Terminal Potential Difference: The Real Voltage Under Load
EMF and Internal Resistance: The Hidden Players
Every power source, be it a simple AA battery or a complex power supply, has two fundamental properties: Electromotive Force (EMF) and internal resistance.
The EMF, represented by the symbol $\mathcal{E}$, is the energy per unit charge provided by the source when no current is flowing. It's the ideal, maximum voltage the source can produce. Think of it as the "advertised" voltage, like the 1.5 V printed on a battery.
However, no power source is perfect. Inside the battery, the chemicals offer a small resistance to the flow of electric charge. This is the internal resistance, denoted by $r$. It acts like a tiny resistor hidden inside the battery itself. When no current flows, this internal resistance has no effect, and the voltage across the terminals is exactly equal to the EMF. But the moment you close a circuit and current ($I$) starts to flow, a voltage drop occurs across this internal resistance.
The relationship between EMF, terminal voltage, current, and internal resistance is given by: $V_{terminal} = \mathcal{E} - I r$
Where:
• $V_{terminal}$ is the terminal voltage (in volts, V).
• $\mathcal{E}$ is the electromotive force (in volts, V).
• $I$ is the current flowing from the source (in amperes, A).
• $r$ is the internal resistance of the source (in ohms, $\Omega$).
Visualizing the Voltage Drop in a Circuit
Imagine a circuit with a battery connected to a single resistor, which we call the load resistor ($R_{load}$). The total resistance in the circuit is the sum of the load resistance and the internal resistance ($R_{total} = R_{load} + r$).
According to Ohm's Law[3], the current in the circuit is $I = \frac{\mathcal{E}}{R_{load} + r}$.
This current $I$ flows through both the internal resistance $r$ and the load resistance $R_{load}$. The voltage drop across the internal resistance is $I r$. This is the "lost" voltage – energy that is converted to heat inside the battery itself and is not available to the external circuit. The remaining voltage, which is the EMF minus this lost voltage, is the terminal voltage: $V_{terminal} = \mathcal{E} - I r$.
Special Cases and Their Significance
The terminal voltage changes depending on the circuit conditions. The table below summarizes three key scenarios.
| Condition | Current ($I$) | Terminal Voltage ($V_{terminal}$) | Explanation |
|---|---|---|---|
| Open Circuit (No load) | $I = 0$ | $V_{terminal} = \mathcal{E}$ | With no current, there is no voltage drop ($I r = 0$) across the internal resistance. The terminal voltage equals the EMF. This is how a voltmeter measures the EMF of a battery. |
| Normal Operation | $I > 0$ | $V_{terminal} < \mathcal{E}$ | Current flows, so a voltage drop ($I r$) occurs inside the source. The terminal voltage is less than the EMF. The higher the current, the greater the drop. |
| Short Circuit | $I = \mathcal{E} / r$ (Maximum) | $V_{terminal} = 0$ | If the terminals are connected with a wire of negligible resistance, the entire EMF is dropped across the internal resistance. The terminal voltage, available to the "outside," becomes zero. This is very dangerous as it produces extremely high current and can cause fires. |
A Practical Example: Dimming Flashlight Bulbs
A common observation that perfectly illustrates terminal voltage is a flashlight with old batteries. A new battery might have an EMF of 1.5 V and a very low internal resistance, say $0.1\ \Omega$. The bulb might have a resistance of $5\ \Omega$.
The current would be: $I = \frac{1.5\ V}{5\ \Omega + 0.1\ \Omega} \approx 0.294\ A$.
The terminal voltage would be: $V_{terminal} = 1.5\ V - (0.294\ A)(0.1\ \Omega) \approx 1.47\ V$. This is very close to the EMF, so the bulb shines brightly.
Now, as the battery is used, its internal resistance increases (due to chemical changes). Suppose the internal resistance rises to $2\ \Omega$ while the EMF remains roughly 1.5 V for a while.
Now the current is: $I = \frac{1.5\ V}{5\ \Omega + 2\ \Omega} \approx 0.214\ A$.
The terminal voltage is now: $V_{terminal} = 1.5\ V - (0.214\ A)(2\ \Omega) \approx 1.07\ V$.
This significant drop in terminal voltage means the bulb receives much less power, causing it to glow dimly. Eventually, the terminal voltage drops so low that the bulb cannot light at all, even though the battery still has a measurable EMF when tested with a voltmeter with no load.
Common Mistakes and Important Questions
Q: Is the terminal voltage always less than the EMF?
For a single source providing power, yes. The formula $V_{terminal} = \mathcal{E} - I r$ shows that as long as current is flowing out of the source ($I > 0$), the terminal voltage must be less than the EMF. The only exception is if the source is being charged (like a rechargeable battery), where current is forced into the positive terminal. In that case, the terminal voltage is $V_{terminal} = \mathcal{E} + I r$, which is greater than the EMF.
Q: Why does a battery get warm when it's used heavily?
The warmth is directly caused by the internal resistance. When a high current is drawn, the power dissipated as heat inside the battery is given by $P = I^2 r$. This energy is wasted and does not power the external device, which is why batteries are designed to have very low internal resistance.
Q: Can the terminal voltage ever be equal to the EMF in a working circuit?
Only if the internal resistance is zero, which is impossible for a real power source. Some sources, like laboratory power supplies, are designed with complex electronics to have a very low and effectively constant terminal voltage regardless of the current drawn, but this is an engineered outcome, not the natural behavior of a simple chemical cell.
The terminal potential difference is a fundamental concept for understanding how real-world power sources behave. It bridges the gap between the ideal electromotive force and the practical voltage delivered to a circuit. The key takeaway is that the voltage available to do useful work is always reduced by the voltage lost to the source's own internal resistance. This explains everyday phenomena like dimming flashlight bulbs and warming batteries, grounding abstract electrical principles in tangible reality. Mastering this relationship is essential for anyone designing, troubleshooting, or simply seeking to understand electronic circuits.
Footnote
[1] EMF (Electromotive Force): Despite its name, it is not a force. It is a potential difference (voltage) and represents the maximum energy per unit charge that a source can provide when no current is flowing. It is measured in volts (V).
[2] Internal Resistance: The inherent resistance to the flow of electric current within a power source itself, such as a battery or generator. It is symbolized by $r$ and measured in ohms ($\Omega$).
[3] Ohm's Law: A fundamental law of circuits stating that the current ($I$) through a conductor between two points is directly proportional to the voltage ($V$) across the two points and inversely proportional to the resistance ($R$) between them. It is expressed by the formula $V = I R$.