Find the current flowing through resistor

[dinospamoni]

Homework Statement

The power supply in the circuit shown has V(t) =
(120V)cos(ωt), where ω = 310 rad/s. Determine the current
flowing through the resistor at time t = 9.7 s, given R = 600 Ω,
C = 18 mF, and I(0) = 0 A. As a reminder, Kirkhoff’s voltage
law for this circuit (Eq. 8-1.3 in the book) reduces to:
dV/dt = R(dI/dt) + I/C

Homework Equations

The Attempt at a Solution

I've tried this about ten times and can't seem to get the right answer:

I found dV/dt = -37200 Sin(wt) (i'll call it v' from now on)

Rearranging the equation to make it in standard form:

dI/dt + (1/RC)I = v'/R

P= 1/RC = .0926

Q=v'/R = -62 Sin(wt)

F = ∫p dt

So e^F = e^.0926 t
and e^-F = e^-.0926 t

This equation was given in class for solving this type of DE:

I = (e^-I)∫Q*e^F dt + c1*e^-F

When plug this into mathematica, it gives me an imaginary answer

Any ideas?

 

[Simon Bridge]

Did you try using phasor diagrams?

 

[dinospamoni]

I have no idea what that is. We were told to use only differential equation methods

 

[Simon Bridge]

OK - I guess you'll learn about phasor analysis after you experience the pain of having to use the integral approach.
$$\renewcommand{\tder}[1]{\frac{d #1}{dt}}

\tder{i}+\frac{1}{RC}i= -\frac{\omega V}{R}\sin\omega t$$

it looks like you tried using an integrating factor?
then you mention an equation "given in class" ... but you don't seem to understand it, so there is confusion. You should back up to where you do understand what is going on.

Have you tried the method of undetermined coefficients?

You could try guessing - it's usually less painful.
You could also have used your understanding of the physics of RC circuits to guide you in guessing i(t). The effect of the capacitor is to change the phase relationship between voltage and current ... so you'd guess a sine wave, different amplitude and phase but same frequency, as v(t). The main trouble is that you need a table of trig identities and it ends up the same as undetermined coefficients.

 

[Nickie]

I would approach this problem by first checking the calculations and equations used to solve it. It is important to double check all calculations and make sure the equations used are appropriate for the problem at hand. In this case, I would also make sure that the units are consistent throughout the calculations.

If the calculations and equations are correct, I would then consider other factors that may affect the current flowing through the resistor. For example, is there any resistance or capacitance in the wires or connections of the circuit that may affect the overall current? Are there any other components in the circuit that may be affecting the current? It is important to consider all possible factors that may contribute to the current in order to accurately solve the problem.

I would also consider using a different method to solve the problem, such as using Kirchhoff's current law to determine the current at a specific point in the circuit. Additionally, I would consult with colleagues or refer to textbooks or other resources for further guidance on how to approach and solve the problem.

In conclusion, as a scientist, I would carefully review and analyze all aspects of the problem, including calculations, equations, and potential factors that may affect the current, in order to provide an accurate and informed response.