A uniform wire is made in to a circular ring. Equivalent resistance between points A and B on the ring which subtend an angle of $$60^\circ$$ at the center of the ring is $$R$$. Find the equivalent resistance between A and B if the angle is increased to $$120^\circ$$ (as a multiple of $$R$$).
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Let the ring's total resistance be $$R_0$$. For $$60^\circ$$ the arcs are $$\frac{R_0}{6}$$ and $$\frac{5R_0}{6}$$ in parallel, giving $$\frac{5R_0}{36} = R$$, so $$R_0 = \frac{36R}{5}$$. For $$120^\circ$$ the arcs are $$\frac{R_0}{3}$$ and $$\frac{2R_0}{3}$$, giving $$\frac{2R_0}{9} = \frac{2}{9} \times \frac{36R}{5} = \frac{8R}{5} = 1.6R$$.