Question
In the network shown, each resistance is equal to $$R.$$ The equivalent resistance between adjacent comers $$A$$ and $$D$$ is
In the network shown, each resistance is equal to $$R.$$ The equivalent resistance between adjacent comers $$A$$ and $$D$$ is
A.
$$R$$
B.
$$\frac{2}{3}R$$
C.
$$\frac{3}{7}R$$
D.
$$\frac{8}{15}R$$
Answer :
$$\frac{8}{15}R$$
Solution :
The equivalent circuit is as shown in figure. The resistance of arm $$AOD\left( { = R + R} \right)$$ is in parallel to the resistance $$R$$ of arm $$AD.$$
Their effective resistance
$${R_1} = \frac{{2R \times R}}{{2R + R}} = \frac{2}{3}R$$
The resistance of arms $$AB,BC$$ and $$CD$$ is
$${R_2} = R + \frac{2}{3}R + R = \frac{8}{3}R$$
The resistance $${R_1}$$ and $${R_2}$$ are in parallel. The effective resistance between $$A$$ and $$D$$ is
$${R_3} = \frac{{{R_1} \times {R_2}}}{{{R_1} + {R_2}}} = \frac{{\frac{2}{3}R \times \frac{8}{3}R}}{{\frac{2}{3}R + \frac{8}{3}R}} = \frac{8}{{15}}R.$$
The equivalent circuit is as shown in figure. The resistance of arm $$AOD\left( { = R + R} \right)$$ is in parallel to the resistance $$R$$ of arm $$AD.$$
Their effective resistance
$${R_1} = \frac{{2R \times R}}{{2R + R}} = \frac{2}{3}R$$
The resistance of arms $$AB,BC$$ and $$CD$$ is
$${R_2} = R + \frac{2}{3}R + R = \frac{8}{3}R$$
The resistance $${R_1}$$ and $${R_2}$$ are in parallel. The effective resistance between $$A$$ and $$D$$ is
$${R_3} = \frac{{{R_1} \times {R_2}}}{{{R_1} + {R_2}}} = \frac{{\frac{2}{3}R \times \frac{8}{3}R}}{{\frac{2}{3}R + \frac{8}{3}R}} = \frac{8}{{15}}R.$$
