31.10
LC Circuits
An LC circuit consists of an inductor and a capacitor, either in series or parallel. Consider a charged capacitor connected with an inductor in series…
An LC circuit comprises an inductor and a capacitor.
Consider an LC circuit where an inductor is connected across a charged capacitor. Here, current flows through the inductor, building up a magnetic field around it. This changes the potential across the capacitor and induces an EMF across the inductor.
These EMFs are used to obtain the equation of the LC circuit. This is equivalent to the equation of a simple harmonic oscillator where displacement is replaced by charge.
Initially, when the circuit is open, no current flows; therefore, the total energy is stored as electrical energy.
When the circuit is closed, current flows through the inductor, transferring all the electrical energy into magnetic energy.
At any arbitrary time, the total energy of the LC circuit is the sum of the electrical and magnetic energy. If there is no energy dissipation, the total energy remains conserved and oscillates between the capacitor and inductor.
This case is similar to a simple harmonic oscillator where energy oscillates between potential and kinetic energy.
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Q1: What happens to energy when a charged capacitor is connected to an inductor?
When a charged capacitor connects to an inductor, electrical energy stored in the capacitor's electric field transfers to the inductor's magnetic field. As the capacitor discharges, current builds through the inductor, creating a magnetic field. Since no resistance exists in an ideal LC circuit, total energy is conserved and oscillates between electrical and magnetic forms.
Q2: How does an LC circuit behave like a simple harmonic oscillator?
An LC circuit exhibits harmonic oscillation because energy continuously exchanges between the capacitor and inductor, similar to how a mass-spring system exchanges kinetic and potential energy. The capacitor charge corresponds to displacement, current corresponds to velocity, inductance corresponds to mass, and capacitance reciprocal corresponds to the spring constant.
Q3: Why does current not change instantaneously when an LC circuit closes?
The inductor opposes changes in current through induced electromotive force (EMF). When the circuit closes, the induced EMF prevents current from rising instantly; instead, it starts at zero and gradually builds to maximum. This gradual buildup occurs while the capacitor discharges and the magnetic field strengthens.
Q4: What is the relationship between capacitor potential and induced EMF during discharge?
During capacitor discharge in an LC circuit, the capacitor potential equals the induced EMF at each instant. As the capacitor potential decreases, the induced EMF also decreases, causing the rate of current change to slow. When capacitor potential reaches zero, the induced EMF becomes zero and current reaches its maximum value.
Q5: How does energy conservation apply to an ideal LC circuit?
In an ideal LC circuit with no resistance, total energy remains constant because no energy dissipates through Joule heating. The maximum energy initially stored in the capacitor equals the maximum energy later stored in the inductor. At any moment, total energy equals the sum of electrical and magnetic energy.
Q6: What role does the magnetic field play in an LC circuit?
The magnetic field in the inductor stores energy and opposes current changes through induced EMF. As current flows through the inductor, the magnetic field builds up, transferring energy from the capacitor. The strength of this magnetic field determines how much energy the inductor stores at any given time.
Q7: How do the components of an LC circuit correspond to mechanical oscillations?
In an LC circuit, the capacitor acts like a spring storing potential energy, while the inductor acts like a mass storing kinetic energy through its magnetic field. Charge oscillates like displacement, and current oscillates like velocity. This mechanical analogy helps explain why LC circuits produce oscillations similar to oscillations in an LC circuit.