Question

Six equal resistances are connected between points $P,Q$ and $R$ as shown in the figure. Then the net resistance will be maximum between:-

• A. $P$ and $Q$
• B. $Q$ and $R$
• C. $P$ and $R$
• D. Any two points

Answer 1

Answer: (A)

Given that all are identical resistors with resistance $r$ .
In between P and R, three resistors are in parallel.
Let $R_1=\frac{r}{3}$
In between Q and R, two resistors are in parallel
Let $R_2=\frac{r}{2}$
In between P and Q, let $R_3=r$

Effective resistance in between points P & Q is $R_{PQ}=\frac{R_3\left(R_1+R_2\right)}{R_1+R_2+R_3}=\frac{r\left(\frac{r}{3}+\frac{r}{2}\right)}{\left(r+\frac{r}{3}+\frac{r}{2}\right)}{R}_{PQ}=\frac{r\left(\frac{5r}{6}\right)}{\frac{11r}{6}}\Rightarrow R_{PQ}=\frac{5r}{11}$
Effective resistance between points Q and R is
$R_{QR}=\frac{R_2\left(R_1+R_3\right)}{R_1+R_2+R_3}=\frac{\frac{r}{2}\left(r+\frac{r}{3}\right)}{\frac{r}{2}+r+\frac{r}{3}}{R}_{QR}=\frac{\frac{r}{2}\left(\frac{4r}{6}\right)}{\left(\frac{11}{6}r\right)}\Rightarrow R_{QR}=\frac{2r}{11}$
Effective resistance between points P and R is $R_{PR}=\frac{R_1\left(R_2+R_3\right)}{R1+R_2+R_3}=\frac{\frac{r}{3}\left(r+\frac{r}{2}\right)}{\frac{r}{3}+r+\frac{r}{2}}{R}_{PR}=\frac{\frac{r}{3}\left(\frac{3r}{2}\right)}{\frac{11r}{6}}\Rightarrow R_{PR}=\frac{3r}{11}$
Hence, $R_{PQ}>R_{PR}>R_{QR}$
i.e, Effective resistance is maximum between points P and Q.