Question

Three point charges \(q, -2q\), and \(q\) are placed along the \(x\)-axis at \(x = -a, 0,\) and \(a\), respectively. As \(a \to 0\) and \(q \to \infty\), while \(qa^2 = Q\) remains finite, the electric field at a point \(P\), at a distance \(x \gg a\) from \(x = 0\), is given by: \[\vec{E} = \frac{qQ}{4 \pi \epsilon_0 x^3} \hat{i}.\] Then, find the relationship between \(\alpha\) and \(\beta\).

• A. \(\alpha = \beta\)
• B. \(\alpha = 2\beta\)
• C. \(\alpha = \frac{2}{3}\beta\)
• D. \(2\alpha = 3\beta\)

Hint:

When dealing with multiple charges, use the principle of superposition to sum the electric fields, and carefully consider the limits of the charges and distances.