Internal Resistance of a Battery Formula: A Complete Guide

📚 What is Internal Resistance?

Every battery, while being a source of power, also presents a small obstacle to the flow of current. This opposition within the battery itself is known as internal resistance ($r$). It's what causes a battery's voltage to drop when you actually start using it. Think of it like this: a perfect battery would deliver its full rated voltage all the time, but real-world batteries lose some voltage due to this internal struggle.

📜 A Brief History

The concept of internal resistance became important as electrical technology advanced. Early batteries were crude, and their internal resistance was a significant factor in their performance. As battery technology improved, understanding and minimizing internal resistance became crucial for efficient power delivery. Scientists and engineers developed models and formulas to characterize and account for this internal resistance, leading to better battery designs and applications.

🔑 Key Principles and the Formula

The voltage you measure across a battery's terminals when it's not connected to anything is its electromotive force (EMF), denoted by $\mathcal{E}$. When you connect a load (like a lightbulb), current ($I$) flows, and the voltage at the terminals ($V$) drops. The relationship is described by the following formula:

$V = \mathcal{E} - Ir$

Where:

• ⚡ $V$ is the terminal voltage (the voltage you measure when the circuit is closed).
• ⚛️ $\mathcal{E}$ is the EMF (electromotive force) of the battery (the voltage when no current is flowing).
• 🌊 $I$ is the current flowing through the circuit.
• 🚧 $r$ is the internal resistance of the battery.

Rearranging the formula to solve for internal resistance ($r$):

$r = \frac{\mathcal{E} - V}{I}$

💡 How to Measure Internal Resistance:

Here’s a simple method to determine internal resistance:

• 🔬 Measure the EMF ($\mathcal{E}$) of the battery using a multimeter when no load is connected.
• 🔌 Connect a known resistor ($R$) to the battery.
• 📏 Measure the voltage ($V$) across the resistor (which is also the terminal voltage of the battery).
• 🧮 Measure the current ($I$) flowing through the resistor.
• ➗ Use the formula $r = (\mathcal{E} - V) / I$ to calculate the internal resistance.

🌍 Real-world Examples

• 🚗 Car Batteries: A car battery with high internal resistance will struggle to start the engine, especially in cold weather, because it can't deliver enough current.
• 📱 Smartphone Batteries: As a smartphone battery ages, its internal resistance increases, leading to shorter battery life and slower charging.
• 🔦 Flashlights: A flashlight with a weak battery (high internal resistance) will produce a dimmer light because a larger portion of the battery's voltage is lost within the battery itself.

📝 Practice Quiz

1. A battery has an EMF of 12V. When connected to a 2Ω resistor, the terminal voltage drops to 11V. Calculate the internal resistance.
2. A battery with an EMF of 9V has an internal resistance of 0.5Ω. What is the terminal voltage when a current of 1A is drawn from it?
3. Explain how increasing the internal resistance of a battery affects its performance.

✅ Conclusion

Understanding internal resistance is key to understanding battery performance. By knowing the formula and its implications, you can better diagnose battery issues and choose the right batteries for specific applications. Hopefully, this guide has made the concept a little less resistant to your understanding! 😉