SCIENCE · INVESTIGATE AND EXPLAIN

Distinguish an instantaneous balance from stored charge

A node can temporarily accumulate charge. The steady junction rule is the zero-accumulation limit of dq/dt=Iin−Iout, not permission to erase an imbalance. Our ideal current sources prescribe inflow 3 mA and outflow (1+βt) mA, starting with zero stored charge. The sources are stipulated to maintain these histories independently of voltage.

Learning outcome: Compare β=0,1,2 and locate the charge maximum and zero crossing when they occur in the displayed interval. Retain current imbalance and charge separately. Explain the independent ideal-source assumption rather than silently assuming steady state.

Starting recap

Review charge continuity, definite integrals, first-order differential equations, initial conditions and signed source work. Distinguish displayed time from physical time.

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Try

At β=1 mA/s and t=2 s, charge is 2 mC and voltage 2 V. Inflow and outflow are both 3 mA, so the instantaneous charging current is zero without the capacitor being uncharged. At t=4 s charge has returned to zero; thereafter its polarity would reverse if the stipulated sources continued.

Open the experiment

Predict

At β=1 and t=2 s, which conclusion follows?

Your prediction and reason

Experiment

Compare β=0,1,2 and locate the charge maximum and zero crossing when they occur in the displayed interval. Retain current imbalance and charge separately. Explain the independent ideal-source assumption rather than silently assuming steady state.

Integrate unequal inflow and outflow

Worked teaching model · synthetic values · Control volume starts uncharged, ideal prescribed inflow 3 mA and outflow (1+beta t) mA. A 1 mF lumped capacitance to reference converts accumulated charge to voltage; ideal current sources are stipulated to maintain their histories regardless of voltage. No claim of steady state or unlimited real-source compliance.

Outflow ramp coefficient: 1 mA/s Elapsed time: 2 s

Reset keeps your prediction, captures and practice history.

Integrated imbalance, not cumulative inflow alone

Unlike graph quantities rescale independently; read units and values, not screen angles. 1: charge continuity (solid). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Integrated imbalance, not cumulative inflow alone

charge continuity

time (s)	stored charge (mC)
0	0
0.05	0.09875
0.1	0.195
0.15	0.28875
0.2	0.38
0.25	0.46875
0.3	0.555
0.35	0.63875
0.4	0.72
0.45	0.79875
0.5	0.875
0.55	0.94875
0.6	1.02
0.65	1.08875
0.7	1.155
0.75	1.21875
0.8	1.28
0.85	1.33875
0.9	1.395
0.95	1.44875
1	1.5
1.05	1.54875
1.1	1.595
1.15	1.63875
1.2	1.68
1.25	1.71875
1.3	1.755
1.35	1.78875
1.4	1.82
1.45	1.84875
1.5	1.875
1.55	1.89875
1.6	1.92
1.65	1.93875
1.7	1.955
1.75	1.96875
1.8	1.98
1.85	1.98875
1.9	1.995
1.95	1.99875
2	2
2.05	1.99875
2.1	1.995
2.15	1.98875
2.2	1.98
2.25	1.96875
2.3	1.955
2.35	1.93875
2.4	1.92
2.45	1.89875
2.5	1.875
2.55	1.84875
2.6	1.82
2.65	1.78875
2.7	1.755
2.75	1.71875
2.8	1.68
2.85	1.63875
2.9	1.595
2.95	1.54875
3	1.5
3.05	1.44875
3.1	1.395
3.15	1.33875
3.2	1.28
3.25	1.21875
3.3	1.155
3.35	1.08875
3.4	1.02
3.45	0.94875
3.5	0.875
3.55	0.79875
3.6	0.72
3.65	0.63875
3.7	0.555
3.75	0.46875
3.8	0.38
3.85	0.28875
3.9	0.195
3.95	0.09875
4	0

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

Calculated quantities under the declared measurement model
Quantity	Value	Unit
Accumulated charge	2	mC
Capacitor voltage	2	V
Net charging current	0	mA
Outflow current	3	mA
Instantaneous capacitor energy rate	0	mW

Zero net current means the stored charge is stationary at that instant, not necessarily zero.

Charge continuity is dq/dt=Iin−Iout. With zero initial charge, q(t)=2t−beta t²/2 in mC. The ideal 1 mF relation gives V=q in volts for this numerical unit choice. If beta is positive, the charge maximum occurs at t=2/beta, whereas return to zero occurs at t=4/beta. Negative later charge is allowed by the stipulated signed-current sources; it is not negative capacitance.

Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.

Zero net current means the stored charge is stationary at that instant, not necessarily zero.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

State the current, source and measurement model

Derive the integral or initial-value solution

Integrate the imbalance to obtain q(t)=2t−βt²/2 in mC. A 1 mF lumped capacitance to a reference gives the same numerical voltage in volts. The derivative vanishes at t=2/β when β is positive, while charge returns to zero at t=4/β. A maximum and a zero crossing are different events.

Test conservation, uncertainty and limits

The energy rate is V dq/dt, which can be positive or negative even though stored energy CV²/2 is nonnegative. Negative later charge is an allowed polarity under the stipulated ideal sources. A real current source has finite compliance, so this synthetic integral does not demonstrate realizability of every long-time voltage.

Worked original case

Investigation

Changed-case practice

Try the changed case before feedback

At β=2 mA/s and t=3 s, find stored charge in mC.

Numerical result in the requested units

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 boundaries

Original derivations and synthetic evidence checked 31 October 2026 against the primary explanation. No live circuit construction, energized switching, component rating or electrical handling instruction is requested. Preserve first predictions, critique, revision and fresh assigned work separately from public practice. Software agreement does not certify measurement technique or learning.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

Which initial state, calibration or constitutive assumption is needed for this conclusion, and what fresh evidence could test it?

Your explanation and limitations

Continue

Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.

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