Question

Find out the effective resistance between A and B in the circuit given will be,

$\begin{align}
  & A.6r \\
 & B.r \\
 & C.2r \\
 & D.\dfrac{r}{2} \\
\end{align}$

Answer 1

Hint: The connection will be in delta connection. Find the effective resistance of the circuit. Then draw the resultant circuit and find the effective resistance across AB. The effective resistance can be found by taking care of series and parallel connection. This will help you in answering this question.

Complete step by step answer:
The circuit can be classified at first. We can draw the first layer as,

The effective resistance of this delta connection can be found as,
The resistance at c will be,
$3r=\dfrac{ab+bc+ac}{c}$
The resistance at a will be,
$3r=\dfrac{ab+bc+ac}{a}$
And the resistance at b will be,
$3r=\dfrac{ab+bc+ac}{b}$
Therefore the new circuit will be shown as,

The $3r$ and $r$ will be in parallel. Therefore the effective resistance will be,
That is,
${r}'=\dfrac{3r\times r}{3r+r}=\dfrac{3r}{4}$
This can be shown in the diagram as,

Now we can see that the resistance between the AC and BC can be in series. Therefore their effective resistance will be found by adding the values of their resistances. That is,
${{r}_{2}}=\dfrac{3r}{4}+\dfrac{3r}{4}=\dfrac{3r}{2}$
This can be shown in the diagram as,

Now the resistances will be in parallel to each other. The resistance between the points A and B should be found. That is we can say that,
${{R}_{AB}}=\dfrac{\dfrac{3r}{2}\times \dfrac{3r}{4}}{\dfrac{3r}{2}+\dfrac{3r}{4}}=\dfrac{18{{r}^{2}}}{4\times 4}\times \dfrac{4}{3r}=\dfrac{2r}{4}=\dfrac{r}{2}\Omega $
Therefore, the value of the resistance has been obtained.

Note:
The effective resistance of the resistors which are connected in series can be found by taking the sum of the value of the resistances of each of the resistances. The reciprocal of the equivalent resistance which has been connected in parallel connection can be found by taking the sum of the reciprocal of the values of the resistances.