Question

The figure shows a potentiometer of total resistance $R_T$ with a sliding contact.

The resistance between the points P and Q of the potentiometer at the position of the contact shown is $R_C$ and the voltage ratio $V_0/V_s$ at this point is 0.5. If the ratio $R_L/R_T=1,\text{the ratio}{R}_C/R_T$ is

• A. $\frac{-1+\sqrt{5}}{2}$
• B. $\frac{1+\sqrt{5}}{2}$
• C. $-1+\sqrt{5}$
• D. $1+\sqrt{5}$

Answer 1

The correct option is A $\frac{-1+\sqrt{5}}{2}$


Given,
$\frac{R_L}{R_T}=1$

$\Rightarrow R_L=R_T$

Also,

$\frac{V_s}{2}=(R_T-R_C)I_s=\left(R_T||R_C\right)I_s$

using (i),

$R_T-R_C=\frac{R_C.R_T}{R_C+R_T}$

$\Rightarrow R_T^2-R_C^2=R_C{R}_T$

$\Rightarrow \frac{R_T}{R_C}-\frac{R_C}{R_T}=1$

$Let\frac{R_C}{R_T}=x$

$\Rightarrow \frac{1}{x}-x=1$

$\Rightarrow x^2+x-1=0$

$\Rightarrow x=\frac{-1\pm \sqrt{5}}{2}$

Since,
$x=\frac{R_C}{R_T}>0$

$\Rightarrow \frac{R_C}{R_T}=\frac{\sqrt{5}-1}{2}$