Two identical particles, each with charge q, are placed on the x axis, one at the origin and the other at x = 5 cm.A third particle, with charge -q, is placed on the x axis so the potential energy of the three-particle system is the same as the potential energy when they are all infinitely far apart.Its x coordinate is:

Question

Quizplus · Accepted Answer

The potential energy (U) of a system of charges is given by the formula $U = k \frac{q_1 q_2}{r}$, where $k$ is Coulomb's constant, $q_1$ and $q_2$ are the charges, and $r$ is the distance between the charges. When the charges are infinitely far apart, their potential energy relative to each other is zero because $r$ approaches infinity, making $U = 0$.For the system to have the same potential energy when the third charge is placed on the x-axis, the potential energy contributions from all pairs of charges must sum to zero. The potential energy between the two identical charges (each with charge $q$) at 0 cm and 5 cm is positive because they repel each other. Placing the third charge (with charge $-q$) at a position where its potential energy with each of the other two charges exactly cancels this positive potential energy will satisfy the condition.The correct position is not immediately obvious without calculation, but the key insight is that the third charge must be placed outside the region between the two positive charges to ensure that the system's total potential energy can be zero. This is because the potential energy between the third charge and each of the positive charges must be negative (since one charge is negative), and it must have a larger magnitude but opposite sign to the potential energy between the two positive charges.Placing the third charge at 13 cm ensures that it is sufficiently far from both charges to have a significant negative potential energy contribution with each that can cancel out the positive potential energy between the two positive charges. This is a conceptual explanation; the exact calculation would involve equating the sum of the potential energies to zero and solving for the position, but the key idea is that the third charge needs to be placed outside the two existing charges to balance the system's potential energy to that of when they are infinitely far apart.

Two Identical Particles, Each with Charge Q, Are Placed on the X