Electromotive Force and Internal Resistance

TL;DR

Electromotive Force (EMF) is the maximum potential difference a source can provide, even when no current flows. Internal resistance in a power source causes some energy to be lost, reducing the actual voltage available at its terminals. This internal resistance makes the terminal voltage drop as more current is drawn from the source.

1. The Mental Model

Imagine a perfect battery as a pump that always pushes water with the same force (EMF). But inside the battery there's a narrow pipe (internal resistance) that resists the water flow, so less "pressure" (voltage) comes out of the battery when you're using a lot of water.

2. The Core Material

When we talk about a power source like a battery or a generator, we often think of it as having a fixed voltage. However, in reality, every power source has some inherent resistance inside it. This is called internal resistance, and it affects the voltage you actually get out of the source.

Let's break down the key terms:

2.1 Electromotive Force (EMF, $\mathcal{E}$)

EMF is the total energy per unit charge that a power source can provide. Think of it as the "strength" of the source when it's just sitting there, not powering anything. It's the maximum potential difference across its terminals when no current is flowing (i.e., in an open circuit). EMF is measured in Volts (V).

2.2 Internal Resistance (r)

This is the opposition to the flow of charge within the power source itself. No power source is perfectly efficient; some energy is always lost as heat due to this internal resistance when current flows. Internal resistance is measured in Ohms ($\Omega$).

2.3 Terminal Voltage (V)

This is the actual voltage you measure across the terminals of the power source when it's connected to a circuit and providing current. Because of internal resistance, the terminal voltage is always less than the EMF when current is flowing.

Here's the crucial relationship:

Terminal Voltage (V) = EMF ($\mathcal{E}$) - (Current (I) $\times$ Internal Resistance (r))

So, $V = \mathcal{E} - Ir$.

This equation shows that as the current (I) drawn from the source increases, the voltage drop across the internal resistance ($Ir$) increases, and consequently, the terminal voltage (V) decreases. If no current flows ($I=0$), then $V = \mathcal{E}$, meaning the terminal voltage equals the EMF.

You can also think of the total resistance in the circuit as the external resistance (R) plus the internal resistance (r). Using Ohm's Law for the entire circuit:

$\mathcal{E} = I(R + r)$

And therefore, $I = \frac{\mathcal{E}}{R + r}$.

Let's visualize the relationship between EMF, internal resistance, and terminal voltage:

graph TD
    A["Power Source (e.g., Battery)"] --> B["EMF ($\mathcal{E}$)"]
    B --> C["Internal Resistance (r)"]
    C --> D{"Circuit Connected?"}
    D -- "No (Open Circuit)" --> E["Terminal Voltage (V) = EMF ($\mathcal{E}$)"]
    D -- "Yes (Current Flows)" --> F["Voltage Drop (Ir) Across Internal Resistance"]
    F --> G["Terminal Voltage (V) = EMF ($\mathcal{E}$) - Ir"]
    G --> H["Energy Loss as Heat"]

3. Worked Example

You have a 1.5 V battery. When you connect it to a light bulb and measure the current, you find it's 0.2 A. The voltage across the battery's terminals at this point is 1.3 V. What is the internal resistance of the battery?

We're given:
* EMF ($\mathcal{E}$) = 1.5 V (this is the battery's stated voltage, usually its EMF)
* Current (I) = 0.2 A
* Terminal Voltage (V) = 1.3 V

We need to find the internal resistance (r).

Using the formula: $V = \mathcal{E} - Ir$

Rearrange to solve for $Ir$:
$Ir = \mathcal{E} - V$

Substitute the values:
$Ir = 1.5 \text{ V} - 1.3 \text{ V}$
$Ir = 0.2 \text{ V}$

Now, solve for $r$:
$r = \frac{0.2 \text{ V}}{I}$
$r = \frac{0.2 \text{ V}}{0.2 \text{ A}}$
$r = 1 \Omega$

So, the internal resistance of the battery is 1 Ohm.

4. Key Takeaways

• EMF is the maximum potential difference a source can provide, measured when no current flows.
• Internal resistance is the opposition to current flow within the power source itself.
• Terminal voltage is the actual voltage available at the source's terminals when current is being drawn.
• As current increases, the voltage drop across internal resistance increases, and terminal voltage decreases.
• The formula $V = \mathcal{E} - Ir$ is fundamental for understanding real-world power sources.
• Internal resistance causes some energy to be dissipated as heat within the source.

Common Mistakes to Avoid:
- Don't confuse EMF with terminal voltage; they are only equal when no current flows.
- Forgetting to account for internal resistance when analyzing circuits with real power sources.
- Assuming a battery's stated voltage (e.g., 1.5 V) is always its terminal voltage under load.
- Not understanding that internal resistance causes a drop in voltage, not an increase.

5. Now Try It

You have a power supply with an EMF of 12 V and an internal resistance of $0.5 \Omega$. If you connect it to a load that draws a current of 2 A, what will be the terminal voltage across the power supply? You have 15 minutes. Success means you can clearly state the terminal voltage and explain how you arrived at your answer using the relevant formula.