Question

If the wire has a resistance per meter $\rho$, the resistance between A and B is $\frac{l\rho }{\sqrt{x}}$. Find the value of $x$.

Answer 1

Resistance per meter =$\rho$

So Let Resistance of $\frac{l}{2}$m wire $=\rho \frac{l}{2}=r⟶\left(1\right)$

Let Resistance $\frac{l}{\sqrt{2}}$m wire $=\rho \frac{l}{\sqrt{2}}=\sqrt{2}r$

equivalent resistance $=\frac{2r\times \sqrt{2}r}{2r+\sqrt{2}r}=\frac{2\sqrt{2}r}{2+\sqrt{2}}={\left(\frac{2}{\sqrt{2+1}}\right)}^r$

Equivalent resistance $=\frac{2\sqrt{2}r\times 2\sqrt{2}r}{2\sqrt{2r}r+2\sqrt{2}r}=\frac{8r^2}{4\sqrt{2}r}=\sqrt{2}r$

and we know that $r=\rho \frac{l}{2}$ (from (1) )

$=\sqrt{2}\times \rho \frac{l}{2}$

Equivalent resistance $=\frac{\rho l}{\sqrt{2}}$