Formula and Method for Internal Resistance
Every real battery, cell, or power supply behaves like an ideal voltage source (EMF, ε) in series with a small internal resistance, r. When no current flows, a voltmeter across the terminals reads the full EMF. As soon as a load draws current I, some voltage is dropped across that internal resistance, so the terminal voltage V you measure is always a bit lower than ε. Kirchhoff's voltage law gives ε = I(R + r) = IR + Ir = V + Ir, since the load voltage V equals IR. Rearranging for the internal resistance gives the formula this calculator uses: r = (ε − V) / I.
How the calculation works
Enter the source's EMF (its open-circuit, no-load voltage), the terminal voltage measured while a known current flows, and that current. The calculator subtracts terminal voltage from EMF to find the voltage dropped inside the source, then divides by the current to get internal resistance in ohms (r = (ε − V) / I). From there it also computes the power wasted as heat inside the source (P = I²r), the useful power delivered to the load (P = IV), and the efficiency of the source at that operating point (V / ε × 100%).
Common mistakes
- Confusing EMF with terminal voltage: EMF is the no-load reading; terminal voltage is what you measure with current flowing. Using the wrong one anywhere in the formula gives an incorrect (often negative) resistance.
- Measuring current and voltage at different moments: internal resistance calculations assume I and V were recorded simultaneously under the same steady load — a battery's terminal voltage can drift as it settles or discharges.
- Ignoring lead and contact resistance: resistance in test leads, connectors, and meter probes adds to your measured r; use short, low-resistance leads for accurate results, especially with low-resistance sources like fresh alkaline or lithium cells.
Real-world applications
- Battery health testing: internal resistance rises as a battery ages or discharges, so measuring it is a standard way to estimate remaining battery life or detect a failing cell.
- Power supply and solar cell design: engineers characterize a source's internal resistance to predict how much its output voltage will sag under load and to size components accordingly.
- Maximum power transfer: a load draws the most power from a source when the load resistance equals the source's internal resistance, a principle used in audio amplifier and RF circuit design.
- Jump-starting and high-current draws: a car starter motor or power tool briefly draws very high current, which is why internal resistance (not just rated voltage) determines whether a battery can deliver that surge.
Frequently Asked Questions
What is the formula for internal resistance?
Internal resistance is r = (ε − V) / I, where ε is the source's EMF (open-circuit voltage), V is the terminal voltage measured while current I flows under load. This comes from Kirchhoff's voltage law applied to a real battery: ε = I(R + r), so ε = V + Ir since V = IR.
What is the difference between EMF and terminal voltage?
EMF (ε) is the voltage a source would produce with no current flowing (open-circuit, measured with a high-impedance voltmeter and no load). Terminal voltage (V) is what you actually measure across the terminals while current flows to a load — it is always lower than EMF because some voltage is dropped across the internal resistance itself (V = ε − Ir).
How do I measure EMF and terminal voltage in practice?
Measure EMF with a voltmeter across the source with no load connected (or a negligibly small current draw). Then connect a known load, let the current stabilize, and record both the current (with an ammeter in series) and the terminal voltage (with a voltmeter across the source) at that same moment.
Why does a battery's terminal voltage drop under heavier load?
Every real battery has some internal resistance from its electrolyte and electrode materials. As current increases, more voltage is dropped across that internal resistance (Ir), leaving less for the external circuit — that's why terminal voltage sags as you draw more current, and why a battery can read full voltage unloaded but sag under a heavy starter motor or flashlight bulb.
Common Tip Percentages
| Service | Typical Tip Range | Notes |
|---|
| Restaurant (sit-down) | 15–20% | 20%+ for exceptional service |
| Food delivery | 10–20% | Consider extra for long distances |
| Taxi / rideshare | 10–20% | More for assistance with luggage |
| Hotel housekeeping | $2–5/night | Leave daily, not just at checkout |
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General Household Budget Ratios (50/30/20 Rule)
| Category | % of Net Income | Examples |
|---|
| Needs | 50% | Rent, groceries, utilities, insurance |
| Wants | 30% | Dining out, entertainment, hobbies |
| Savings / Debt | 20% | Emergency fund, retirement, extra payments |
Common Measurement Conversions for Everyday Use
| Quantity | US / Imperial | Metric Equivalent |
|---|
| 1 cup | 8 fl oz | 237 mL |
| 1 pint | 16 fl oz | 473 mL |
| 1 quart | 32 fl oz | 946 mL |
| 1 gallon | 128 fl oz | 3.785 L |
| 1 lb | 16 oz | 453.6 g |
| 1 ft | 12 in | 30.48 cm |