Series RLC circuit connected to a DC battery

[Meow12]

Homework Statement

A series RLC circuit is connected to a DC battery via a switch. A long time after the switch is closed, what is the current in the circuit?

Relevant Equations

Kirchhoff's voltage law gives $\displaystyle iR+\frac{q}{C}+L\frac{di}{dt}=\mathcal{E}$

How do I solve the differential equation? Please give me a hint.

 

[Meow12]

I guess I could write $\displaystyle i=\frac{dq}{dt}$ to get
$\displaystyle R\frac{dq}{dt}+\frac{q}{C}+L\frac{d^2q}{dt^2}=\mathcal{E}$

Then how do I proceed?

 

[Tom.G]

For the dt terms, what does t approach, After a long time?

-OR-

A couple clues to a different approach:

A series RLC circuit is connected to a DC battery...

A long time after the switch is closed...

Now:
What is the voltage on a Capacitor after being connect to a voltage source for A long time...?

 

[Meow12]

Now:
What is the voltage on a Capacitor after being connect to a voltage source for A long time...?

I think it will be $\mathcal{E}$, the emf of the DC battery.

 

[Tom.G]

I just edited my post at the same time you responded. Take a look.

 

[Meow12]

For the dt terms, what does t approach, After a long time?

$t\to\infty$

 

[Tom.G]

Yes to both V_c and t.

With the Capacitor voltage equal to and in series with the battery voltage, what is the voltage applied to the R and the L?

 

[Meow12]

Yes to both V_c and t.

With the Capacitor voltage equal to and in series with the battery voltage, what is the voltage applied to the R and the L?

I guess $V_R=V_L=0$. And since $V_R=iR$, this means $i=0$?

I would've preferred to solve the differential equation, though.

 

[Tom.G]

Correct.

Sorry about the shortcut, too much practical field experience I guess.
I'll see if I can get one of the more math-oriented folks here to help on that approach.
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Just entered a request for the math help.

 

[Orodruin]

I guess I could write $\displaystyle i=\frac{dq}{dt}$ to get
$\displaystyle R\frac{dq}{dt}+\frac{q}{C}+L\frac{d^2q}{dt^2}=\mathcal{E}$

Then how do I proceed?

This is a regular 2nd order ODE with constant coefficients. You can solve it with any applicable method for solving ODEs. However, the easiest way is to note that a constant solution $q = \mathcal E C$ solves the ODE and is time-independent. Unless the ODE has a oscillatory solution with constant amplitude, that will therefore be the large time solution.

 

[Orodruin]

To add to that: You can of course solve the entire ODE and take the limit $t\to \infty$ as well. Start by defining $w = q - \mathcal E C$ to obtain a homogeneous ODE for $w$. Solve for the general case with the ansatz $w = e^{kt}$ and determine $k$ (since you have a second order ODE, there will be two solutions for $k$, the general solution is a linear combination -- $A e^{k_1t} + B e^{k_2t}$ -- of those as the ODE for $w$ is a linear and homogeneous ODE).

 

[BvU]

More in favour of the conventional approach

(useful in a lot of contexts)

$\$

 

[hutchphd]

This problem for q is the damped harmonic oscillator. The equation is ubiquitous and holds for nearly any dynamic sysytem close to equilibrium. Masses on springs or an approximated pendulum It is solved in every good freshman physics text, with colors and pictures. Learn it inside and out. Soon you'll be doing quantum field theory.