Question

ABCD is a square of side $a$ metres and is made of wires of resistance $x$ ohms/metre. Similar wires are connected across the diagonals AC and BD. The effective resistance between the corners A and C will be

• A. $(2-\sqrt{2})ax$
• B. $6ax$
• C. $3ax$
• D. $(3-\sqrt{2})ax$

Answer 1

The correct option is A $(2-\sqrt{2})ax$

To find :

Effective Resistance across AC.

Sol :-

Let E.R = Effective Resistance

$\frac{1}{E.R}=\frac{1}{2ax}+\frac{1}{2ax}+\frac{1}{\sqrt{2}ax}$

$=\frac{1}{ax}[\frac{1}{2}+\frac{1}{2}+\frac{1}{\sqrt{2}}]$

$=\frac{1}{ax}[1+\frac{1}{\sqrt{2}}]$

$=\frac{1}{ax}\left[\frac{1+\sqrt{2}}{\sqrt{2}}\right]$

$=\frac{1+\sqrt{2}}{\sqrt{2}ax}\times \frac{1-\sqrt{2}}{1-\sqrt{2}}$

$=\frac{1-2}{(\sqrt{2}-2)ax}$

$E.R=\frac{(\sqrt{2}-2)ax}{1-2}$

$=(2-\sqrt{2})ax$

$\therefore$ option (a)