gen-define-capacitor-charging-in-an-rc-circuit-and-give-one-shor-fe18

Define capacitor charging in an RC circuit and give one short worked example.

# Capacitor Charging in an RC Circuit ## Definition **Capacitor charging in an RC circuit** is the process where a capacitor accumulates charge when connected to a voltage source through a resistor. The charge builds up exponentially over time according to: **Key Equations:** - **Charge**: Q(t) = Q₀(1 - e^(-t/RC)) - **Voltage**: V_C(t) = V₀(1 - e^(-t/RC)) - **Current**: I(t) = (V₀/R)e^(-t/RC) Where: - **τ = RC** is the time constant - V₀ is the source voltage - Q₀ = CV₀ is the maximum charge --- ## Worked Example **Problem:** A 10 μF capacitor is charged through a 50 kΩ resistor connected to a 12 V battery. Find: (a) The time constant (b) The voltage across the capacitor after 0.5 seconds **Solution:** **(a) Time constant:** ``` τ = RC = (50 × 10³ Ω)(10 × 10⁻⁶ F) τ = 0.5 s ``` **(b) Voltage at t = 0.5 s:** ``` V_C(t) = V₀(1 - e^(-t/τ)) V_C(0.5) = 12(1 - e^(-0.5/0.5)) V_C(0.5) = 12(1 - e^(-1)) V_C(0.5) = 12(1 - 0.368) V_C(0.5) = 12(0.632) V_C(0.5) ≈ 7.58 V ``` **Answer:** The capacitor reaches approximately 63.2% of full charge (7.58 V) after one time constant.