P}(\mathrm{r})\left[\mathrm{Q} /\left(\pi \mathrm{R}^{4}\right)\right] \mathrm{r}$ be the charge density distribution for a solid sphere of radius \(\mathrm{R}\) and total charge \(\mathrm{Q}\). For a point ' \(\mathrm{P}\) ' inside the sphere at distance \(\mathrm{r}_{1}\) from the centre of the sphere the magnitude of electric field is (A) $\left[\mathrm{Q} /\left(4 \pi \epsilon_{0} \mathrm{r}_{1}^{2}\right)\right]$ (B) $\left[\left(\mathrm{Qr}_{1}^{2}\right) /\left(4 \pi \in{ }_{0} \mathrm{R}^{4}\right)\right]$ (C) $\left[\left(\mathrm{Qr}_{1}^{2}\right) /\left(3 \pi \epsilon_{0} \mathrm{R}^{4}\right)\right]$
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