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Q.
Six equal resistances are connected between points $P,Q$ and $R$ as shown in the figure. Then the net resistance will be maximum between:-
NTA AbhyasNTA Abhyas 2022
Solution:
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 $R_{PQ}=\frac{R_{3} \left(R_{1} + R_{2} \right)}{R_{1} + R_{2} + R_{3}}=\frac{ r \left(\frac{r}{3}+\frac{r}{2}\right)}{\left(r + \frac{r}{3} + \frac{r}{2}\right)}R_{PQ}=\frac{r \left(\frac{5 r}{6}\right)}{\frac{11 r}{6}}\Rightarrow R_{PQ}=\frac{5 r}{11}$
Effective resistance between points Q and R is
$R_{QR}=\frac{R_{2} \left(R_{1} + R_{3}\right)}{R_{1} + R_{2} + R_{3}}=\frac{\frac{r}{2}\left(r + \frac{r}{3}\right)}{\frac{r}{2}+r+\frac{r}{3}}R_{QR}=\frac{\frac{r}{2}\left(\frac{4 r}{6}\right)}{\left(\frac{11}{6}r\right)}\Rightarrow R_{QR}=\frac{2 r}{11}$
Effective resistance between points P and R is $R_{PR}=\frac{R_{1} \left(R_{2} + R_{3}\right)}{R 1 + R_{2} + R_{3}}=\frac{\frac{r}{3}\left(r + \frac{r}{2}\right)}{\frac{r}{3}+r+\frac{r}{2}}R_{PR}=\frac{\frac{r}{3}\left(\frac{3 r}{2}\right)}{\frac{11 r}{6}}\Rightarrow R_{PR}=\frac{3 r}{11}$
Hence, $R_{PQ}>R_{PR}>R_{QR}$
i.e, Effective resistance is maximum between points P and Q.
