Question

A charge \(Q_{1}\) is positioned on the \(x\) -axis at \(x=a\). Where should a charge \(Q_{2}=-4 Q_{1}\) be placed to produce a net electrostatic force of zero on a third charge, \(Q_{3}=Q_{1}\), located at the origin? a) at the origin c) at \(x=-2 a\) b) at \(x=2 a\) d) at \(x=-a\)

Short answer

In this exercise, we are given that Q2 = -4Q1 and Q3 = Q1, and we need to find the position of Q2 to produce a net electrostatic force of zero on Q3 (located at the origin). After applying Coulomb's law and setting the sum of forces to zero, we found that there are two possible positions of Q2, -2a and 2a. However, according to the given exercise options, the most appropriate answer is the position of Q2 at x = -2a (option c).

Step-by-step solution

Define the charges and distances

Let's define the charges as: - \(Q_{1}\) - \(Q_{2} = -4Q_{1}\) - \(Q_{3} = Q_{1}\) Let's define the position of \(Q_{1}\) on the \(x\)-axis as \(x=a\). We need to find the position of \(Q_{2}\), let's denote it with \(x=b\). Since \(Q_{3}\) is at the origin, the distance between \(Q_{1}\) and \(Q_{3}\) is \(a\), and the distance between \(Q_{2}\) and \(Q_{3}\) is \(|b|\).

Apply Coulomb's law on the forces between the charges

To find the electrostatic force, we need to apply Coulomb's law: - \(F_{13} = k \frac{Q_{1}Q_{3}}{a^2} = k \frac{Q_{1} \cdot Q_{1}}{a^2} = k \frac{Q_{1}^2}{a^2}\) - \(F_{23} = k \frac{Q_{2}Q_{3}}{b^2} = k \frac{Q_{1} \cdot (-4Q_{1})}{b^2} = -4 k \frac{Q_{1}^2}{b^2}\)

Set the sum of forces to zero

We are asked to find the position of \(Q_2\) to produce a net force on \(Q_3\) of zero. Therefore, we can write: \(F_{13} + F_{23} = 0\) \( k \frac{Q_{1}^2}{a^2} - 4 k \frac{Q_{1}^2}{b^2} = 0\)

Solve for the position of Q2

Now, we can solve for \(b\) using the previous equation: \(\frac{Q_{1}^2}{a^2} = 4 \frac{Q_{1}^2}{b^2}\) \(\frac{1}{a^2} = 4 \frac{1}{b^2}\) \(a^2 = \frac{b^2}{4}\) \(b^2 = 4a^2\) \(b = \pm 2a\) Since we have two possible options \((-2a, 2a)\) and we are asked to find the most appropriate one that matches the given options in our exercise, we can conclude: The position of \(Q_{2}\) to produce a net electrostatic force of zero on \(Q_{3}\) (located at the origin) is at \(x = -2a\) (option c).