A regular hexagon is formed by six wires each | vedclass

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Question

(A) Let the resistance of each wire be $r$. The hexagon consists of $6$ outer wires and $6$ radial wires connected to the center.Due to symmetry, if c

Vedclass · Accepted Answer

$\frac{4}{5}r$

Vedclass · Answer

(A) Let the resistance of each wire be $r$. The hexagon consists of $6$ outer wires and $6$ radial wires connected to the center.Due to symmetry, if current enters at corner $A$ and leaves at the opposite corner $B$, the potential at the nodes symmetric with respect to the axis $AB$ will be the same.The circuit can be simplified by considering two parallel branches between $A$ and $B$.One branch consists of two resistors of $r$ in series, giving $2r$.The other branch consists of the remaining network which simplifies to an equivalent resistance of $\frac{4}{3}r$.Thus, the equivalent resistance $R_{eq}$ is given by the parallel combination of $2r$ and $\frac{4}{3}r$:$R_{eq} = \frac{2r 	imes \frac{4}{3}r}{2r + \frac{4}{3}r} = \frac{\frac{8}{3}r^2}{\frac{10}{3}r} = \frac{8}{10}r = \frac{4}{5}r$.

$A$ regular hexagon is formed by six wires each of resistance $r \Omega$ and the corners are joined to the centre by wires of the 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:

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