Question

Three point charges +q, –2q and +q are placed along x-axis at points x = –1 m, x = 0 m and x = 2 m respectively. The potential energy of the system is

• A. $-\frac{q^2}{\pi\epsilon_0}$
• B. $-\frac{2q^2}{3\pi\epsilon_0}$
• C. $\frac{q^2}{4\pi\epsilon_0}$
• D. $-\frac{5q^2}{4\pi\epsilon_0}$

Hint:

Always draw a quick 1D axis and plot the points to avoid silly mistakes when calculating the relative distances between charges.

Answer 1

The Correct Option is B

Concept:
• The electrostatic potential energy of a system of multiple point charges is the total work done in bringing these charges from infinity to their respective positions in the assembly.

• For a system of three point charges $q_1$, $q_2$, and $q_3$ separated by distances $r_{12}$, $r_{23}$, and $r_{13}$ respectively, the total potential energy $U$ is given by the algebraic sum of the potential energies of all unique pairs.

• The foundational formula used is $U = \frac{1}{4\pi\epsilon_0} \left[ \frac{q_1 q_2}{r_{12}} + \frac{q_2 q_3}{r_{23}} + \frac{q_1 q_3}{r_{13}} \right]$.

Step 1: Identify the charges and their coordinates
Let the three charges be defined as follows:
$q_1 = +q$ located at position $x_1 = -1 \text{ m}$.
$q_2 = -2q$ located at position $x_2 = 0 \text{ m}$.
$q_3 = +q$ located at position $x_3 = 2 \text{ m}$.

Step 2: Calculate the separation distances between each pair
The distance between $q_1$ and $q_2$ is $r_{12} = |x_2 - x_1| = |0 - (-1)| = 1 \text{ m}$.
The distance between $q_2$ and $q_3$ is $r_{23} = |x_3 - x_2| = |2 - 0| = 2 \text{ m}$.
The distance between $q_1$ and $q_3$ is $r_{13} = |x_3 - x_1| = |2 - (-1)| = 3 \text{ m}$.

Step 3: Apply the potential energy formula for the system
Substitute the charges and their calculated distances into the potential energy equation:
\[ U = \frac{1}{4\pi\epsilon_0} \left[ \frac{(+q)(-2q)}{1} + \frac{(-2q)(+q)}{2} + \frac{(+q)(+q)}{3} \right] \]
Carefully compute the products in the numerators:
\[ U = \frac{1}{4\pi\epsilon_0} \left[ \frac{-2q^2}{1} + \frac{-2q^2}{2} + \frac{q^2}{3} \right] \]
Simplify the fractions inside the bracket:
\[ U = \frac{1}{4\pi\epsilon_0} \left[ -2q^2 - q^2 + \frac{q^2}{3} \right] \]
Combine the integer terms:
\[ U = \frac{1}{4\pi\epsilon_0} \left[ -3q^2 + \frac{q^2}{3} \right] \]

Step 4: Find the final simplified expression
Find a common denominator to add the terms inside the bracket:
\[ -3q^2 + \frac{q^2}{3} = \frac{-9q^2 + q^2}{3} = \frac{-8q^2}{3} \]
Now substitute this back into the total energy expression:
\[ U = \frac{1}{4\pi\epsilon_0} \left[ \frac{-8q^2}{3} \right] \]
Cancel out the common factor of $4$ in the numerator and the denominator:
\[ U = -\frac{2q^2}{3\pi\epsilon_0} \]

Step 5: Conclusion
The calculated potential energy perfectly matches option (B). The negative sign indicates that the system is bound and energy would need to be supplied to break the charges apart to infinity.