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(III) You are designing a wire resistance heater to heat an enclosed volume of gas. For the apparatus to function properly, this heater must transfer heat to the gas at a very constant rate. While in operation, the resistance of the heater will always be close to the value $R=R_{0},$ but may fluctuate slightly causing its resistance to vary a small amount $\Delta R\left(\ll R_{0}\right) .$ To maintain the heater at constant power, you design the circuit shown in Fig. $44,$ which includes two resistors, each of resistance $r .$ Determine the value for $r$ so that the heater power will remain constant even if its resistance $R$ fluctuates by a small amount. [Hint: If $\Delta R \ll R_{0},$ then $\Delta P \approx \Delta R\left.\frac{d P}{d R}\right|_{R=R_{0}} \cdot ]$

          (III) You are designing a wire resistance heater to heat an enclosed volume of gas. For the apparatus to function properly, this heater must transfer heat to the gas at a very constant rate. While in operation, the resistance of the heater will always be close to the value $R=R_{0},$ but may fluctuate slightly causing its resistance to vary a small amount $\Delta R\left(\ll R_{0}\right) .$ To maintain the heater at constant power, you design the circuit shown in Fig. $44,$ which includes two resistors, each of resistance $r .$ Determine the value for $r$ so that the heater power will remain constant even if its resistance $R$ fluctuates by a small amount. [Hint: If $\Delta R \ll R_{0},$ then $\Delta P \approx \Delta R\left.\frac{d P}{d R}\right|_{R=R_{0}} \cdot ]$

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