Introduction to CP8: E.m.f. and Internal Resistance

Welcome! In this chapter, we explore a reality of physics that often surprises students: batteries (cells) are not perfect. If you have ever noticed your phone getting warm while it's working hard, or your car headlights dimming slightly when you start the engine, you have seen internal resistance in action. This core practical (CP8) is all about measuring that "hidden" resistance and the total energy a cell can provide. Since this is part of your Unit 3 (Practical Skills in Physics I) assessment, we will focus on the setup, the measurements, and how to analyze your data to get the right results.

The Core Concepts: E.m.f. vs. Terminal P.D.

Before we jump into the experiment, let’s clear up the terminology. Don't worry if these terms feel similar at first; the difference is simply about where the energy is going.

1. Electromotive Force (e.m.f., \(\varepsilon\)): This is the total energy the cell provides to each coulomb of charge. Think of it as the "theoretical maximum" voltage the cell can give when no current is flowing. It is measured in Volts (V).

2. Terminal Potential Difference (terminal p.d., \(V\)): This is the actual voltage delivered to the external circuit (like a bulb or a resistor). It is always slightly less than the e.m.f. when a current is flowing because some energy is "lost" inside the battery itself.

3. Internal Resistance (\(r\)): This is the resistance to the flow of charge inside the cell materials. As current flows, this resistance causes a voltage drop, often called "lost volts."

The Golden Equation:
\(\varepsilon = V + Ir\)
Where:
\(\varepsilon\) = e.m.f. (V)
\(V\) = Terminal potential difference (V)
\(I\) = Current (A)
\(r\) = Internal resistance (\(\Omega\))

Analogy: Imagine you win \$100 (\(\varepsilon\)). However, to collect it, you have to pay a \$5 delivery fee (\(Ir\)). The amount you actually get to spend in the shops is \$95 (\(V\)). The internal resistance is like that annoying delivery fee!

The Experiment: Step-by-Step

In the exam, you might be asked to describe or evaluate this procedure. Here is how it is done in the lab.

Apparatus

  • Cell (the source we are testing).
  • Variable Resistor (or Rheostat) to change the current.
  • Ammeter to measure current (\(I\)).
  • Voltmeter to measure terminal p.d. (\(V\)).
  • Switch (very important to prevent the cell from overheating!).
  • Connecting wires.

The Setup

Connect the cell in series with the ammeter, the switch, and the variable resistor. Connect the voltmeter in parallel directly across the terminals of the cell. This allows the voltmeter to measure the terminal p.d. (\(V\)) specifically.

Method

  1. Close the switch and adjust the variable resistor to its highest resistance (this keeps the current low to start).
  2. Record the current (\(I\)) from the ammeter and the terminal p.d. (\(V\)) from the voltmeter.
  3. Adjust the variable resistor to decrease the resistance, which increases the current.
  4. Take a range of readings (at least 6 different sets of \(V\) and \(I\)).
  5. Crucial Tip: Open the switch between readings! This prevents the internal resistance from changing due to temperature increases and keeps your battery from running flat.

Analyzing the Data: The Graph

This is the part most likely to appear in your Unit 3 exam. We rearrange our golden equation into the form of a straight-line graph equation: \(y = mx + c\).

Starting with \(\varepsilon = V + Ir\), we move things around:
\(V = -rI + \varepsilon\)

If we plot \(V\) on the y-axis and \(I\) on the x-axis:
- The gradient (slope) of the line is \(-r\) (negative internal resistance).
- The y-intercept (where the line crosses the vertical axis) is \(\varepsilon\) (the e.m.f.).

Quick Review Box:
Gradient = \(-r\)
y-intercept = \(\varepsilon\)
Always use a large triangle to calculate your gradient to reduce uncertainty!

Uncertainties and Errors

In Unit 3, you are expected to comment on the quality of the data.

Systematic Errors

If your voltmeter or ammeter has a zero error (it doesn't show 0.00 when disconnected), all your readings will be shifted. Check the meters before you start!

Random Errors and Precision
  • Repeat Readings: To improve reliability, take repeat readings for each setting of the variable resistor and calculate a mean.
  • Uncertainty Calculation: For repeat readings, the uncertainty is half the range (\(\frac{\text{max} - \text{min}}{2}\)).
  • Resolution: The resolution is the smallest change the instrument can detect. For a digital voltmeter, this might be \(0.01\text{ V}\).
Heating Effects

If the cell stays connected for too long, it warms up. Since resistance changes with temperature, this would make your "internal resistance" value drift during the experiment. Always use a switch!

Common Exam Mistakes to Avoid

1. Forgetting the minus sign: When you calculate a negative gradient from your graph, remember that the internal resistance \(r\) is a positive value. If your gradient is \(-1.5\), then \(r = 1.5\ \Omega\).

2. Misidentifying the Intercept: The e.m.f. is the value of \(V\) when \(I = 0\). If your graph axis doesn't start at zero (a "broken axis"), you cannot read the e.m.f. directly from the intercept; you must calculate it using \(y = mx + c\).

3. Significant Figures: Always give your final answer to the same number of significant figures as your raw data (usually 2 or 3).

Key Takeaways

  • E.m.f. (\(\varepsilon\)) is the total energy per charge; Terminal P.D. (\(V\)) is what the circuit actually gets.
  • The difference between them is the "lost volts" (\(Ir\)) due to internal resistance (\(r\)).
  • The practical involves varying a resistor and measuring \(V\) and \(I\).
  • A graph of \(V\) against \(I\) gives a straight line with gradient \(-r\) and intercept \(\varepsilon\).
  • Minimize errors by using a switch and taking a wide range of readings.

Next Step: Why not try drawing a quick sketch of a \(V-I\) graph? Label where you would find the e.m.f. and how you would find the internal resistance. It’s a classic exam question!