AC Circuits & Resonance
AC Resonance
In an AC circuit, the inductor and capacitor fight each other — one resists changes in current, the other resists changes in voltage, and their effects push in opposite directions. At one special frequency their opposing effects cancel out exactly, leaving minimal resistance to the current, so the current surges to its biggest value. This is resonance, and it is how a radio tunes into one station while ignoring the rest.
The formula
ω₀ = 1 / √(L · C)
- ω₀ — resonant angular frequency (rad/s): the frequency where the inductor and capacitor cancel and current peaks
- L — inductance (H): how strongly the inductor resists changes in current
- C — capacitance (F): how much charge the capacitor stores per volt across it
Worked example
A radio tuning circuit has an inductor of 1 henry and a capacitor of 0.25 farads. What is the resonant frequency?
- L = 1 H
- C = 0.25 F
- ω₀ = 1 / √(L · C)
- ω₀ = 1 / √(1 H · 0.25 F)
- ω₀ = 1 / √0.25 = 1 / 0.5
ω₀ = 2 rad/s
Test yourself
Same inductor, L = 1 H, but now C = 4 F. What is the new resonant frequency?
- Correct answer: ω₀ = 0.5 rad/s
- ω₀ = 2 rad/s
- ω₀ = 8 rad/s
Yes! ω₀ = 1 / √(1 · 4) = 1 / 2 = 0.5 rad/s.
If you increase the inductor's inductance L while keeping C the same, the resonant frequency ω₀…
- increases
- Correct answer: decreases
- stays exactly the same
Right — ω₀ is one over the square root of L times C, so a bigger L makes ω₀ smaller.
Where you see this
Turning a radio dial is tuning a circuit's resonant frequency: every station broadcasts at its own frequency, and the little LC circuit inside responds with a big current only at the one it is tuned to, ignoring the rest of the dial. One knob, one equation, and a clear station out of a crowded band.
Common mistakes
The scaling trips people: L and C both sit under the square root in the denominator of ω₀ = 1 / √(L · C), so a bigger inductor or capacitor makes the resonant frequency lower, not higher — with L = 1 H, going from C = 1 F to C = 4 F drops ω₀ from 1 rad/s to 0.5 rad/s. And at resonance the current is at its maximum because the opposition is at its minimum — nothing is being amplified from nowhere; energy is arriving from the AC source.
How it connects
This is where the whole electromagnetism module converges: Coulomb's charges, the capacitor's storage, the inductor's induced back-voltage from Faraday — at ω₀ = 1 / √(L · C) their oppositions cancel and the current surges. It is the electrical sibling of mechanical resonance: a swing pushed at its natural frequency, or the beats of the waves module slowing to nothing at a perfect match.