Question

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

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

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: \(U = \frac{(k*q1*q2)}{r}\).

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.

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 \(3*U = 3*\frac{(k*q*q)}{\ell} = \frac{(3 * k* q^{2})}{\ell}\).

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)\).

Key concepts

Coulomb's Law

Coulomb's Law is a fundamental principle of physics that describes the interaction between two electric charges. Imagine you have two small electrically charged particles. Coulomb's Law tells us how much force one charge exerts on the other. This force depends on:

• The magnitudes of the charges (\(q_1\) and \(q_2\))
• The distance between the charges (\(r\))

The formula is: \[ F = k \frac{q_1 q_2}{r^2} \]where:

• \(F\) is the force between the charges
• \(k\) is Coulomb's constant \( (\approx 8.99 \times 10^9 \text{ N m}^2/\text{C}^2) \)
• \(q_1\) and \(q_2\) are the magnitudes of the charges
• \(r\) is the distance between the centers of the two charges

In our exercise, we're dealing with potential energy instead of force, which still relies on the same principles of distance and charge magnitude.

Equilateral Triangle

An equilateral triangle is a special type of triangle where all three sides are equal in length. Because all sides and angles are equal, it has unique symmetry properties.
In the context of our exercise, this equality in sides means that any calculations involving distances between points can be much simpler. Since the side lengths are \(\ell\) cm for each, it helps make calculations straightforward because every pair of charges is separated by the same distance.
Here's what’s beneficial about using an equilateral triangle when three charges are involved:

• The distance in any pair of charges remains constant.
• Symmetry simplifies physics calculations since the layout is regular and predictable.

Each pair of charges contributes equally to the total potential energy because the conditions are perfectly identical for each pair.

Electrical Potential Energy

Electrical potential energy is the energy stored in the system of charged particles due to their positions relative to each other. When you have charges, they can interact with each other through electric forces, and this interaction involves potential energy.
The formula for electrical potential energy (\(U\)) of two point charges is:\[ U = k \frac{q_1 q_2}{r} \]where:

• \(U\) is the potential energy.
• \(k\) is Coulomb’s constant.
• \(q_1\) and \(q_2\) are the charges.
• \(r\) is the distance between the charges.

In an equilateral triangle configuration with three charges at each vertex:- Each pair of charges contributes equally to the total potential energy.- You need to calculate the potential energy for each pair first, and then sum them up for the entire system.
This ensures a straightforward and efficient way to compute total potential energy for multiple interacting charges.