In a potentiometer experiment, when three cells \(\mathrm{A}, \mathrm{B}\) and \(\mathrm{C}\) are connected in series, the balancing length is found to be \(420 \mathrm{~cm}\). If cells A and B are connected in series the balancing length is \(220 \mathrm{~cm}\) and for cells B and C connected in series the balancing length is \(320 \mathrm{~cm}\). The emf of cells A, B and C are respectively in the ratio of

\(\begin{aligned} & \mathrm{E}_{\mathrm{A}}+\mathrm{E}_{\mathrm{B}}+\mathrm{E}_{\mathrm{C}}=\mathrm{x}(420) \quad[\mathrm{x}=\text { potential gradient }] \\ & \mathrm{E}_{\mathrm{A}}+\mathrm{E}_{\mathrm{B}}=\mathrm{x}(220) \quad \therefore \mathrm{E}_{\mathrm{C}}=\mathrm{x}(420)-\mathrm{x}(220)=\mathrm{x}(200) \\ & \mathrm{E}_{\mathrm{B}}+\mathrm{E}_{\mathrm{C}}=\mathrm{x}(320) \quad \therefore \mathrm{E}_{\mathrm{B}}=\mathrm{x}(320)-\mathrm{x}(200)=\mathrm{x}(120) \\ & \mathrm{E}_{\mathrm{A}}=\mathrm{x}(220)-\mathrm{E}_{\mathrm{B}}=\mathrm{x}(100) \\ & \therefore \mathrm{E}_{\mathrm{A}}: \mathrm{E}_{\mathrm{B}}: \mathrm{E}_{\mathrm{C}}=100: 120: 200=1: 1.2: 2\end{aligned}\)