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In a potentiometer circuit, two wires of same material of resistivity \( \rho \), one of radius of cross-section ' a' and other of radius of cross-section ' \( 2 \mathrm{a} \) ' are joined in series. They are of length \( \ell \) and \( 2 \ell \) respectively. This combination acts as the potentiometer wire of length \( 3 \ell \). The emf of the cell in the primary circuit is \( \varepsilon \) and internal resistance is \( \frac{\rho \ell}{2 \pi \mathrm{a}^{2}} \). This cell is connected to the potentiometer wire by a conducting wire of negligible resistance with positive terminal of the cell connected to one end (call it \( \mathrm{A} \) ) of longer wire. The negative terminal of the cell is connected to one end of the smaller wire. The remaining ends of the two wires are joined together. Find:
The maximum voltage which can be balanced on the potentiometer wire.
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