The Differential equation for RC-series circuit is
Rdtdq+C1q=E(t) .
Now, R=500 ohm, C = 10−4 F, E(t) = 100 V.
So, 500dtdq+104q=100⟹dtdq+20q=0.2
This is standard first-order differential equation.
So, Integrating factor = e∫20dt=e20t .
Hence solution is qe20t=∫(0.2)e20tdt+c
⟹qe20t=(0.2)20e20t+c=0.01e20t+c
Now, q(0)=0⟹0.01+c=0⟹c=−0.01
⟹qe20t=0.01(e20t−1)⟹q=e20t0.01(e20t−1)=0.01(1−e−20t)
And current i(t)=dtdq=0.2e−20t
Comments
Dear Helen, thank you for correcting us.
Hi, Thank you for the solution :) I think you made a slight mistake though. Isn't i(t) supposed to be i(t)=0.2e^-20t . You forgot the -ve sign.