I need help answering all of these t =0 In the circuit shown, the switch is dose

Question

I need help answering all of these

t =0 In the circuit shown, the switch is dosed at time rite the Kirchhoff loop equation for the circuit: Write the differential equation that describes the trantient behave for of the circuit (you don't have to solve the equation): Charge on the capacitor Q Voltage across the capacitor Vc Voltage across the resistor Vg, Current in the circuit I Identify the following after a long time (t>5 time constants): Charge on the capacitor Q Voltage across the capacitor Vc Voltage across the resistor VR Current in the circuit I

Explanation / Answer

writing kvl we get

Vc+Vr = Vi

where Vi is initial voltage on capacitor

Vc = decaying voltage on capacitor

Vr= resistor voltage

but i = C*dVc/dt

Vr = i*R = RC* dVc/dt

so

Vc+RC*dVc/dt = Vi

solve for Vc we get

Vc= Vi*e^(-t/RC)

at t=0

voltage across capacitorVc= Vi

so charge = CVc = C*Vi

voltage across resistor = Vi-Vc = Vi-Vi (at t=0) = 0; Vr=0

current = C*dVc/dt = - (Vi / R) ;here negative sign indicates decay of charge..

for t> 5RC

charge ~~ zero

voltage ~~ zero

current~~ zero

voltage across resistor = Vi-Vc = Vi-0 = Vi

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