Potentiometer Formulas

1. Potential gradient (k)

The fall of potential per unit length of potentiometer wire:
k = \(\frac{E}{L}\) or k = ip volt/m L
where i is the current flowing through the wire and ρ is the resistance of
unit length of wire. The current in the primary circuit is –
i = \(\frac{\mathrm{E}}{\mathrm{R}+\mathrm{r}+\mathrm{R}_{\mathrm{h}}}\) and ρ = \(\frac{R}{L}\)

2. Measurement of potential difference or emf

The balancing length for emf E₀ of cell is l₀ and that for unknown potential I difference V is l, then
E₀ = kl₀
or k = \(\frac{E_{0}}{\ell_{0}}\)
and V = kl = \(\left(\frac{\mathrm{E}_{0}}{\ell_{0}}\right)\)l
Comparison of emfs:

\(\frac{E_{1}}{E_{2}}=\frac{\ell_{1}}{\ell_{2}}\)

3. Measurement of small resistance

The potential difference across resistance R
iR = kl₁
The potential difference across the resistance (R + r)
= i(R + r) = kl₂
Thus r = \(\left(\frac{\ell_{2}-\ell_{1}}{\ell_{1}}\right)\)R

4. Internal resistance of a cell

If emf of a cell is E and terminal voltage is V then internal resistance.

r = \(\left(\frac{\mathrm{E}-\mathrm{V}}{\mathrm{V}}\right)\)R
E = kl₁ and V = kl₂
r = \(\left(\frac{\ell_{2}-\ell_{1}}{\ell_{1}}\right)\)R

5. Calibration

To compare apparent value with actual value measured by potentiometer.
Error = apparent value – actual value
A graph between error (e) and observed (apparent) value is called calibration curve.
Actual values of voltage or current are measured by potentiometer. Hence the calibration of voltmeter and ammeter are done by it.

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