Question

Resistance of each side is $R$. Find equivalent resistance between two opposite points as shown in the figure. 

• A. $\dfrac{4}{5}R$
• B. $\dfrac{8}{5}R$
• C. $\dfrac{8}{10}R$
• D. $\dfrac{2}{5}R$

Hint:

In symmetric resistor networks, identify equipotential points to reduce the circuit easily.

Answer 1

The Correct Option is A

Step 1: Use symmetry of the network.
The hexagonal resistor network is symmetric about the line joining the opposite points. Hence, current divides equally in equivalent branches.
Step 2: Simplification of the internal connections.
Each diagonal divides resistance $R$ into two equal parts of $R/2$. These resistances can be combined using series and parallel rules.
Step 3: Equivalent resistance of the inner triangle.
\[ R_{\text{inner}} = \dfrac{R \times R}{R + R} = \dfrac{R}{2} \]
Step 4: Combining inner and outer resistances.
The reduced network gives two resistances $2R$ and $\dfrac{4R}{3}$ in parallel.
Step 5: Final equivalent resistance.
\[ R_{\text{eq}} = \dfrac{2R \times \dfrac{4R}{3}}{2R + \dfrac{4R}{3}} = \dfrac{8R^2}{10R} = \dfrac{4}{5}R \]