Question

Consider a system where n point charges, 91, 92,..., In, are placed on the circum-ference of a ring of radius R. A point charge Q is placed at the center of the ring. Find the total electrostatic potential energy of the system.

Answer 1

The total electrostatic potential energy of the system is given by:

$\frac{1}{4\pi\epsilon_0} \left( \frac{Q}{R} \sum_{i=1}^{n} q_i + \sum_{1 \le i < j \le n} \frac{q_i q_j}{r_{ij}} \right)$

where:

• $Q$ is the point charge at the center of the ring.
• $q_i$ are the point charges on the circumference of the ring.
• $R$ is the radius of the ring.
• $r_{ij}$ is the distance between the point charges $q_i$ and $q_j$ on the circumference.
• $\epsilon_0$ is the permittivity of free space.

Answer 2

The total electrostatic potential energy of a system of point charges is the sum of the potential energies of all distinct pairs of charges. For a system of N charges, this is given by $U = \sum_{1 \le i < j \le N} \frac{k q_i q_j}{r_{ij}}$, where $k = \frac{1}{4\pi\epsilon_0}$ is Coulomb's constant, $q_i$ and $q_j$ are the magnitudes of the charges, and $r_{ij}$ is the distance between them.

In this system, there are $n+1$ charges: $Q$ at the center and $q_1, q_2, \ldots, q_n$ on the circumference of a ring of radius R.

We can split the sum over all pairs into two parts:

1. Pairs consisting of the central charge $Q$ and one of the charges $q_i$ on the ring.
2. Pairs consisting of two charges $q_i$ and $q_j$ on the ring.

For the pairs involving the central charge $Q$:

The distance between the charge $Q$ at the center and any charge $q_i$ on the circumference is equal to the radius of the ring, $R$. The potential energy for the pair $(Q, q_i)$ is $U_{Q, q_i} = \frac{k Q q_i}{R}$. There are $n$ such pairs: $(Q, q_1), (Q, q_2), \ldots, (Q, q_n)$. The total potential energy from these pairs is the sum: $U_{Q, ring} = \sum_{i=1}^{n} U_{Q, q_i} = \sum_{i=1}^{n} \frac{k Q q_i}{R} = \frac{k Q}{R} \sum_{i=1}^{n} q_i$.

For the pairs involving two charges on the ring, $q_i$ and $q_j$ ($i \neq j$):

These charges are located on the circumference. Let $r_{ij}$ be the distance between the charge $q_i$ and the charge $q_j$ on the circumference. This distance depends on the specific positions of $q_i$ and $q_j$ on the ring. The potential energy for the pair $(q_i, q_j)$ is $U_{q_i, q_j} = \frac{k q_i q_j}{r_{ij}}$. We need to sum this over all distinct pairs $(i, j)$ where $i$ and $j$ range from 1 to $n$ and $i < j$. The total potential energy from these pairs is: $U_{ring} = \sum_{1 \le i < j \le n} U_{q_i, q_j} = \sum_{1 \le i < j \le n} \frac{k q_i q_j}{r_{ij}}$. The distance $r_{ij}$ is the chord length between the positions of $q_i$ and $q_j$ on the circle. If the angular positions of $q_i$ and $q_j$ are $\theta_i$ and $\theta_j$, then $r_{ij} = 2R \sin\left(\frac{|\theta_i - \theta_j|}{2}\right)$. However, without knowing the specific positions, we must leave $r_{ij}$ as the distance between $q_i$ and $q_j$.

The total electrostatic potential energy of the system is the sum of the potential energies of all these pairs: $U_{total} = U_{Q, ring} + U_{ring}$ $U_{total} = \frac{k Q}{R} \sum_{i=1}^{n} q_i + \sum_{1 \le i < j \le n} \frac{k q_i q_j}{r_{ij}}$.

Here, $k = \frac{1}{4\pi\epsilon_0}$, and $r_{ij}$ is the distance between the point charges $q_i$ and $q_j$ on the circumference of the ring.