A circuit contains a battery, a variable resistor Rv, a fixed resistor Rf, and a single-pole switch S arranged as follows: Battery → Rv → node P. From node P the circuit can follow two paths that rejoin at node Q before returning to the battery:
• Path 1 (always present): directly from P to Q through a wire of negligible resistance.
• Path 2 (optional): from P through the fixed resistor Rf, then through the switch S, then to Q. The switch S can be either open or closed.
Assume ideal wires and battery.
(a) With the switch S open, describe whether changing Rv has any effect on the current supplied by the battery. Explain your reasoning in terms of the available closed loops.
(b) With the switch S closed, explain qualitatively how increasing Rv affects (i) the current through Path 1, (ii) the current through Path 2, and (iii) the total current delivered by the battery.
(c) Provide a concise explanation for why the effect of adjusting Rv depends on the position of the switch, referencing the concepts of open and closed circuits.