In a DC RC charging circuit, when is a capacitor considered 'fully charged' in practical terms (i.e., what condition indicates the charging process is essentially complete)?

Introduction:In RC charging, a capacitor does not reach the final value instantaneously; instead, its voltage approaches the supply asymptotically. This question tests practical engineering judgment about when a capacitor is considered ‘‘fully charged’’ and the relationship between the capacitor voltage, the source voltage, and the current in a first-order RC circuit.

Given Data / Assumptions:

• DC source of constant voltage Vs.
• Series RC network with time constant τ = R * C.
• Charging starts from an initially uncharged state.
• ‘‘Fully charged’’ is a practical criterion, not a strict mathematical limit.

Concept / Approach:For a step input, capacitor voltage follows Vc(t) = Vs * (1 − e^(−t/τ)) and current follows i(t) = (Vs/R) * e^(−t/τ). As t increases, Vc tends to Vs while current decays toward zero. Engineers typically declare ‘‘fully charged’’ at about 5τ, where Vc ≈ 0.993 * Vs and i ≈ 0.007 * (Vs/R).

Step-by-Step Solution:

Verification / Alternative check:The exponential decay of current i(t) confirms the approach to zero. Measuring Vc with a voltmeter near Vs while the series current falls to near the meter's noise floor supports the 5τ rule of thumb.

Why Other Options Are Wrong:

• Initial inrush current equality: This occurs at t = 0, the opposite of ‘‘fully charged.’’
• Vc = 0.707 * Vs: This corresponds to filter cutoff ideas, not RC charging completion (at 1τ, Vc ≈ 0.632 * Vs, not 0.707).
• Current proportional to plate area: Plate area affects capacitance value, not the ‘‘fully charged’’ condition definition.

Common Pitfalls:Confusing time constant with full charge, mixing AC filter corner concepts with DC charging, and assuming the capacitor ever reaches Vs exactly (it approaches it asymptotically).

Final Answer:When the capacitor voltage is essentially equal to the DC source voltage (about after 5τ) and the charging current is approximately zero.