A real battery is an ideal EMF in series with a small internal resistance. Change the values, flip the switch and watch current, potential steps and power change.
Sweeps R1 over 3 decades with the circuit solver, finds the peak numerically and compares it with the closed form R1* = r + Rrest.
Not run yet.
Educational model. Wires and the closed switch are ideal (0.001 Ω); resistors are ideal and temperature-independent; the 0.25 W rating is a typical value for small carbon-film resistors; internal-resistance presets are typical orders of magnitude, not a specific cell.
The science
Real battery. Terminal voltage V = E − I r. The internal resistance r (milliohms for a car battery, a few ohms for a 9 V block, more as cells age) makes the voltage sag as the current grows.
Ohm and Kirchhoff. V = I R; resistors in series add, R = R1 + R2; in parallel 1/R = 1/R2 + 1/R3. The simulator does not use those shortcuts: it solves Kirchhoff's current law at every node (nodal analysis, a 3×3 linear system) and only then compares with the closed forms in the checks.
Power. P = I V = I2 R = V2 / R. The source power splits into heat in r and in the resistors (Tellegen's theorem: Σ P = 0 over all elements).
Maximum power transfer. For a load R fed by a source E with series resistance r: P(R) = E2 R / (R + r)2, maximal at R = r where Pmax = E2 / 4r and half of the total power is wasted in r (efficiency 50 %). Here the load is R1 and r is replaced by everything else in series: R* = r + R2∥R3. Power plants and audio amplifiers choose the opposite regime (efficient, R >> r); radio receivers and sensors often match.
Drift speed. Free electrons in copper move slowly: v = I / (n e A) with n ≈ 8.5×1028 m−3, e = 1.602×10−19 C; for 1 A in a 1 mm² wire v ≈ 0.07 mm/s. The signal (the field) spreads near light speed, which is why a lamp lights at once. Electrons flow opposite to conventional current: use the flow button.
Honest simplifications. DC steady state only (no capacitors, inductors, transients); ideal wires; constant resistances (real resistors heat up and change); the field-line display is a decorative dipole pattern whose brightness follows the terminal voltage, not a field computation; bead speed is log-scaled and bead density is fixed.
Models built with Blender (bench, battery, resistor, switch; baked textures) and rendered with Three.js. Not affiliated with any battery or component maker.