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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 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 length, measured from point \( A \), where cell of emf \( \frac{\varepsilon}{2} \) will balance.
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