A simple potential divider setup consists of a circuit with a cell and a resistor. The components of another circuit are connected at one end to a terminal of the cell, and the other end to a sliding contact which can vary resistance in parallel to the second circuit, thereby varying the potential difference across the components in the second circuit.
The resistance arising from the resistance in the potential divider in parallel with the resistance in the second circuit, resulting from its components, is RA. The total resistance, RB, is the resistance not involved in the second circuit plus RA. The ratio of the RA to RB is proportional to the potential supplied to the components in the second circuit.
One of the advantages of the potential divider compared to the variable resistor is that whereas variable resistors have a maximum resistance where some current will always flow, dividers are able to vary the current from maximum to practically zero. Zero voltage occurs in the second circuit when the contact is moved so the resistor is not involved in it. Now the components in the second circuit can be considered as in parallel with a conductor of no resistance. As the total resistance (RT) is equal to the sum of the individual inverse of the resistances in parallel:
1/RT = 1/R1 + 1/R2
RT = (R1 * R2) / (R1 + R2)
As R2 = 0:
RT = (R1 * 0) / R1
The total resistance is shown to be zero. As v = ir, the voltage is also zero. Of course this will never actually be achieved as there will always be a small resistance in the wires.