Capacitor current–voltage relationship: If a given capacitor experiences a voltage that changes at a constant rate with respect to time (dv/dt = constant), what happens to the current through the capacitor?

Introduction:Many timing and waveform-shaping circuits depend on the fundamental law of capacitors relating current and the time rate of change of voltage. Understanding the qualitative behavior when dv/dt is constant is crucial for recognizing why integrators produce ramps and why differentiators produce spikes.

Given Data / Assumptions:

• Ideal capacitor with capacitance C (no leakage or series resistance).
• Voltage across the capacitor varies such that dv/dt is constant.
• Steady operating conditions (no dielectric absorption effects considered).

Concept / Approach:The governing equation is i_C = C * dv/dt. If dv/dt is constant, multiplying by the fixed C yields a constant current. This is exactly what happens in an op-amp integrator receiving a constant input level—the capacitor current is constant, so the capacitor voltage changes linearly with time.

Step-by-Step Solution:Start with i_C = C * dv/dt.Let dv/dt = k (constant). Then i_C = C * k (a constant).Therefore, the capacitor current neither increases nor decreases; it remains constant while dv/dt remains constant.If the sign of dv/dt changes (slope reversal), the current changes sign but its magnitude remains constant for the same |dv/dt|.

Verification / Alternative check:In a triangular-wave generator, the integrator sees a constant voltage level, producing a constant capacitor current and therefore a linear (constant-slope) ramp—direct experimental confirmation of the rule.

Why Other Options Are Wrong:

• Increase/decrease/decrease logarithmically: Would require dv/dt itself to change with time.
• Oscillate sinusoidally: That would imply a sinusoidal dv/dt, not a constant.

Common Pitfalls:

• Confusing constant voltage (dv/dt = 0, giving zero current) with constant slope (nonzero current).
• Overlooking series resistance that slightly distorts the ideal constant-current condition in real circuits.

Final Answer:be constant