Question

Three equal charges are placed on the three corners of a square as shown below. If the magnitude of force between ' \(\mathrm{q}_1\) ' and ' \(\mathrm{q}_2\) ' is ' \(\mathrm{F}_{12}\) ' and that between ' \(\mathrm{q}_1\) ' and ' \(\mathrm{q}_3\) ' is \(\mathrm{F}_{13}\), then the ratio of \(F_{13}\) to \(F_{12}\) is

• A. \(\frac{1}{\sqrt{2}}\)
• B. \(\sqrt{2}\)
• C. \(\frac{1}{2}\)
• D. 2

Hint:

Find the distances between the charges: one pair is along a side and the other along the diagonal. Use the inverse-square law.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
Coulomb's force between two equal charges is \(F = \dfrac{kq^2}{r^2}\).

Step 2: Distances
The charges \(q_1\) and \(q_2\) are on a side, so \(r_{12} = a\).
The charges \(q_1\) and \(q_3\) are on opposite corners, so \(r_{13} = \sqrt{2}\,a\).

Step 3: Calculation
\(F_{12} = \dfrac{kq^2}{a^2}\) and \(F_{13} = \dfrac{kq^2}{2a^2}\)
\(\dfrac{F_{13}}{F_{12}} = \dfrac{a^2}{2a^2} = \dfrac{1}{2}\)

Final Answer:
The ratio of \(F_{13}\) to \(F_{12}\) is one half. \[ \boxed{\text{(C) }\dfrac{1}{2}} \]