Six equal resistances are connected between points $\mathrm{P}, \mathrm{Q}$ and $\mathrm{R}$ as shown in…
Six equal resistances are connected between points $\mathrm{P}, \mathrm{Q}$ and $\mathrm{R}$ as shown in figure. Then net resistance will be maximum between :
$\mathrm{P}$ and $\mathrm{R}$
P and Q
Q and R
Any two points
Solution
Resistance between $\mathrm{P}$ and $\mathrm{Q}$
$
r_{\mathrm{PQ}}=r_{11}\left(\frac{r}{3}+\frac{r}{2}\right)=\frac{r \times \frac{5}{6} r}{r+\frac{5}{6} r}=\frac{5}{11} r
$
Resistance between Q and R
$
\mathrm{r}_{\mathrm{QR}}=\frac{\mathrm{r}}{2} 11\left(\mathrm{r}+\frac{\mathrm{r}}{3}\right)=\frac{\frac{\mathrm{r}}{2} \times \frac{4}{3} \mathrm{r}}{\frac{\mathrm{r}}{2}+\frac{4}{3} \mathrm{r}}=\frac{4}{11} \mathrm{r}
$
Resistance between $\mathrm{P}$ and $\mathrm{R}$
$
\mathrm{r}_{\mathrm{PR}}=\frac{\mathrm{r}}{3} \mathrm{l}\left(\frac{\mathrm{r}}{2}+\mathrm{r}\right)=\frac{\frac{\mathrm{r}}{3} \times \frac{3}{2} \mathrm{r}}{\frac{\mathrm{r}}{3}+\frac{3}{2} \mathrm{r}}=\frac{3}{11} \mathrm{r}
$
Hence, it is clear that $\mathrm{r}_{\mathrm{PQ}}$ is maximum
