Real circuits have resistance, so LC oscillations do not last forever: each cycle the resistor turns a little of the energy into heat. The result is a damped oscillation, the electrical twin of a damped spring.
Why it works
Add the resistor's voltage to the loop rule: $L\ddot{q} + R\dot{q} + \dfrac{q}{C} = 0$. Compare $m\ddot{x} + b\dot{x} + kx = 0$: the resistance plays the role of the drag constant. So everything you know about damped springs carries over. For small $R$ the charge still oscillates, but inside a shrinking envelope:
$$q = Q_0e^{-Rt/2L}\cos(\omega' t + \phi), \qquad \omega' = \sqrt{\frac{1}{LC} - \left(\frac{R}{2L}\right)^2}.$$
Key idea
Underdamped ($R < 2\sqrt{L/C}$): oscillation with amplitude falling as $e^{-Rt/2L}$ and energy as $e^{-Rt/L}$. Critically damped at $R = 2\sqrt{L/C}$; overdamped above it, with no oscillation at all. Light damping barely changes the frequency.
Worked examples
Worked example: How much does the frequency change?
A series circuit has $L = 0.10\ \text{H}$, $C = 10\ \mu\text{F}$ and $R = 20\ \Omega$. Find $\omega_0$, $\omega'$ and the critical resistance.
- $\omega_0 = 1/\sqrt{LC} = 1000\ \text{rad/s}$ and $R/2L = 100\ \text{s}^{-1}$.
- $\omega' = \sqrt{1.00 \times 10^{6} - 1.00 \times 10^{4}} = 995\ \text{rad/s}$, only 0.50% lower.
- Critical resistance: $2\sqrt{L/C} = 200\ \text{Ω}$, so this circuit is well underdamped.
Answer: $\omega' \approx 995\ \text{rad/s}$
Worked example: How fast it dies away
For the same circuit, how long until the amplitude of the charge oscillation halves? How long for the energy to halve?
- Amplitude: $e^{-Rt/2L} = \tfrac{1}{2}$ gives $t = \dfrac{2L\ln 2}{R} = 6.93\ \text{ms}$.
- Energy goes as the amplitude squared, $e^{-Rt/L}$, so it halves in half that time: $3.47\ \text{ms}$.
Answer: $6.93\ \text{ms}$ and $3.47\ \text{ms}$
Worked example: Too much resistance
What happens if the same circuit has $R = 300\ \Omega$?
- $300\ \Omega$ is above the critical $200\ \Omega$, so $(R/2L)^2 > 1/(LC)$ and $\omega'$ would be the square root of a negative number.
- There is no oscillation: the charge drains away without swinging back (overdamped).
Answer: Overdamped: no oscillation
Common mistake
"Adding resistance changes the frequency a lot." For light damping the correction $(R/2L)^2$ is tiny next to $1/(LC)$: in the first example it shifts the frequency by about half a percent. What you notice is the amplitude dying away. Check by comparing the two terms under the square root before deciding the frequency has changed.