Introduction to LC Circuits: The Electrical Pendulum

Welcome to the final topic of Unit 13! So far, you have seen how resistors paired with capacitors (RC circuits) or inductors (LR circuits) cause currents to grow or decay exponentially. But what happens when we connect a capacitor directly to an inductor with no resistor in between? We get an LC Circuit.

In an LC circuit, energy doesn't just fade away; it "sloshes" back and forth between the electric field of the capacitor and the magnetic field of the inductor. This creates electromagnetic oscillations. Think of it as the electrical version of a swinging pendulum or a mass on a spring. If you can understand a simple harmonic oscillator in mechanics, you already understand the logic of an LC circuit!

The Mechanical Analogy: Mass-Spring vs. LC Circuit

To help visualize this, let’s compare the two systems. In mechanics, a mass on a spring moves back and forth, trading potential energy for kinetic energy. In an LC circuit, the "motion" is the flow of charge.

  • The Capacitor (\(C\)): Stores energy in an electric field. This is like the spring potential energy. A charged capacitor wants to "push" charge out, just like a compressed spring wants to push a mass.
  • The Inductor (\(L\)): Stores energy in a magnetic field. This is like the kinetic energy of a moving mass. Inductors have "electrical inertia"—they resist changes in current, just like a heavy mass resists changes in its velocity.
  • The Resistance (\(R = 0\)): In an ideal LC circuit, we assume there is no resistance. This is like a frictionless floor where the pendulum never stops swinging.

Did you know? This "sloshing" is how early radio transmitters worked! By choosing the right \(L\) and \(C\), engineers could create oscillations at specific frequencies to carry music and news through the air.

Deriving the Equation of Motion (Calculus Focus)

To understand the physics, we apply Kirchhoff’s Loop Rule to a simple loop containing a capacitor and an inductor. Starting from the capacitor and moving around the loop, the sum of the potential differences must be zero:

\( \Delta V_{cap} + \Delta V_{ind} = 0 \)

Using our definitions for potential difference across a capacitor (\(V = \frac{Q}{C}\)) and the induced emf in an inductor (\(\mathcal{E} = -L \frac{dI}{dt}\)), we get:

\( \frac{Q}{C} - L \frac{dI}{dt} = 0 \)

Since current is the rate of flow of charge, and the charge on the capacitor is decreasing as the current increases, we use the relationship \( I = -\frac{dQ}{dt} \). Taking the derivative again, we find that \( \frac{dI}{dt} = -\frac{d^2Q}{dt^2} \). Substituting this back into our loop equation:

\( \frac{Q}{C} - L \left( -\frac{d^2Q}{dt^2} \right) = 0 \)

\( L \frac{d^2Q}{dt^2} + \frac{1}{C} Q = 0 \)

If you rearrange this to isolate the second derivative, it looks exactly like the equation for simple harmonic motion (\( a = -\omega^2 x \)):

\( \frac{d^2Q}{dt^2} = -\left( \frac{1}{LC} \right) Q \)

Key Takeaway: The charge \(Q\) on a capacitor in an LC circuit behaves mathematically exactly like the position \(x\) of a mass on a spring!

The Oscillatory Solutions

Because the equation above is a second-order linear differential equation, the solution is a sinusoidal function (sine or cosine). We define the angular frequency (\(\omega\)) of the oscillation as:

\( \omega = \frac{1}{\sqrt{LC}} \)

The charge as a function of time \(t\) is usually written as:

\( Q(t) = Q_{max} \cos(\omega t + \phi) \)

(Where \(Q_{max}\) is the maximum charge and \(\phi\) is the phase constant, usually \(0\) if we start with a fully charged capacitor at \(t=0\).)

Since current \(I\) is the derivative of charge, we can find the current function:

\( I(t) = -\frac{dQ}{dt} = \omega Q_{max} \sin(\omega t + \phi) \)

Notice that the maximum current is \( I_{max} = \omega Q_{max} \). This shows that the current and charge are "out of phase"—when the charge is at its maximum, the current is zero, and vice versa.

Energy Transformations in LC Circuits

In an ideal LC circuit, the total energy (\(U_{total}\)) is conserved. It simply shifts between two forms:

1. Electric Potential Energy (\(U_C\)): Stored in the capacitor's electric field.
\( U_C = \frac{1}{2} \frac{q^2}{C} = \frac{Q_{max}^2}{2C} \cos^2(\omega t) \)

2. Magnetic Potential Energy (\(U_L\)): Stored in the inductor's magnetic field.
\( U_L = \frac{1}{2} L I^2 = \frac{1}{2} L (\omega Q_{max})^2 \sin^2(\omega t) \)

At any moment in time:
\( U_{total} = U_C + U_L = \text{Constant} \)

Step-by-Step Energy Cycle:
  1. \(t = 0\): The capacitor is fully charged. All energy is in the electric field (\(U_C\)). Current is zero.
  2. The "Quarter Cycle": The capacitor discharges through the inductor. The electric field weakens, and the magnetic field strengthens.
  3. The Peak Current: When the capacitor is completely empty (\(Q = 0\)), the current is at its maximum. All energy is now in the magnetic field (\(U_L\)).
  4. The "Half Cycle": The inductor's "inertia" keeps the current flowing, which charges the capacitor back up, but with opposite polarity.

Quick Tip: If an exam question asks for the maximum current and gives you the maximum charge, don't panic! Just use energy conservation: \( \frac{1}{2} \frac{Q_{max}^2}{C} = \frac{1}{2} L I_{max}^2 \). It's often faster than using the \(\omega\) formulas.

Common Mistakes to Avoid

  • Radians vs. Degrees: When calculating \(\cos(\omega t)\), always make sure your calculator is in Radian mode. \(\omega\) is in radians per second.
  • Units: Remember that \(L\) is in Henries (\(H\)) and \(C\) is in Farads (\(F\)). If you are given millihenries (\(mH\)) or microfarads (\(\mu F\)), convert them to base units before calculating \(\omega\).
  • Phase Confusion: Remember that when charge is maximum (start of the cycle), current is zero. This is why we use \(\cos\) for charge and \(\sin\) for current.

Chapter Summary / Quick Review

The Main Idea: LC circuits oscillate because inductors resist changes in current and capacitors store charge.

The Frequency: The angular frequency is \( \omega = \frac{1}{\sqrt{LC}} \). The regular frequency is \( f = \frac{\omega}{2\pi} \).

Conservation of Energy: Maximum energy in the capacitor equals maximum energy in the inductor: \( \frac{Q_{max}^2}{2C} = \frac{1}{2} L I_{max}^2 \).

Calculus Link: The differential equation \( L \frac{d^2Q}{dt^2} + \frac{Q}{C} = 0 \) is the foundation for all LC circuit analysis.