Please be as detailed as possible!!! We first consider the case where E(t) = E0,
ID: 2973142 • Letter: P
Question
Please be as detailed as possible!!!
We first consider the case where E(t) = E0, a constant applied voltage. Verify that the function Q1(t) = E0C(l-e-t/(RC)) is a solution to the initial value problem with E(t) = E0, dQ1/dt = ~1/RCQ1 + 1/RE0, Q1(0) = 0, where R, C, and E0 are constants. That is, by plugging t = 0 into the formula (1) show that the initial condition is satisfied, and then by differentiating the formula (1) and comparing with the right-hand side of the differential equation show that Q1(t) satisfies the differential equation. (In other words, do not try to find the solution of the initial-value problem, but rather just check that the given function solves the problem.)Explanation / Answer
Consider 1/RC = k and E0/R =p dQ/dt = -kQ + p dQ/(p/k -Q) = k dt Integrating both sides -ln(p/k -Q) = kt + c p/k - Q = Ae^-kt Q = p/k-Ae^-kt P/k = E0*C so Q = E0*C - A*e^(-t/RC) t = 0 ; Q=0 so A = E0*C so Q =E0*C * [ 1 - e^(-t/RC)]