Mastering Internal Resistance and EMF in A-Level Physics

Introduction to EMF and Internal Resistance

In your A-Level Physics studies, you have likely encountered the ideal battery, which provides a constant voltage regardless of the circuit conditions. However, real-world components behave differently. Every power source, whether it is a chemical cell or a generator, possesses internal resistance. This is the resistance to the flow of charge within the source itself, caused by the materials and chemical processes inside.

Understanding internal resistance is crucial for your exams because it explains why the voltage across a battery terminals drops when you draw a large current. By mastering the concepts of Electromotive Force (EMF) and 'lost volts', you will be able to accurately model real-world circuits and predict how they behave under load.

Defining EMF and Terminal Potential Difference

Electromotive Force (EMF), denoted by the symbol $\epsilon$, is defined as the total energy supplied by a source per unit charge. It represents the total potential difference the battery would provide if no current were flowing. Its unit is the Volt (V), which is equivalent to Joules per Coulomb (J C⁻¹).

In contrast, the terminal potential difference ($V$) is the voltage measured across the terminals of the battery when a current ($I$) is flowing. Because of the internal resistance ($r$), some energy is dissipated as heat inside the battery whenever current flows. This leads to the fundamental relationship:

$$\epsilon = V + Ir$$

Where $Ir$ represents the 'lost volts'—the potential difference across the internal resistance.

The Concept of Lost Volts

When a circuit is closed, current flows through both the external load resistor ($R$) and the internal resistance ($r$) of the battery. The internal resistance acts as a resistor in series with the rest of the circuit. Consequently, the voltage available to the external circuit is always less than the EMF.

We define the 'lost volts' ($v$) as the potential difference across the internal resistance:

$$v = Ir$$

Therefore, the terminal potential difference is given by:

$$V = \epsilon - Ir$$

As the current $I$ increases, the lost volts increase, causing the terminal potential difference to decrease. This is why your phone battery might show a lower percentage when you are running power-intensive apps compared to when the device is idle.

Worked Example 1: Calculating Terminal Voltage

A battery has an EMF of 12.0 V and an internal resistance of 0.50 $\Omega$. It is connected to an external load resistor of 5.5 $\Omega$. Calculate the terminal potential difference.

Step 1: Find the total resistance of the circuit. Since the internal resistance and load resistor are in series: $R_{total} = R + r = 5.5 + 0.5 = 6.0 , \Omega$

Step 2: Calculate the current in the circuit using Ohm's Law. $I = \frac{\epsilon}{R_{total}} = \frac{12.0}{6.0} = 2.0 , \text{A}$

Step 3: Calculate the terminal potential difference. $V = \epsilon - Ir = 12.0 - (2.0 \times 0.5) = 12.0 - 1.0 = 11.0 , \text{V}$

Worked Example 2: Determining Internal Resistance

A cell with an EMF of 1.5 V drives a current of 0.20 A through a circuit. The terminal potential difference is measured to be 1.4 V. Calculate the internal resistance of the cell.

Step 1: Identify the lost volts. $v = \epsilon - V = 1.5 - 1.4 = 0.1 , \text{V}$

Step 2: Use the formula for lost volts ($v = Ir$) to find $r$. $0.1 = 0.20 \times r$ $r = \frac{0.1}{0.20} = 0.50 , \Omega$

Common Mistakes

1. Confusing EMF with Terminal PD: Remember that EMF is the total energy per unit charge available from the source, while terminal PD is what is actually available to the external circuit. They are only equal when $I = 0$.
2. Ignoring Internal Resistance in Series: Students often forget to add $r$ to the total resistance when calculating the current in a circuit. Always treat $r$ as a resistor in series with the load.
3. Sign Errors: When rearranging $\epsilon = V + Ir$, ensure you correctly move terms across the equals sign. The lost volts are always subtracted from the EMF to find the terminal voltage.

Frequently Asked Questions

What happens to the terminal voltage if the external resistance is zero? If the external resistance is zero (a short circuit), the current becomes $I = \epsilon / r$. The terminal voltage drops to zero because all the EMF is dropped across the internal resistance.

Does internal resistance change? In many A-Level problems, $r$ is treated as a constant. In reality, it can change as a battery discharges or as its temperature changes, but for exam purposes, assume it is constant unless stated otherwise.

Why is internal resistance important? It limits the maximum current a battery can supply and causes the battery to heat up during use, which is a significant factor in battery efficiency and safety.

Conclusion

Mastering the relationship between EMF, internal resistance, and terminal potential difference is a cornerstone of A-Level electricity. By understanding how to account for 'lost volts', you can solve complex circuit problems with ease. To see these concepts in action with interactive visualisations, head over to MathInstructor AI and generate a free animated lesson on this topic today.