Close a source, internal-loss and load-power ledger

Distinguish emf from loaded voltage

An ideal emf E supplies energy per charge inside the source. A simple nonideal battery model places internal resistance r in series. With load R, loop balance gives I = E/(r + R), while terminal voltage is E − Ir = IR during discharge. At zero current the terminal value approaches emf. The displayed model does not describe chemical aging or a safe physical power supply.

Track each energy destination

Source power is EI. Internal and load resistors receive I²r and I²R, whose sum equals EI. The equivalent forms VI, I²R and V²/R use the voltage across the particular resistor, not the source emf indiscriminately. Thermal conversion can change real component temperature; our stated fixed parameters exclude that feedback.

Compare power and efficiency

Optional differentiation of load power E²R/(r + R)² gives a maximum at R = r. At that point half the source power is lost internally, so maximum delivered power is not maximum load fraction. For comparable idealized bulbs, larger dissipated power suggests greater brightness qualitatively; no universal conversion from watts to perceived brightness is supplied.

Respect omitted and mechanical elements

A wire’s resistance may be negligible compared with other large resistances, but not when it is the only element limiting current. An ideal zero-resistance load directly across an ideal voltage source makes the model singular rather than predicting a realizable finite current. For a separate ideal motor packet, 3 V at 2 A gives 6 W electrical input; if mechanical conversion is stipulated 75%, mechanical output is 4.5 W and the remaining 1.5 W needs an explicit loss destination.

Worked original case

With E = 12 V, r = 2 ohm and R = 4 ohm, I = 2 A and terminal voltage is 8 V. Source power is 24 W, split into 8 W internal and 16 W load. Using 12 V rather than 8 V in load power would give the wrong account.

Investigation

Compare loads below, equal to and above the internal resistance. Retain terminal voltage, source power, each resistor power and load fraction. Differentiate the load-power expression if choosing the optional maximum challenge, then contrast the result with efficiency.

Changed-case practice

Sources and evidence boundary

Original teaching checked 31 October 2026 against the current official framework, Unit 11 and OpenStax electrical energy and power. All diagrams and numerical records are virtual, supplied or generated. No physical wiring, battery connection, short circuit, meter operation or electrical handling is requested. Public feedback and retained self-reports do not certify independent reasoning, actual laboratory participation or institutional credit.