Account for electrical energy across a full schedule

Define topology, recipient and initial conditions

Power is an energy-transfer rate, whereas energy depends on both rate and duration. The fictional device spends separate intervals at 24 W and 40 W, then the remainder of a 24-hour day at 2 W standby. The intervals must not overlap or exceed the day. For these constant segments, add PΔt; actual varying powers would require a finer measured schedule.

Derive the quantitative response

With active durations a and b hours, energy in watt-hours is 24a+40b+2(24−a−b). The standby subtraction matters: extending the 24 W interval by one hour replaces a 2 W standby hour and adds 22 Wh, not 24 Wh. One kWh is 3.6×10⁶ J and is not a unit of power. The displayed tariff is explicitly a toy .20 accounting units per kWh, not a claim about a current utility price.

Check evidence and the limits of the model

For a constant potential difference, energy transferred by charge is qV; equal charge moved at different voltages does not transfer equal energy. An ohmic conductor has P=IV=I²R=V²/R under its stated conditions. Material resistance also depends on geometry and conditions: R=ρL/A is a uniform-sample relation, and a small-range linear temperature approximation needs a stated coefficient and reference. Neither nominal power nor this ideal schedule establishes actual heating, device duty limits, fuse choice or safe household wiring. Those practical decisions are outside this virtual exercise.

A changed material and geometry packet

For a hypothetical uniform wire, doubling length at unchanged cross-sectional area doubles R. Keeping length fixed while halving diameter quarters area and multiplies R by four; doing both gives an eightfold increase. This is a geometry comparison, not a wire-sizing recommendation. Separately a supplied small-range temperature law R=R0[1+αΔT] with R0=10 Ω and α=.004 K⁻¹ gives 10.8 Ω after a 20 K rise. Predict/check: a 10 K rise gives 10.4 Ω, but extrapolating this local law to arbitrary temperature is not justified.

Worked original case

At a=1 h and b=.5 h, standby lasts 22.5 h. Energies are 24,20,45 Wh, totaling 89 Wh=.089 kWh=320400 J. Mean input power is 89/24≈3.70833 W; toy cost is .0178 accounting units.

Investigation

Change the first active interval while keeping the second fixed. Retain both added active energy and displaced standby energy. Compare a hypothetical second schedule with the same energy but a different peak power, and explain why energy equality does not establish equal heating peaks.

Changed-case practice

Try the changed case before feedback

The device spends 2 h at 24 W and 1 h at 40 W, with 2 W standby for the rest of the day. Find total energy in Wh.

Numerical result in the requested unitsCompare this result

This practice is public and unsaved. A matching number does not verify understanding. Retain your explanation, assumptions and changed-case evidence in the separate course workspace.

Sources and evidence boundary

Original numerical cases and prose checked 30 October 2026 against the relevant primary explanation. All records are synthetic and all diagrams mathematical. No practical wiring, battery testing, capacitor handling, mains measurement or equipment safety rating is requested. Public feedback does not verify independent mastery; retain first predictions, contrasting recipes and criterion-based critique in the separate course evidence workspace.

The separate material/geometry comparison is checked against resistance and resistivity.