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Wize University Physics Textbook (Master) > DC Circuits

LC Circuits

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LC Circuits


An LC circuit contains an inductor and a capacitor. Due to the absence of a dissipative element (e.g. resistor) the electrical energy stored in the capacitor transforms to the magnetic energy in the inductor and vice versa, in an oscillatory way.

In the following LC circuit, the capacitor has an initial charge QQ at time t=0t=0.





According to Kirchhoff’s voltage rule, we have 0=qC−L didt0=\dfrac{q}{C}-L\ \dfrac{di}{dt} with i=−dqdti=-\dfrac{dq}{dt} .

Therefore the charge is:
 q(t)=Qcos⁡(ωt+ϕ) \boxed {\ q(t)=Q\cos(\omega t+\phi) \ }

  • q(t)q(t) is the charge as a function of time
  • QQ is the maximum charge
  • ϕ\phi is the phase constant
  • ω\omega is the angular frequency

The angular frequency is given by:

 ω=1LC \boxed{ \ \omega=\dfrac{1}{\sqrt{LC}} \ }

  • CC is the capacitance
  • LL is the inductance

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The current can be obtained by taking the derivative of the charge equation with respect to time:

 i(t)=Isin⁡(ωt+ϕ) \boxed{ \ i(t)=I\sin\left(\omega t+\phi\right) \ }

with I=ωQI=\omega Q .


If we assume that at time t=0t=0 the capacitor is fully charged, the phase constant ϕ\phi in above equations is zero.





Wize Concept
Energy is conserved, and it oscillates between being stored in the magnetic and electric fields: UE(t)+UB(t)=constantU_E(t)+U_B(t)=constant

The maximum electric and magnetic potential energies are equal:

 Q22C=LI22 \boxed{ \ \dfrac{Q^2}{2C}=\dfrac{L I^2}{2} \ }



Exam Tip
During one full period of oscillation, the capacitor is fully charged - then empty - then fully charged again (in the opposite way) - then empty - then full again.








SHM : LC Circuits :

position x(t)x(t) charge q(t)q(t)
xmax=Ax_{max}=A qmax=Qq_{max}=Q
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velocity v(t)v(t) current i(t)i(t)
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mass mm inductance LL
spring constant kk capacitance (reciprocal) 1/C1/C
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kinetic energy K=12mv2K=\dfrac{1}{2}mv^2 inductor energy U=LI22U=\dfrac{LI^2}{2}

potential energy U=12kx2U=\dfrac{1}{2}kx^2 capacitor energy U=Q22CU=\dfrac{Q^2}{2C}
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Example: Maximum Current


A 11 µF capacitor is charged by a 2020 V battery. The fully charged capacitor is then discharged through a 55 mH inductor. Determine the maximum current in the resulting oscillations.

The maximum current is given by:

Imax⁡=ωQI_{\max}=\omega Q

We know that ω=1LC\omega=\dfrac{1}{\sqrt{LC}} and Q=CVQ=CV , so substituting these in the equation above we have:


Imax=CVLC=VCLI_{max}=\dfrac{CV}{\sqrt{LC}}=V\sqrt\dfrac{C}{L}

=201×10−65×10−3=20\sqrt\dfrac{1\times10^{-6}}{5\times10^{-3}}

=0.283=0.283 (A)


NOTE: this is equivalent to using conservation of energy: LImax22=Q22 Cmax\dfrac{LI_{max}^2}{2}=\dfrac{Q^2}{2\ C_{max}}

Practice: Period of Oscillation


Consider an LC circuit with maximum current of 8.08.0 mA through the inductor and 2.02.0 µC of maximum charge on the capacitor.

a) What is the period of oscillation?

b) How long will it take between the capacitor being fully charged to being fully discharged for the first time?