Question

A series circuit has a capacitor of 0.25⋅10⁻⁶ F, a resistor of 5⋅10³ Ω, and an inductor of 1H. The initial charge on the capacitor is zero. If a 12 V battery is connected to the circuit and the circuit is closed at t=0, determine the charge on the capacitor at t=0.001 s, at t=0.01 s, and at any time t.

Answer 1

The charge on the capacitor at t=0 is zero as it is initially uncharged. The charge on the capacitor at t=0.001 s is approximately 2.38×10^-6 C, and at t=0.01 s it is approximately 2.23×10^-6 C, calculated using the formula for the charging of a capacitor in an RC circuit.

Step-by-step explanation:

The subject of the question is a RC circuit, which consists of a resistor and a capacitor, and the dynamics when a DC voltage source is applied. The student is asked to calculate the charge on the capacitor at different time intervals after a battery is connected in series with a resistor and an initially uncharged capacitor. As the switch is closed at t = 0 s, the charging of the capacitor follows an exponential behavior governed by the time constant τ = RC, where R is the resistance and C is the capacitance.

The charge on the capacitor at any time t after the circuit is closed is given by the equation q(t) = C * V * (1 - e^{-t/τ}), where V is the voltage of the battery, C is the capacitance, and t is the time. The time constant for the RC circuit in this question is τ = RC = 5×10^3 Ω * 0.25×10^-6 F = 1.25×10^-3 s.

At t = 0.001 s, the charge on the capacitor is q(0.001) = 0.25×10^-6 F * 12 V * (1 - e^{-0.001/(1.25×10^-3)}) ≈ 2.38×10^-6 C. At t = 0.01 s, the charge on the capacitor is q(0.01) = 0.25×10^-6 F * 12 V * (1 - e^{-0.01/(1.25×10^-3)}) ≈ 2.23×10^-6 C.