When you're using a cell (like a battery) to power something, not all of the energy it provides goes to the gadget. Some gets lost in the battery itself, kinda like how not every drop of gasoline in a car goes to speeding down the highway—some just helps the engine keep running. So

• Total Power from the Cell = Power for the gadget (external circuit) + Power wasted in the cell.
• Math Magic: P = I²R + I²r

Here

• P = Total Power
• I = Current
• R = Resistance in the gadget
• r = Internal resistance (within the cell)

🎢 Fun Analogy: Imagine a water slide where R is the resistance due to the slide's twists and turns and r is the friction of the slide itself. The total thrill (or resistance) is a combo of both!

• Equation: \(\frac {e^2}{(R + r)^2} × R\)

  Here, ε is like the battery's "oomph" or electromotive force (emf).

📈 Graph Alert: If you graph the power delivered to the gadget against R, the "peak fun" happens when the slide friction (r) is equal to the slide's twists and turns (R). In nerd terms, the peak of the curve is when r = R.

💡 Lightbulb Moment! When internal and external resistances are equal, we say they are "matched." This is like saying our slide's friction perfectly complements its design. This balance is super important in the world of electronics.

🌎 Real-world Example: Think of a professional singer with a microphone. If both are perfectly matched, the sound is pristine. But if they're mismatched, you might get feedback or distortion.