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PhysicsMediumClass 12

RC Circuits — Charging and discharging

Current Electricity

19

JEE Qs

8%

Hard

75

min

Master the initial (t=0+) and final (t=∞) states of the capacitor and how to correctly calculate the time constant (τ=RC) to efficiently solve most RC circuit problems.

🧮 Key Formulas

Time constant (τ): τ = RC
Charging a capacitor (with initial voltage 0, charged by a source V_S):
Voltage across capacitor: V_C(t) = V_S * (1 - e^(-t/RC))
Current in the circuit: I(t) = (V_S/R) * e^(-t/RC)
Voltage across resistor: V_R(t) = V_S * e^(-t/RC)
Discharging a capacitor (initially charged to V_0, through a resistor R):
Voltage across capacitor: V_C(t) = V_0 * e^(-t/RC)
Current in the circuit: I(t) = -(V_0/R) * e^(-t/RC) (negative sign indicates current direction opposite to charging)
Voltage across resistor: V_R(t) = V_0 * e^(-t/RC) (magnitude)
Charge on capacitor: Q(t) = C * V_C(t)
Energy stored in capacitor: U = 0.5 * C * V^2 = Q^2 / (2C)

✅ Key Points for JEE

  • 1At t=0 (just after connection), a capacitor acts like a short circuit to current. Its voltage cannot change instantaneously.
  • 2At t=infinity (steady state), a capacitor acts like an open circuit for DC sources, blocking current completely, and its voltage equals the source voltage (if fully charged).
  • 3The time constant τ = RC dictates the rate of charging/discharging. After one time constant (t=τ), the capacitor charges to approximately 63.2% of its final voltage or discharges to approximately 36.8% of its initial voltage.
  • 4The behavior of voltage and current in RC circuits is exponential, governed by first-order differential equations derived from Kirchhoff's Voltage Law.
  • 5For complex circuits, find the Thevenin equivalent resistance (R_eq) across the capacitor's terminals to correctly calculate the time constant: τ = R_eq * C.

⚠️ Common Mistakes

  • Incorrectly assuming capacitor voltage can change instantaneously at t=0; it's the current that's maximum at t=0.
  • Confusing the formulas for charging and discharging or misapplying initial/final conditions for voltage and current.
  • Calculating the equivalent resistance 'R' for τ incorrectly, especially in circuits with multiple resistors or sources.
  • Forgetting that the current through the resistor (and thus the voltage across it) starts high and decays to zero during charging, while the capacitor voltage builds up.

📝 Practice Questions

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NCERT Chapters

  • Class 12 Physics Ch 2: Electrostatic Potential and Capacitance
  • Class 12 Physics Ch 3: Current Electricity