The ends of six wires, each of resistance R (= 10 \(\Omega\)) are joined as shown in the figure. The points A and B of the arrangement are connected in a circuit. Find the value of the effective resistance offered by it to the circuit.
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In symmetric circuits, identify series and parallel combinations systematically to simplify the network step by step.
The provided circuit is a symmetric triangle with individual side resistances of \( R \). The objective is to determine the effective resistance between points A and B. Step 1: Series Combination of Resistors Each pair of resistors connected in series results in an equivalent resistance of:\[R_{\text{eq1}} = R + R = 2R\] Step 2: Parallel Combination of Resistors The three \( 2R \) equivalent resistances are now connected in parallel. Applying the formula for parallel resistors:\[\frac{1}{R_{\text{eq}}} = \frac{1}{2R} + \frac{1}{2R} + \frac{1}{2R} = \frac{3}{2R}\]Consequently, the overall effective resistance is:\[R_{\text{eq}} = \frac{2R}{3}\] Step 3: Calculation with Given Resistance Value Substituting the given value \( R = 10 \, \Omega \):\[R_{\text{eq}} = \frac{2 \times 10}{3} = 6.67 \, \Omega\]
