Introduction: Why Your Phone Gets Warm

Have you ever noticed your phone getting warm after playing a game for a long time, or noticed that a car's headlights dim slightly when the engine starts? This happens because of a fundamental concept in physics: Internal Resistance. While we often imagine batteries as perfect "pumps" of energy, in reality, they have to use up some of their own energy just to get the electricity moving through themselves. In this chapter, we will explore the difference between the energy a battery promises to give and the energy it actually delivers.

1. Understanding E.m.f. and Terminal P.D.

To master this topic, you first need to distinguish between two types of "voltage":

Electromotive Force (e.m.f.)

The e.m.f. (represented by the symbol \( \epsilon \)) is the total energy supplied by a source (like a battery or cell) to each unit of charge that passes through it. It is measured in Volts (V), where \( 1 \text{ V} = 1 \text{ J C}^{-1} \). Think of this as the "theoretical maximum" voltage the battery could provide if it were perfect.

Terminal Potential Difference (Terminal p.d.)

The terminal p.d. (represented by \( V \)) is the potential difference across the terminals of the source when a current is flowing. This is the "actual" voltage delivered to the rest of the circuit (the external resistance). Because some energy is always lost inside the battery, the terminal p.d. is almost always less than the e.m.f.

The Core Difference:
- e.m.f. = Work done on the charge (converting chemical energy to electrical energy).
- p.d. = Work done by the charge (converting electrical energy into other forms like heat or light in the circuit).

2. Internal Resistance: The "Energy Thief"

Every power source has some resistance due to the materials it is made of (like the chemicals inside a cell). This is called internal resistance (represented by a lowercase \( r \)).

When current \( I \) flows through a battery, some potential difference is "used up" overcoming this internal resistance. We call these "lost volts".

The Calculation for Lost Volts:
Using Ohm's Law (\( V = IR \)), the lost volts can be calculated as:
\( \text{Lost Volts} = I \times r \)

Don't worry if this seems tricky! Just remember that the battery is essentially a "perfect" voltage source in series with a small resistor (\( r \)).

3. The Fundamental Equation

Based on the Conservation of Energy, the total energy supplied by the source must equal the energy used in the external circuit plus the energy lost inside the source.

The relationship is written as:
\( \epsilon = V + Ir \)

Since the terminal p.d. (\( V \)) is also equal to \( I \times R \) (where \( R \) is the external resistance of the circuit), we can also write:
\( \epsilon = I(R + r) \)

Key Takeaway: If the current (\( I \)) increases, the "lost volts" (\( Ir \)) also increase. This means the terminal p.d. (\( V \)) must decrease. This is why a battery's output voltage drops when you draw more current from it!

4. Core Practical 8: Determining e.m.f. and \( r \)

In your exam, you may be asked how to find the e.m.f. and internal resistance of a cell experimentally. This is Core Practical 8.

The Setup

1. Connect a cell in series with an ammeter and a variable resistor (rheostat).
2. Connect a voltmeter in parallel across the terminals of the cell.

The Procedure

1. Change the resistance of the variable resistor to get different values for current (\( I \)) and terminal p.d. (\( V \)).
2. Record at least 6 different pairs of readings for \( V \) and \( I \).
3. Use a switch between readings to prevent the cell from heating up, which would change its internal resistance!

Analyzing the Data

We rearrange the equation \( \epsilon = V + Ir \) into the form of a straight-line graph (\( y = mx + c \)):
\( V = -rI + \epsilon \)

If you plot Terminal p.d. (\( V \)) on the y-axis and Current (\( I \)) on the x-axis:
- The y-intercept equals the e.m.f. (\( \epsilon \)).
- The gradient (slope) of the line equals \( -r \) (negative internal resistance).

Memory Trick: The graph always slopes downwards because as current goes up, the "lost volts" steal more of the available voltage!

5. Important Practical Considerations

When working on these problems or practicals, keep these Pearson Edexcel standards in mind:

  • Precision and Resolution: A standard digital voltmeter usually has a resolution of \( 0.01 \text{ V} \). Ensure your readings reflect the precision of the instrument.
  • Systematic Errors: If your voltmeter isn't calibrated correctly (a "zero error"), your intercept for e.m.f. will be inaccurate.
  • Calculations: When calculating the gradient, always use a large triangle on your graph to reduce percentage uncertainty.

Quick Review Box

e.m.f. (\( \epsilon \)): Total energy per unit charge supplied by the source.
Terminal p.d. (\( V \)): Voltage measured across the battery terminals when current flows.
Internal Resistance (\( r \)): The resistance inside the battery itself.
Lost Volts (\( Ir \)): The voltage used up inside the battery.
The Equation: \( \epsilon = V + Ir \)

Common Mistake to Avoid: Many students think that e.m.f. is a force because of the name. It is not a force; it is energy per unit charge (Potential Difference), measured in Volts!