LC Circuit

LC Circuit consists of a charged capacitor and an inductor in series We will once again be using the loop rule for this system:

$$-L\frac{di}{dt}-\frac{q}{c}=0⟹L\frac{d^{2}q}{d{t}^2}+\frac{q}{c}=0$$ $$q(t)=Q_{max}\cos (\omega t=\varphi )$$

where $\omega =\frac{1}{\sqrt{LC}}$ This is similar to mass on a spring harmonic oscillations, but the energy storage is all electric and is being transformed into magnetic energy. Cool stuff.

Electric field ⇒ Spring (potential) energy Magnetic field ⇒ Magnetic (kinetic) energy

It is very interesting to see how themes play out throughout physics. The same harmonic oscillation problems also come up in QP1, and I am sure they will appear in IQM and Theoretical Mech.

For our electric system:

$$U_E=\frac{q^2}{2C}=\frac{Q^2}{2C}{\cos }^2(\omega t+\varphi )$$ $$U_B=\frac{1}{2}{L}^2=\frac{1}{2}L{I}^2{\sin }^2(\omega t+\varphi )$$ $$U_E+U_B=constant$$