Question

In the following star circuit diagram, the equivalent resistance between the points $A$ and $H$ will be

• A. $1.944r$
• B. $0.973r$
• C. $0.486r$
• D. $0.243r$

Answer 1

The correct option is B $0.973r$

In given figure, CFHJD is a pentagon, hence angle subtended by JH is $\frac{360}{5}={72}^{\circ }$
From figure(a) we can write,
$JH=2MJ=2\left(rsin{18}^{\circ }\right)$
$JH=0.62r$

Now, the resistance of each triangle is
$\frac{1}{R_T}=\frac{1}{2r}+\frac{1}{0.62r}$

$\frac{1}{R_T}=\frac{2.62r}{\left(2r\right)\left(0.62r\right)}$

$R_T=\frac{\left(2r\right)\left(0.62r\right)}{2.r}$

$R_T=0.4732r$

Now, the circuit can be converted into the equivalent one as shown in figure (b).

Current will not flow through branch CD hence, its resistance can be neglected.

The equivalent resistance between A and H is:
$\frac{1}{R}=\frac{1}{r+0.4732r+0.4732r}+\frac{1}{r+0.4732r+0.4732r}$

$\frac{1}{R}=\frac{1}{1.946r}+\frac{1}{1.946r}$

$\frac{1}{R}=\frac{2}{1.946r}$

$R=0.973r$