Introduction: The "Perfect" Battery vs. Reality

Have you ever noticed that your phone or laptop gets warm when you’re using it heavily? Or perhaps you’ve noticed that the headlights of an old car dim slightly just as the engine starts up? These things happen because power supplies—like batteries and cells—aren't perfect. In this chapter, we are going to look at why batteries have their own "internal tax" on energy, known as internal resistance, and the total energy they can provide, known as electromotive force (emf).

Don't worry if these terms sound a bit technical! By the end of these notes, you’ll see that it’s all just a simple balancing act of energy.

1. What is Electromotive Force (emf)?

In the previous chapter, Basics of Electricity, we defined potential difference (pd) as the work done per unit charge. Electromotive force (\(\varepsilon\)) is very similar, but it describes the source of the energy.

The Definition: The emf of a source is the total energy transferred by the source to each coulomb of charge that passes through it.

The Units: Even though it has "force" in the name, emf is NOT a force. It is measured in Volts (\(V\)), just like potential difference. One volt is one joule per coulomb (\(1V = 1 J C^{-1}\)).

Analogy: Think of emf as the "Total Budget" a battery has. It is the maximum amount of energy the battery can possibly give to the charges before it starts "spending" some on itself.

2. Internal Resistance: The Battery's "Tax"

Inside a battery, chemicals are reacting to move electrons. However, the materials inside the battery (the chemicals and metal plates) aren't perfect conductors—they have their own resistance. This is called internal resistance (\(r\)).

  • Internal Resistance (\(r\)): The resistance to current flow inside the power source itself.
  • Terminal Potential Difference (\(V\)): The voltage measured across the terminals of the battery when a current is flowing. This is the "useful" voltage that actually reaches the rest of your circuit.
  • "Lost Volts" (\(v\)): The energy wasted per coulomb of charge overcoming the internal resistance inside the battery.

The Key Concept: When a current flows, some energy is always wasted inside the battery as heat. This is why:
Terminal pd = emf – Lost Volts

Quick Review:

If no current is flowing (an "open circuit"), no volts are lost. In this specific case, the Terminal pd is equal to the emf.

3. The Fundamental Equations

To solve problems in Physics 7407, you need to be comfortable with the relationship between these values. We use the symbol \(\varepsilon\) (the Greek letter epsilon) for emf.

The standard equation is:
\(\varepsilon = I(R + r)\)

Where:
\(\varepsilon\) = Electromotive force (V)
\(I\) = Current (A)
\(R\) = External resistance of the circuit (\(\Omega\))
\(r\) = Internal resistance of the battery (\(\Omega\))

We can expand this to see the "Lost Volts" clearly:
\(\varepsilon = IR + Ir\)
Since \(V = IR\) (the terminal pd), we can also write:
\(\varepsilon = V + Ir\)

Common Mistake to Avoid: Students often forget that \(I\) is the same everywhere in a simple series circuit. The current flowing through the external resistor \(R\) is exactly the same current flowing through the internal resistance \(r\).

4. Required Practical 6: Investigating emf and Internal Resistance

As part of your AS Level, you must investigate how the terminal pd (\(V\)) changes as you change the current (\(I\)).

The Setup:

1. Connect a cell in series with an ammeter and a variable resistor.
2. Connect a voltmeter in parallel across the terminals of the cell.
3. By changing the resistance of the variable resistor, you can change the current flowing through the circuit.

The Math behind the Graph:

We can rearrange our formula \(\varepsilon = V + Ir\) into the format of a straight-line graph (\(y = mx + c\)):
\(V = -rI + \varepsilon\)

  • If we plot Terminal pd (\(V\)) on the \(y\)-axis and Current (\(I\)) on the \(x\)-axis...
  • The gradient of the graph will be \(-r\) (negative internal resistance).
  • The y-intercept (where the line hits the vertical axis) will be the emf (\(\varepsilon\)).

Study Tip: The graph always slopes downwards. This makes sense: as you draw more current (\(I\)), more volts are "lost" inside the battery (\(Ir\)), so the remaining terminal pd (\(V\)) must go down!

5. Summary and Key Takeaways

Don't let the symbols intimidate you. Just remember these three points:

1. Emf (\(\varepsilon\)) is the total energy per unit charge available from the source.

2. Internal resistance (\(r\)) causes "lost volts" (\(Ir\)) which are dissipated as heat inside the battery.

3. Terminal pd (\(V\)) is what's left over for the external circuit to use.

Quick Formula Check:

\(\varepsilon = V + Ir\)
\(\text{Total Energy} = \text{Useful Energy} + \text{Wasted Energy}\)

Did you know? High-voltage power supplies (like those used in school labs) often have a very high internal resistance. This is a safety feature—if you accidentally short-circuit it, the internal resistance "gobbles up" most of the energy, preventing a dangerously high current from flowing!

Next Steps: You can now use these principles to solve circuit problems involving resistors in series and parallel, which you learned in the Circuits chapter.