RC circuits show exponential change in time.

A capacitor does not change its voltage instantaneously. In an RC circuit, current is initially largest and then decreases as the capacitor voltage approaches its final value.

• Charging: capacitor voltage rises toward a steady value
• Discharging: capacitor voltage falls toward zero (or a new value)
• Initial conditions control the early-time behavior
• Qualitative shapes of V(t) and I(t)

The time constant τ = RC sets how quickly the circuit responds. After one τ, the system has completed a fixed fraction of its total change.

• τ = RC as the characteristic timescale
• After 1τ: about 63% toward the final value (charging)
• After 1τ: about 37% remaining (discharging)
• How changing R or C stretches time

The governing equations produce exponential time dependence. You do not need heavy calculus to use the results, but you should interpret them correctly.

• General form: A + Be^−t/τ
• Charging: approach to final value
• Discharging: decay toward zero (or a new baseline)
• Reading graphs: slope is steepest at early times

During charging, energy comes from the source and ends partly in the capacitor’s field and partly as thermal energy in the resistor. During discharging, stored field energy is converted to heat in the resistor.

• Where energy is stored: electric field in the capacitor
• Where energy is lost: resistor as thermal energy
• Power vs time during charging/discharging
• Conceptual “energy accounting” view

Practice reading exponential time graphs and using τ to estimate or compute circuit behavior.

• Identify charging vs discharging from V(t) and I(t)
• Compute τ and interpret what it means physically
• Use exponential forms to find V or I at a given time
• Energy accounting questions (field vs heat)
• Exam-style RC transient sets