I'm just really lost on this topic and honestly don't have too much of an idea.
(a) Using Kirchoff's loop rule, find the differential equation satisfied by the charge q(t).
(b) Verify that q(t) = q_max e^-(a/t) cos(ωt) satisfies the differential equation for particular values of a and w and find these values.
(c) The resistance R_c for which w = 0 yields critical damping. If L = 10 mH and C = 0.2 mF, determine R_c.
(d) Sketch q(tL)for R < R_c, R = R_c, and R > R_c.
The Attempt at a Solution
a) using the loop law all voltages must add to give 0 so you use an equation that includes q for each of the voltages, giving
b) so if we are to solve you place it at time =0 because we want q(t)=0? therefore q(t) = q_max*e^-(a/t) and solve for "a" that way. but what original value of q(t) would be used, on top of this how would i then solve for ω.
c) i think the equation ω_0=1/√LC is used but i don't know how that translates the the question as it uses ω and not ω_0.
d) it should be exponential but i've been told me someone who is always right that i'm wrong.
any help would be greatly appreciated as im really lost.