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As I was studying about potentiometer I came across how to use it to compare emf of two cells.

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It was written my text book that if current I flows through potentiometer wire then

$$\mathcal{E}= v\cdot L$$

where $v$ is potential drop per unit length. Now

$$\mathcal{E}_1= v\cdot L_1$$

and

$$\mathcal{E}_2= v\cdot L_2$$

But on what basis did we take $v$ as a constant $v=\mathcal{E}/L$? If current through battery is zero then

$$\mathcal{E}=\mathrm{voltage}=IR/L$$

$$=I\rho/A,\quad \mathrm{(\rho=resistivity)}$$

Thus $v$ depend on $I$ and if we change $\mathcal{E}$ then $I$ must change and thus $v$ will also change. But here $v$ is taken as constant.

My book

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