A battery has an EMF (ε) of 12V and an internal resistance of 2Ω. You connect it to an external resistor of 4Ω. You clip a voltmeter across the battery’s terminals to measure the voltage.
What does the voltmeter read?
The voltmeter reads 8V — not 12V! The battery’s EMF is 12V, but it has to push current through its own internal resistance first. With I = 12V/(4Ω + 2Ω) = 2A flowing, the internal resistance drops Ir = 2A × 2Ω = 4V. Only 8V is left at the terminals for the external circuit. The battery literally wastes 4V heating itself.
Loading simulation…
Adjust the external resistance and watch the terminal voltage change. At very high resistance, the terminal voltage approaches the EMF (tiny current). At very low resistance, the terminal voltage plummets (huge current). Find the sweet spot where maximum power is delivered to the external load.
V = ε − Ir
The terminal voltage is always LESS than the EMF when current flows. More current means more internal voltage drop. Only at zero current does V = ε.
Cold temperatures increase a battery’s internal resistance. That’s why cars struggle to start in winter — more voltage is lost internally, leaving less for the starter motor.
Your phone shows 15% but dies under heavy load. High current (gaming, camera) increases the Ir drop, causing the terminal voltage to sag below the minimum the phone can operate at.
Power plants have internal impedance too. During peak demand (high I), the voltage delivered to homes can sag — the same physics as a battery, just on a massive scale.
Solar panels in space have internal resistance that increases with radiation damage over years. Mission planners must account for rising Ir drops throughout the spacecraft’s lifetime.
“A battery’s label tells you its EMF. What you actually get depends on how hard you push it. V = ε − Ir is the reality check every circuit needs.”
Battery Workshop
Explore the difference between EMF and terminal voltage. Connect loads of varying resistance to a battery with adjustable internal resistance. Watch the terminal voltage sag under heavy load, and discover the maximum power transfer point where R equals r.
Loading simulation…
Start with a low external resistance — notice how the terminal voltage drops far below the EMF. Slowly increase R and watch V recover toward ε. Now find the maximum power transfer point: sweep R until the power delivered to the load peaks. It happens when R = r! But notice the efficiency at that point is only 50%. Try cranking up the internal resistance to simulate an old, degraded battery — the terminal voltage sags even more.
Every real voltage source has internal resistance. The EMF is a theoretical maximum; the terminal voltage is reality. Understanding V = ε − Ir explains why batteries die under load, why car starters need thick cables, and why maximum power delivery always costs 50% efficiency.