Physics Required Practical: EMF and Internal Resistance

A power source like a battery doesn't deliver its full potential voltage to a circuit; some is lost within the battery itself. This required practical allows you to quantify that loss by determining a cell's electromotive force (EMF)—its maximum potential difference when no current flows—and its internal resistance—the opposition to current flow within the source. Mastering this experiment is crucial for understanding why real-world power supplies behave differently from ideal ones and is a core assessment skill for A-Level Physics.

Core Concepts: EMF, Terminal Voltage, and Internal Resistance

Every electrical source, from a AA battery to a lab power pack, possesses an electromotive force (EMF), denoted by $ϵ$. The EMF is the energy transferred per unit charge when other forms of energy (chemical, in a battery) are converted into electrical energy. It represents the maximum possible potential difference the source can provide. However, when a current ($I$) flows, energy is dissipated within the source due to its internal resistance ($r$). This reduces the voltage available to the external circuit, known as the terminal potential difference ($V$).

The relationship between these quantities is given by the EMF equation: $$ϵ=V+Ir$$ Rearranging this into the form $y=mx+c$ provides the basis for our graphical analysis: $$V=ϵ-Ir$$ Here, a plot of terminal voltage ($V$) on the y-axis against current ($I$) on the x-axis will yield a straight line. The y-intercept (where $I=0$) gives the EMF, $ϵ$, and the negative gradient of the line equals the internal resistance, $r$.

Experimental Setup and Procedure

The aim is to collect simultaneous readings of terminal voltage and current for a range of external circuit resistances. You will need a cell (e.g., a 1.5V D-cell), a variable resistor (rheostat), a voltmeter, an ammeter, and connecting wires.

First, construct a series circuit containing the cell, the ammeter, and the variable resistor. Connect the voltmeter directly across the terminals of the cell to measure the terminal potential difference $V$. It is vital that the voltmeter is connected in parallel with the cell alone, not the entire circuit, to measure the voltage at its terminals. Before closing the circuit, set the variable resistor to its maximum value to minimize the initial current and protect the equipment.

To collect data, systematically decrease the external resistance by adjusting the rheostat. For each setting, record the precise ammeter reading (current, $I$) and the corresponding voltmeter reading (terminal voltage, $V$). Take a minimum of 6-8 pairs of readings across the widest possible range of current, ensuring the current does not become excessively high to avoid overheating the cell or components. Open the circuit between readings to prevent unnecessary cell run-down.

Graph Plotting and Analysis

Plot your results with $V$ on the vertical axis and $I$ on the horizontal axis. Draw a line of best fit through your data points. Because the relationship is $V=ϵ-Ir$, your graph should be a straight line with a negative gradient.

To find the EMF ($ϵ$), extrapolate the line of best fit backwards until it intercepts the y-axis (where $I=0$). The value of $V$ at this intercept is the EMF of the cell. This represents the terminal voltage when no current is drawn—the maximum potential difference.

To calculate the internal resistance ($r$), determine the gradient of the line. Since the equation is $V=ϵ-Ir$, the gradient is $-r$. Therefore, $r=-\text{gradient}$. Choose two widely separated points on the line of best fit (not data points) and use the formula: $$\text{gradient}=\frac{\Delta V}{\Delta I}$$ The internal resistance $r$ will be the negative of this value, and its unit is ohms ($\Omega$).

Common Pitfalls

Systematic Error from Voltmeter Loading: A significant source of error arises if the voltmeter has a relatively low resistance. While it measures the terminal voltage across the cell, it also provides a parallel path for current. This means the current measured by the ammeter ($I$) is not only the current through the variable resistor; a small portion goes through the voltmeter. This leads to an overestimation of the current in your calculations. Consequently, the gradient of your $V$-$I$ graph becomes steeper, causing you to overestimate the internal resistance. Using a high-resistance digital voltmeter minimizes this effect.

Cell Run-Down During Extended Measurements: If the circuit is left closed between readings or if high currents are drawn for prolonged periods, the cell's internal resistance can increase and its EMF can decrease as it depletes. This makes the later readings inconsistent with the earlier ones, causing scatter on your graph and making it difficult to determine a reliable line of best fit. To mitigate this, open the circuit (break the connection) immediately after taking each reading and work efficiently to complete data collection quickly.

Poor Graphical Technique: A common analytical mistake is forcing the line of best fit through the origin or incorrectly determining the gradient. The line should be a balanced fit through the data points, and it must be extrapolated to the y-axis to find the EMF, not assumed to pass through (0,0). When calculating the gradient, always use points from the line of best fit itself, not individual data points, to average out random errors. Failing to use a sharp pencil and a clear ruler can also introduce unnecessary inaccuracies in your final values.

Summary

• The electromotive force (EMF) of a cell is its maximum potential difference and is found by extrapolating the $V$-$I$ graph to the y-intercept where current is zero.
• The internal resistance of the cell is the cause of lost voltage within the source and is equal to the negative gradient of the terminal voltage against current graph.
• The fundamental equation linking these quantities is $ϵ=V+Ir$, which rearranges to $V=ϵ-Ir$ for graphical analysis.
• The main experimental errors are voltmeter loading (leading to an overestimation of $r$) and cell run-down (causing non-linear scatter in results), both of which must be managed during the procedure.
• Accurate graphical work, including a correctly plotted line of best fit and careful gradient calculation using points from that line, is essential for determining precise and reliable values.