Q.
In the arrangement of resistances as shown, what is the equivalent resistance between $A$ and $B$ ?
NTA AbhyasNTA Abhyas 2020
Solution:
We can simplify the circuit diagram
And we can see that between A and C terminal 2r and 2r resistances are parallel combination and we can calculate by using the formmula $\left(\right.R_{e q}=\frac{R_{1} \times R_{2}}{R_{1} + R_{2}}\left.\right)$
Hence $\frac{2 r \times 2 r}{2 r + 2 r}=r$
Similarly we can find the parallel combination between B and C terminal $\frac{2 r \times 2 r}{2 r + 2 r}=r$
Again we can see that between C and D terminal r and r resistances are parallel combination and again we calculate by using formula $\left(\right.R_{e q}=\frac{R_{1} \times R_{2}}{R_{1} + R_{2}}\left.\right)$
hence we get $\frac{r \times r}{r + r}=\frac{r}{2}$
And now we get a simplifying circuit
Hence we can understand that it follows the wheatstone bridge concept and balance the following equation which is $\frac{P}{Q}=\frac{R}{S}$
where P = Q = R = S = r Ω hence no current will be drawn from $\frac{r}{2}$ Ω resistance i.e
Equilent resistance is $\frac{2 r \times 2 r}{2 r + 2 r}=r$
(Because r and r are in series combination so $r+r=2r$ )
And we can see that between A and C terminal 2r and 2r resistances are parallel combination and we can calculate by using the formmula $\left(\right.R_{e q}=\frac{R_{1} \times R_{2}}{R_{1} + R_{2}}\left.\right)$
Hence $\frac{2 r \times 2 r}{2 r + 2 r}=r$
Similarly we can find the parallel combination between B and C terminal $\frac{2 r \times 2 r}{2 r + 2 r}=r$
Again we can see that between C and D terminal r and r resistances are parallel combination and again we calculate by using formula $\left(\right.R_{e q}=\frac{R_{1} \times R_{2}}{R_{1} + R_{2}}\left.\right)$
hence we get $\frac{r \times r}{r + r}=\frac{r}{2}$
Hence we can understand that it follows the wheatstone bridge concept and balance the following equation which is $\frac{P}{Q}=\frac{R}{S}$
where P = Q = R = S = r Ω hence no current will be drawn from $\frac{r}{2}$ Ω resistance i.e
Equilent resistance is $\frac{2 r \times 2 r}{2 r + 2 r}=r$
(Because r and r are in series combination so $r+r=2r$ )