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 of its resistance is

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

Answer 1

Answer: (C)

For above circuit
$\frac{1}{R^{\prime }}=\frac{1}{R}+\frac{1}{X}=\frac{R+X}{RX}$
$R^{\prime }=\frac{RX}{R+X}$
The current in the circuit
$i=\frac{E}{R+R^{\prime }}=\frac{E}{\left(R+\frac{RX}{R+X}\right)}$
$V_{RX}=\frac{E\cdot \frac{RX}{R+X}}{\left(R+\frac{RX}{R+X}\right)}=\frac{EX}{R+2X}$
Power dissipated in the circuit
$P_X=\frac{V_{RX}^2}{X}=\frac{E^2\cdot X^2}{X(R+2X{)}^2}$
$=\frac{E^{2}X}{(R+2X{)}^2}$
$\frac{d{P}_X}{dX}=E^2\frac{(R-2X)}{(R+2X{)}^3}$
$\Rightarrow d{P}_X=\frac{E^2(R-2X)}{(R+2X{)}^3}\cdot dX$
$\left(d{P}_x\right)$ will be zero for all $(dX)$ if
$X=\frac{R}{2}\left[d{P}_x=\frac{E^2\left(R-2\times \frac{R}{2}\right)}{{\left(R+2\times \frac{R}{2}\right)}^3}\right]$
$=0$