Challenge Question 1 Solution
Analysis: Three charged beads are in equilibrium on a loop. The tension $T$ is uniform.
Step 1: Force Balance
The Coulomb force along the side connecting two charges must be balanced by the Tension $T$.
For side $l_1$ (between $q_2, q_3$): $T = \frac{k q_2 q_3}{l_1^2} \implies l_1 = \sqrt{\frac{k q_2 q_3}{T}}$
For side $l_2$ (between $q_3, q_1$): $T = \frac{k q_3 q_1}{l_2^2} \implies l_2 = \sqrt{\frac{k q_3 q_1}{T}}$
For side $l_3$ (between $q_1, q_2$): $T = \frac{k q_1 q_2}{l_3^2} \implies l_3 = \sqrt{\frac{k q_1 q_2}{T}}$
Step 2: Total Length Constraint
Total length $l = l_1 + l_2 + l_3$. $$l = \frac{\sqrt{k}}{\sqrt{T}} (\sqrt{q_2 q_3} + \sqrt{q_3 q_1} + \sqrt{q_1 q_2})$$ $$\frac{\sqrt{k}}{\sqrt{T}} = \frac{l}{\sum \sqrt{q_i q_j}}$$
Step 3: Solve for Individual Lengths
Substitute $\frac{\sqrt{k}}{\sqrt{T}}$ back into the expressions for $l_1, l_2, l_3$.
Part (b) Condition:
For the beads to form a triangle, the Triangle Inequality must hold (sum of any two sides > third side).
e.g., $l_1 < l_2 + l_3 \implies \sqrt{q_2 q_3} < \sqrt{q_1 q_3} + \sqrt{q_1 q_2}$.