21.2 Electromotive Force: Terminal Voltage

Electromotive Force and Terminal Voltage

Electromotive force (EMF) and terminal voltage describe how a power source behaves when it's actually doing work versus sitting idle. EMF is the maximum voltage a source can provide (when nothing is connected), while terminal voltage is what you actually get once current starts flowing. The gap between them comes down to one thing: internal resistance.

Voltage vs electromotive force

Despite the name, electromotive force isn't actually a force. It's a voltage, measured in volts (V). The symbol is $\mathcal{E}$.

• EMF ($\mathcal{E}$) is the potential difference a power source produces in an open circuit, meaning nothing is connected and no current flows. It's determined by the source's internal properties (battery chemistry, solar cell materials, etc.).
• Terminal voltage ($V_T$) is the potential difference measured across the source's terminals when it's connected to a load and current is flowing. This is always less than or equal to the EMF.

The relationship between them is:

$V_T = \mathcal{E} - Ir$

• $I$ is the current flowing through the circuit
• $r$ is the internal resistance of the power source

The term $Ir$ is the voltage drop across the internal resistance. Think of it this way: some of the source's voltage gets "used up" just pushing current through the source itself, so the load never sees the full EMF.

Effects of internal resistance

Internal resistance ($r$, measured in ohms $\Omega$) is the resistance within the power source itself. Every real battery, generator, or solar cell has some amount of it.

Here's how it affects circuit performance:

• Reduces terminal voltage. From $V_T = \mathcal{E} - Ir$, a larger $r$ means a bigger voltage drop inside the source, leaving less voltage for the load.
• Reduces current. The total current in the circuit follows $I = \frac{\mathcal{E}}{R + r}$, where $R$ is the load resistance. Internal resistance adds to the total resistance the current must flow through, so the current decreases.
• Reduces power to the load. Since $P = I^2R$, lower current means less power delivered to whatever you've connected. A flashlight bulb dims, a motor slows down.

The practical takeaway: a battery with high internal resistance (like an old or cheap battery) delivers noticeably less voltage and power to your device than a fresh one with low internal resistance, even if both have the same rated EMF.

Advantages of parallel voltage sources

When you connect identical voltage sources in parallel (positive to positive, negative to negative), the voltage across the load stays the same as a single source, but the current capacity increases.

• Increased current capacity. Each source contributes current, so $I_{\text{total}} = I_1 + I_2 + \ldots + I_n$. The load can draw more current without overloading any single source.
• Reduced effective internal resistance. Multiple internal resistances in parallel combine to a smaller total internal resistance, which means less voltage drop and a terminal voltage closer to the EMF.
• Redundancy. If one source fails, the remaining sources keep supplying power. This is why backup generator systems and solar panel arrays use parallel connections for reliability.
• Load sharing. Each source handles only a fraction of the total current, reducing stress on individual sources and extending their lifespan.

Circuit Analysis and Power Dissipation

These concepts tie directly into analyzing real circuits:

• The electric potential difference between two points in a circuit is what drives current flow. Current always flows from higher to lower potential.
• Power dissipation occurs whenever current flows through a resistance. Electrical energy converts to heat (or other forms) in both the load and the internal resistance. The power dissipated in any resistor is $P = I^2R$.
• In a circuit with internal resistance, total power from the source splits between the load ($P_{\text{load}} = I^2R$) and the internal resistance ($P_{\text{internal}} = I^2r$). This means not all the energy from the source reaches the load.