Question

The value of effective resistance between points $\text{A}$ and $\text{B}$ is? $(R=2\text{K}\Omega )$

• A. $(1+\sqrt{3})\text{K}\Omega$
• B. $2(1+\sqrt{3})\text{K}\Omega$
• C. $(1-\sqrt{3})\text{K}\Omega$
• D. $2(1-\sqrt{3})\text{K}\Omega$

Answer 1

The correct option is B $2(1+\sqrt{3})\text{K}\Omega$

This is an infinite grid of resistance repeating itself.

Here $\text{A}$ is equvalent to $\text{C}$ & $\text{B}$ is equivalent to $\text{D}$.

So, if $\left(R_{eq}\right)$ $\text{A}$ & $\text{B}$ is $X$, then
$\left(R_{eq}\right)$ between $\text{C}$ & $\text{D}$ must also be $X$

We can accordingly reduce the circuit as shown below,


$\Rightarrow X=2R+\frac{RX}{R+X}$

$\Rightarrow RX+X^2=2R^2+3RX$

$X^2-2RX-2R^2=0$

$X=\frac{2R\pm \sqrt{4R^2+8R^2}}{2}$

$X=\frac{2R+R\sqrt{12}}{2}$

$\Rightarrow X=(1+\sqrt{3})R\text{K}\Omega$

$\Rightarrow X=(1+\sqrt{3})2\text{K}\Omega$

Hence, option $\left(b\right)$ is the correct answer.