Question

Consider the circuit shown in the figure. The value of the resistance $X$ for which the thermal power generated in it is practically independent of small variation its resistance is

• A. $X=R$
• B. $X=\frac{R}{3}$
• C. $\frac{R}{2}$
• D. $X=2R$

Answer 1

The correct option is C $\frac{R}{2}$

$i=\frac{E}{(R+\frac{Rx}{R+x})}$
$V_{Rx}=\frac{E\frac{Rx}{R+x}}{R+\frac{Rx}{R+x}}=\frac{Ex}{(R+2x)}$
$P_x=\frac{{V_{Rx}}^2}{x}=\frac{E^{2}x}{{(R+2x)}^2}$
$\frac{d{P}_x}{dx}=E^2\frac{R-2x}{{(R+2x)}^3}$
$\Rightarrow d{P}_x=E^2\frac{R-2x}{{(R+2x)}^3}.dx$
$\left(d{P}_x\right)$ will be zero for all $\left(dx\right)$ if
$X=\frac{R}{2}$