Question 115: Current Electricity

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

Accepted Answer

Because the hexagon is regular and the entry/exit points (A and D ) are symmetrical, we can identify points with equal potential relative to the axis AD :
1. Points B and F are at the same potential due to symmetry relative to the axis $\mathrm{AD} \Rightarrow \mathrm{V}_{\mathrm{B}}=\mathrm{V}_{\mathrm{F}}$.
2. Points C and E are also at the same potential $\Rightarrow \mathrm{V}_{\mathrm{C}}=\mathrm{V}_{\mathrm{E}}$.
3. The centre point O lies exactly halfway between the entry and exit points on the direct path AOD.
$R_{B{O_1} C}=r+r=2 r$ and this is in parallel with point $B$ and $C$,
$$ R_{B C}=\frac{r 	imes 2 r}{r+2 r}=\frac{2 r}{3} $$
Now, $R_{A B}, R_{B C}$ and $R_{C D}$ are in series,
$$ R_{A B C D}=r+\frac{2 r}{3}+r=\frac{8 r}{3} $$
Similarly, due to symmetry, $\mathrm{R}_{	ext {AFED }}=\frac{8 \mathrm{r}}{3}$
$R_{A O}$ and $R_{O D}$ are in series, $R_{O D}=r+r=2 r$
The resulting circuit is,
Now, all are in parallel, so the equivalent resistance between A and D is,
$$ \frac{1}{R}=\frac{1}{R_{A B C D}}+\frac{1}{R_{A O B}}+\frac{1}{R_{A F E D}}=\frac{1}{(8 r / 3)}+\frac{1}{(2 r)}+\frac{1}{(8 r / 3)}=\frac{3}{8 r}+\frac{1}{2 r}+\frac{3}{8 r}=\frac{3+4+3}{8 r} $$
$$\Rightarrow $$ $$ \frac{1}{R}=\frac{10}{8 r} \Rightarrow R=\frac{4 r}{5} $$
Therefore, the equivalent resistance between any two corners opposite to each other is $\frac{4 \mathrm{r}}{5}$.
Hence, the correct option is (B).

Answer

$\frac{3}{5} r$

Answer

$\frac{4}{5} r$

Answer

$\frac{5}{8} r$

Answer

$\frac{3}{4} r$

Current Electricity

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

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