Potentiometer

• The working principle is based on the fact that for a uniform wire carrying a constant current, the potential drop ($V$) across any segment is directly proportional to the length ($l$) of that segment.

• This is mathematically expressed as $V \propto l$, or $V = kl$, where $k$ is the potential gradient (potential drop per unit length).

• The potential gradient is defined as $k = \frac{V_{total}}{L_{total}}$, where $V_{total}$ is the voltage across the entire wire of length $L_{total}$.

• When the jockey is moved to a position where the galvanometer shows zero deflection (the null point), the potential difference across the wire segment equals the EMF of the test cell.

Comparing EMFs of Two Cells

• To compare two cells with EMFs $E_1$ and $E_2$, find the balancing lengths $l_1$ and $l_2$ for each cell individually using the null method.

• Since $E_1 = kl_1$ and $E_2 = kl_2$, the ratio of their EMFs is given by the ratio of their balancing lengths: $\frac{E_1}{E_2} = \frac{l_1}{l_2}$

Measuring Internal Resistance

• First, find the balancing length $l_1$ for the cell on open circuit (EMF $E$). Then, connect a known resistance $R$ in parallel with the cell and find the new balancing length $l_2$ (Terminal Voltage $V$).

• The internal resistance $r$ can then be calculated using the formula: $r = R \left( \frac{l_1}{l_2} - 1 \right)$

• Check the Driver Cell: Always ensure the EMF of the driver cell in the primary circuit is greater than the EMF of the cell being measured in the secondary circuit; otherwise, a null point cannot be reached.

• Polarity Check: The positive terminals of both the driver cell and the test cell must be connected to the same end of the potentiometer wire (usually point A).

• Sensitivity Analysis: To increase the sensitivity of a potentiometer, you must decrease the potential gradient ($k$). This is achieved by either increasing the length of the wire or decreasing the current in the primary circuit using a rheostat.

• Uniformity Assumption: Calculations assume the wire has a perfectly uniform cross-sectional area and composition; any variation will lead to a non-linear potential gradient.