Question

A regular hexagon is formed by six wires each of resistance \( r \,\Omega \) and the corners are joined to the centre by wires of same resistance. If the current enters at one corner and leaves at the opposite corner, the equivalent resistance of the hexagon between the two opposite corners will be

• A. \( \dfrac{4}{5} r \)
• B. \( \dfrac{3}{4} r \)
• C. \( \dfrac{3}{5} r \)
• D. \( \dfrac{5}{8} r \)

Hint:

In symmetric resistor networks, always look for identical current paths—symmetry greatly simplifies calculations.

Answer 1

The Correct Option is A

Step 1: Use symmetry of the hexagonal network.
Since the hexagon is perfectly symmetrical and each corner is connected to the centre by equal resistances, the current distribution on symmetric paths will be equal. This allows us to group equivalent paths and simplify the circuit.
Step 2: Analyze current flow between opposite corners.
When current enters at one corner and exits from the opposite corner, the circuit splits into multiple identical branches due to symmetry. The resistances on equivalent paths combine in parallel and series combinations.
Step 3: Simplify the equivalent resistance.
After reducing the symmetric branches and combining the resistances properly, the net equivalent resistance between the opposite corners comes out to be \[ R_{\text{eq}} = \frac{4}{5} r \]
Step 4: Final conclusion.
Thus, the correct equivalent resistance is \( \dfrac{4}{5} r \).