Second order derivatives and resonance

[GwtBc]

Homework Statement

2. Consider an electric circuit consisting of an inductor with inductance L Henrys, a resistor with resistance R Ohms and a capacitor with capacitance C Farads, connected in series with a voltage source of V Volts. The charge q(t) Coulombs on the capacitor at time t ≥ 0 seconds satisfies the differential equation:
$L\frac{\mathrm{d} ^2q}{\mathrm{d} t^2} + R\frac{\mathrm{d} q}{\mathrm{d} t} + \frac{q}{C} = V$
Also, the current in the circuit i(t) Amps satisfies:
$i = \frac{\mathrm{d} q}{\mathrm{d} t}$
. Suppose that in a particular circuit, L = 0.4 Henrys, R = 0 Ohms, C = 0.1 Farads and V = 110 sin(ωt) Volts, where ω ∈ R. Initially the charge on the capacitor is 1 Coulomb and there is no current in the circuit.
(a) Write down the differential equation satisfied by q(t) in this circuit.
(b) Determine the value(s) of ω so that resonance occurs in the circuit.
(c) In the case where there is no resonance,
i. Solve the differential equation to find the charge on the capacitor at any time.
ii. Determine the transient and steady state solutions for the charge, if they exist.
iii. Find the current in the circuit at any time.

Homework Equations

The Attempt at a Solution

Hey everyone. Thanks for reading through and trying to help! Basically I just want to know if I'm thinking along the right track. So for part a, I apply the condition for resonance by solving the homogeneous second order equation, and then for part c I solve it across the domain where omega is not equal to the values I found in part b and I should be getting an answer in terms of omega? I'm just confused because the wording of the question makes it sound like I need to be getting a solution in terms of t only (i.e. omega should cancel out or something).

 

[Ray Vickson]

Homework Statement

2. Consider an electric circuit consisting of an inductor with inductance L Henrys, a resistor with resistance R Ohms and a capacitor with capacitance C Farads, connected in series with a voltage source of V Volts. The charge q(t) Coulombs on the capacitor at time t ≥ 0 seconds satisfies the differential equation:
$L\frac{\mathrm{d} ^2q}{\mathrm{d} t^2} + R\frac{\mathrm{d} q}{\mathrm{d} t} + \frac{q}{C} = V$
Also, the current in the circuit i(t) Amps satisfies:
$i = \frac{\mathrm{d} q}{\mathrm{d} t}$
. Suppose that in a particular circuit, L = 0.4 Henrys, R = 0 Ohms, C = 0.1 Farads and V = 110 sin(ωt) Volts, where ω ∈ R. Initially the charge on the capacitor is 1 Coulomb and there is no current in the circuit.
(a) Write down the differential equation satisfied by q(t) in this circuit.
(b) Determine the value(s) of ω so that resonance occurs in the circuit.
(c) In the case where there is no resonance,
i. Solve the differential equation to find the charge on the capacitor at any time.
ii. Determine the transient and steady state solutions for the charge, if they exist.
iii. Find the current in the circuit at any time.

Homework Equations

The Attempt at a Solution

Hey everyone. Thanks for reading through and trying to help! Basically I just want to know if I'm thinking along the right track. So for part a, I apply the condition for resonance by solving the homogeneous second order equation, and then for part c I solve it across the domain where omega is not equal to the values I found in part b and I should be getting an answer in terms of omega? I'm just confused because the wording of the question makes it sound like I need to be getting a solution in terms of t only (i.e. omega should cancel out or something).

You need to show your actual work, formulas and all. It is impossible to help you---or to judge if you need help---from the generaties you have written.

 

[vela]

I should be getting an answer in terms of omega? I'm just confused because the wording of the question makes it sound like I need to be getting a solution in terms of t only (i.e. omega should cancel out or something).

It'll depend on $\omega$.