Series RLC circuit connected to a DC battery

AI Thread Summary
To solve the differential equation for a series RLC circuit connected to a DC battery, the equation can be expressed as R(dq/dt) + (q/C) + L(d²q/dt²) = E. After a long time, the voltage across the capacitor approaches the battery's emf (E), leading to zero voltage across the resistor (V_R) and inductor (V_L), resulting in zero current (i). A constant solution, q = EC, satisfies the ODE, indicating a time-independent solution. The discussion suggests that the problem can be approached as a damped harmonic oscillator, which is a common topic in physics. Understanding this concept is essential for further studies in dynamic systems and quantum field theory.
Meow12
Messages
46
Reaction score
20
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.
 
Physics news on Phys.org
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?
 
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...?
 
Tom.G said:
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.
 
I just edited my post at the same time you responded. Take a look.
 
Tom.G said:
For the dt terms, what does t approach, After a long time?
##t\to\infty##
 
Yes to both Vc 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?
 
Tom.G said:
Yes to both Vc 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.
 
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.
.
.
.
Just entered a request for the math help.
 
  • #10
Meow12 said:
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.
 
  • #11
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).
 
  • #12
  • #13
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.
 
  • Like
Likes DaveE, BvU, WWGD and 1 other person

Similar threads

Replies
3
Views
2K
Replies
15
Views
5K
Replies
12
Views
7K
Replies
6
Views
2K
Replies
2
Views
1K
Replies
21
Views
1K
Replies
3
Views
1K
Back
Top