1. Use maximum power transfer idea

   Heat generated in the variable resistor $r$ is $P_r = I^2 r$ and for a linear resistive network, this power is maximum when $r = R_{\text{th}}$ where $R_{\text{th}}$ is the Thevenin resistance of the remaining circuit seen from the terminals of $r$.

2. Find the equivalent resistance seen by $r$

   From the given circuit, when we remove $r$ and look into its terminals, the remaining network reduces to an equivalent resistance of $R_{\text{th}} = \frac{R}{2}.$

3. Condition for maximum heat in $r$

   Therefore, for maximum heat generation, $r = R_{\text{th}} = \frac{R}{2}.$

4. Compare with $r = fR$

   Given $r = fR,$ so $fR = \frac{R}{2} \implies f = \frac{1}{2}.$

5. Check options

   $f = \frac{1}{2}$ corresponds to Option B.

Final Answer: $\boxed{\frac{1}{2}}$