A slide-wire potentiometer circuit consists of a driver cell of constant emf \(\mathcal{E}_0\) connected across a uniform resistance wire of length \(L\). A secondary branch containing a battery of unknown emf \(\mathcal{E}_x\) and unknown internal resistance \(r_x\) is connected to the wire through a sensitive galvanometer. When the battery is on open circuit, a balance point of zero galvanometer current is obtained at a distance \(x_1\) from the left end of the wire. When a resistor of known resistance \(R\) is connected directly across the terminals of the battery, a new balance point is obtained at a distance \(x_2\). Assuming the driver cell and lead wires have negligible resistance, which of the following expressions correctly gives the internal resistance \(r_x\) of the battery?