In a series RC circuit with a DC source V0, the voltage across the capacitor as a function of time during charging is:

In a series RC circuit with a DC source V0, the voltage across the capacitor as a function of time during charging is:

• A. Vc(t) = V0(1 − e^(−t/RC))
• B. Vc(t) = V0 e^(−t/RC)
• C. Vc(t) = V0 t / (RC)
• D. Vc(t) = V0

Answer: (A)

When a DC source charges a series RC circuit, the capacitor voltage rises exponentially toward the supply voltage because the charging current decreases as the capacitor stores more charge. The loop equation V0 = IR + Vc, together with i = C dVc/dt, gives RC dVc/dt + Vc = V0. Solving this with the capacitor initially uncharged (Vc(0) = 0) yields Vc(t) = V0 [1 − e^(−t/RC)]. The time constant RC sets the charging speed: after t = RC, the capacitor is about 63% of V0, and after a few time constants it gets very close to V0. The other forms don’t fit this charging behavior: a decaying exponential would describe discharging with the source removed; a linear ramp would require a constant current, which isn’t the case here; and a constant Vc = V0 would imply the capacitor instantly reaches the final value, which isn’t possible with finite RC.

When a DC source charges a series RC circuit, the capacitor voltage rises exponentially toward the supply voltage because the charging current decreases as the capacitor stores more charge. The loop equation V0 = IR + Vc, together with i = C dVc/dt, gives RC dVc/dt + Vc = V0. Solving this with the capacitor initially uncharged (Vc(0) = 0) yields Vc(t) = V0 [1 − e^(−t/RC)]. The time constant RC sets the charging speed: after t = RC, the capacitor is about 63% of V0, and after a few time constants it gets very close to V0.

The other forms don’t fit this charging behavior: a decaying exponential would describe discharging with the source removed; a linear ramp would require a constant current, which isn’t the case here; and a constant Vc = V0 would imply the capacitor instantly reaches the final value, which isn’t possible with finite RC.