Internal Resistance and EMF Measurement

Every battery or power supply has a hidden, inherent limit to its performance, a concept crucial for designing efficient circuits and diagnosing why devices fail under load. Understanding internal resistance—the opposition to current flow within the source itself—is key to explaining why the measured voltage across a battery’s terminals drops when you switch on a circuit.

The Fundamental Relationship: EMF, Terminal Voltage, and Lost Volts

The electromotive force (EMF), denoted by $\mathcal{E}$, is the maximum potential difference a source can provide when no current is flowing. It represents the total energy per unit charge supplied by the source, such as the chemical energy converted in a battery. However, when a source is connected to a circuit and current $I$ flows, you measure a lower terminal voltage $V$ across its external terminals.

The difference is caused by the source’s internal resistance $r$. This resistance acts like a small resistor in series with the ideal source. As current flows, a voltage drop occurs across this internal resistance. This drop is often called the lost volts. The core equation linking these quantities is:

$$\mathcal{E}=V+Ir$$

Here, $V=IR$ is the terminal voltage across the external load resistance $R$, and $Ir$ represents the lost volts. When the circuit is open (no current, $I=0$), the terminal voltage equals the EMF: $V=\mathcal{E}$. Under load, the terminal voltage is always less than the EMF because some voltage is "lost" internally. A high internal resistance means a significant voltage drop even at moderate currents, which is why a weak battery might show a normal voltage on a multimeter but fail to spin a motor.

Experimental Determination: The Voltage-Current Graph Method

A direct method to find both the EMF and internal resistance of a source is to plot terminal voltage $V$ against current $I$. Rearranging the fundamental equation gives the graph equation:

$$V=\mathcal{E}-Ir$$

This is in the linear form $y=c+mx$. You set up a circuit with the source connected to a variable external load (like a rheostat). Measure the terminal voltage $V$ and the corresponding current $I$ for several different resistance values.

• Y-intercept: When you extrapolate the line of best fit back to where $I=0$, the y-intercept gives the EMF $\mathcal{E}$. This represents the terminal voltage under zero current, the open-circuit condition.
• Gradient: The gradient of the line is $-r$. The negative sign indicates that voltage decreases as current increases; the magnitude of the gradient is the internal resistance $r$.

This graphical method is superior to single measurements because it averages out errors and visually confirms the linear relationship predicted by the model. An alternative practical involves measuring $V$ and the external load $R$, then using $I=V/R$ to calculate current before plotting.

Power Delivery and Maximum Power Transfer

The power delivered to the external load resistor $R$ is $P=I^{2}R$. Using the relationship $I=\mathcal{E}/(R+r)$ from Ohm's Law for the complete circuit, power can be expressed as:

$$P={\left(\frac{\mathcal{E}}{R+r}\right)}^{2}R$$

This leads to a critical question: for what value of the external load $R$ is the power transfer from source to load maximized? By differentiating the power expression with respect to $R$ (or by analyzing symmetry), you find that maximum power is delivered to the external load when its resistance equals the internal resistance of the source: $R=r$.

This condition for maximum power transfer has major practical implications. While it maximizes power in the load, note that efficiency is only 50% at this point because an equal amount of power is dissipated as heat inside the source itself ($I^{2}r$). Therefore, design choices depend on the goal: high efficiency (requiring $R\gg r$, like in most household wiring) or maximum power transfer (requiring $R=r$, common in audio amplifier design or some communication systems).

Practical Significance in Real Circuits

The concept of internal resistance is not merely theoretical; it explains everyday phenomena and guides engineering decisions.

• Batteries Under Load: A car battery must deliver hundreds of amps to the starter motor. A battery with high internal resistance will have such large lost volts ($Ir$) that the terminal voltage collapses, causing the starter to turn slowly or fail. "Flat" batteries often have increased internal resistance due to chemical aging.
• Power Supplies: Laboratory power supplies are designed with extremely low internal resistance to maintain a nearly constant terminal voltage regardless of the current drawn, approximating an ideal voltage source.
• Voltage Sag and Safety: Short circuits cause extremely high current. Even a small internal resistance $r$ can lead to a massive $Ir$ drop, causing the terminal voltage to plunge. More critically, the power dissipated inside the source ($I^{2}r$) becomes enormous, generating heat that can cause batteries to leak, rupture, or ignite. This is why internal resistance limits the current a real source can provide.

Common Pitfalls

1. Confusing EMF with Terminal Voltage: The most frequent error is assuming the labeled voltage on a battery (e.g., 9V) is always the voltage you get in a circuit. Remember, the label indicates the EMF. The terminal voltage $V$ is always $\mathcal{E}-Ir$ when current flows. Correction: Explicitly distinguish between the open-circuit voltage (EMF) and the under-load voltage (terminal voltage).

1. Misinterpreting the $V-I$ Graph Gradient: Students often forget the negative sign in the gradient or try to calculate internal resistance as $V/I$ at a point. This is incorrect because $V/I$ gives the external resistance $R$, not $r$. Correction: Internal resistance is the magnitude (absolute value) of the gradient of the $V$ vs. $I$ graph: $r=|\text{gradient}|=(\mathcal{E}-V)/I$ for any point on the line.

1. Ignoring the Conditions for Maximum Power: It’s easy to mistakenly state that maximum power is delivered when the external resistance is maximized. In reality, very high $R$ minimizes current and thus power. Correction: Recall that maximum power transfer is a balance condition: $R=r$. Use the power formula $P=I^{2}R$ and consider how both $I$ and $R$ change together as the load varies.

1. Overlooking Efficiency Trade-offs: Applying the maximum power theorem ($R=r$) without considering the context is a mistake. In systems where energy conservation or heat management is critical (like battery-powered devices), a high-efficiency setup ($R\gg r$) is preferable, even if it doesn't deliver the absolute maximum possible power.

Summary

• The electromotive force (EMF) $\mathcal{E}$ is the ideal, open-circuit voltage of a source. The terminal voltage $V$ is the measurable voltage under load and is given by $V=\mathcal{E}-Ir$, where $Ir$ represents the lost volts across the internal resistance $r$.
• You can determine both $\mathcal{E}$ and $r$ experimentally by plotting terminal voltage $V$ against current $I$. The y-intercept equals the EMF, and the magnitude of the gradient equals the internal resistance.
• The power delivered to an external load $R$ is maximized when $R=r$, but this condition is only 50% efficient. For high efficiency, the load resistance should be much larger than the internal resistance.
• Internal resistance has critical practical consequences, explaining voltage sag in batteries under heavy load, the design of power supplies, and the thermal hazards of short circuits.