Determining Internal Resistance

Conservation of Energy: In a closed loop, the total energy supplied by the source (emf) must equal the sum of the energy dissipated in the external load and the energy lost within the source itself.

The Governing Equation: The relationship between these variables is expressed as $V = \varepsilon - Ir$. This formula shows that as the current $I$ increases, the terminal voltage $V$ decreases linearly.

Ohm's Law for the Whole Circuit: By substituting $V = IR$ (where $R$ is the external load resistance), the equation can be rewritten as $\varepsilon = I(R + r)$. This allows for the calculation of current based on the total resistance of the system.

The Graphical Method: This is the most accurate way to determine $r$. By varying an external rheostat (variable resistor), multiple pairs of $V$ and $I$ readings are recorded and plotted on a graph of $V$ vs. $I$.

Linear Regression Analysis: Since the equation $V = -rI + \varepsilon$ follows the linear form $y = mx + c$, the y-intercept of the best-fit line directly provides the value of the emf ($\varepsilon$).

Calculating the Gradient: The internal resistance $r$ is determined by calculating the magnitude of the gradient (slope) of the $V-I$ graph. Because the voltage drops as current increases, the slope is negative, so $r = -\text{gradient}$.

Single Point Calculation: If only one set of measurements is available, $r$ can be found using $r = \frac{\varepsilon - V}{I}$, provided the emf is already known from an open-circuit measurement.

Identify the Intercepts: In exam questions featuring a $V-I$ graph, always look at the y-intercept first. This is almost always the emf of the cell, which is a common starting point for further calculations.

Check the Gradient Units: Ensure that the units for voltage and current are in Volts and Amperes before calculating the gradient. If current is in milliamperes (mA), the resulting resistance will be off by a factor of 1000.

Sanity Check Values: Internal resistance for standard laboratory cells is typically small (often between $0.1 \Omega$ and $5 \Omega$). If your calculation results in thousands of Ohms for a simple battery, re-check your algebra or unit conversions.

Watch for the X-intercept: The point where the line crosses the x-axis represents the 'short-circuit current'. At this point, $V = 0$, meaning all the emf is being dropped across the internal resistance ($I_{sc} = \frac{\varepsilon}{r}$).

Assuming $V$ is Constant: A frequent mistake is treating the terminal voltage as a fixed value. Students must remember that $V$ changes whenever the external resistance $R$ or the current $I$ changes.

Misinterpreting the Slope: Some learners confuse the slope with the total resistance ($R+r$). The slope of a $V-I$ graph is specifically the internal resistance $r$, not the external or total resistance.

Voltmeter Placement: If a voltmeter is placed incorrectly, it might measure the voltage across a specific component rather than the terminals of the source. Always ensure the voltmeter is in parallel with the power source to measure terminal voltage.