How to emf will be compared by the help of potentiometer

Dear Student,
emf of three cells compared with the help of potentiometer:
First label the cells as 1,2 and 3

Lets take 1 and 2 first.


or , $E_2=\frac{l_2}{l_1}{E}_1$

Next lets take 1 and 3

$$\begin{gathered} we will get a new relation \\ {E}_3=\frac{l_3}{l_1\text{'}}{E}_1 \\ now, \\ {E}_1:E_2:E_3=E_1:\frac{l_2}{l_1}{E}_1:\frac{l_3}{l_1\text{'}}{E}_1 \\ or, E_1:E_2:E_3=1:\frac{l_2}{l_1}:\frac{l_3}{l_1\text{'}} \\ multiplying by l_1{l}_1\text{'} \\ {\mathit{E}}_{\mathbf{1}}\mathbf{:}{\mathit{E}}_{\mathbf{2}}\mathbf{:}{\mathit{E}}_{\mathbf{3}}\mathbf{=}\mathbf{ }{\mathit{l}}_{\mathbf{1}}{\mathit{l}}_{\mathbf{1}}\mathbf{\text{'}}\mathbf{:}\mathbf{ }{\mathit{l}}_{\mathbf{2}}{\mathit{l}}_{\mathbf{1}}\mathbf{\text{'}}\mathbf{:}\mathbf{ }{\mathit{l}}_{\mathbf{1}}{\mathit{l}}_{\mathbf{3}} \end{gathered}$$ Regards