Current Electricity - Potentiometer and Cells

Electromotive Force (EMF): The potential difference across the terminals of a cell when no current is being drawn from it ($I = 0$). It is denoted by $E$.

Terminal Potential Difference ($V$): The potential difference across the terminals of a cell when current $I$ flows through the circuit. It is related to EMF by $V = E - Ir$, where $r$ is the internal resistance.

Internal Resistance ($r$): The resistance offered by the electrolyte and electrodes inside a cell to the flow of current. It depends on the nature of the electrolyte, area of electrodes, and distance between them.

Cells in Series: When cells are connected in series, the equivalent EMF is the sum of individual EMFs ($E_{eq} = E_1 + E_2$) and equivalent internal resistance is $r_{eq} = r_1 + r_2$.

Cells in Parallel: For two cells connected in parallel, the equivalent EMF is $E_{eq} = \frac{E_1 r_2 + E_2 r_1}{r_1 + r_2}$ and the equivalent internal resistance is $\frac{1}{r_{eq}} = \frac{1}{r_1} + \frac{1}{r_2}$.

Potentiometer Principle: The potential drop across any portion of a wire of uniform cross-section is directly proportional to the length of that portion ($V \propto l$), provided a constant current flows through it. This gives the potential gradient $k = \frac{V}{L}$.

Comparison of EMFs: Two cells of EMFs $E_1$ and $E_2$ can be compared using a potentiometer by finding their respective balancing lengths $l_1$ and $l_2$, giving $\frac{E_1}{E_2} = \frac{l_1}{l_2}$.

Measurement of Internal Resistance: A potentiometer can measure the internal resistance $r$ of a cell using the formula $r = R \left( \frac{l_1}{l_2} - 1 \right)$, where $l_1$ is the open circuit balancing length and $l_2$ is the closed circuit balancing length with external resistance $R$.

Sensitivity of Potentiometer: A potentiometer is more sensitive if it has a smaller potential gradient $k$. This can be achieved by increasing the length of the potentiometer wire or decreasing the current in the primary circuit.