# Current on Infinite Periodic LC Circuit

• Kyuubi
In summary, the conversation discusses the equation of motion for In(t) and its relationship with the infinite spring mass system equation solution. The spring mass system considers A to be the equilibrium length of the springs, while in the infinite periodic LC circuit, A represents the unknown parameter needed to convert In(t) to I(nA,t). The conversation also mentions using a Taylor Expansion to model the system as a long thin rod that has been pushed.
Kyuubi
Homework Statement
Show that the current on an infinite periodic LC circuit obeys the wave equation in the long wavelength limit with the speed of the wave being the speed of light.
Relevant Equations
Equations of motion of current in an LC circuit.

(In)''=1/LC(-2In +In+1 +In -1)

Note here In means i sub n. As in the current on the nth inductor.
I wrote down the equation of motion for In(t) and I'm trying to match it with infinite spring mass system equation solution. In the spring mass system, we consider A to be the equilibrium length of the springs, and we can thus write Xn(t) = X(nA,t) and put it back into the equation of motion while taking a Taylor Expansion. This allows us to model the system as a long thin rod that's been pushed. But in the infinite periodic LC Circuit, what exactly is my A? What will help me turn my In(t) to I(nA,t)? In other words, how can I change my equation of motion

Sorry I'm new to this. Is there a way to delete a thread or close it? I think I've solved my problem.

## 1. What is an infinite periodic LC circuit?

An infinite periodic LC circuit is a type of electrical circuit that consists of an inductor (L) and a capacitor (C) connected in a loop or series. The circuit has no resistance and is considered to be infinite in size and duration, meaning that it does not decay over time.

## 2. How does an infinite periodic LC circuit work?

An infinite periodic LC circuit works by storing energy in the magnetic field of the inductor and the electric field of the capacitor. When the circuit is first turned on, the capacitor begins to charge and the inductor builds up a magnetic field. As the capacitor reaches its maximum charge, the inductor's magnetic field reaches its maximum strength. This cycle then repeats, resulting in a constant flow of energy within the circuit.

## 3. What is the resonance frequency of an infinite periodic LC circuit?

The resonance frequency of an infinite periodic LC circuit is the frequency at which the circuit naturally oscillates. It is determined by the values of the inductance and capacitance in the circuit and can be calculated using the formula f = 1/(2π√(LC)), where f is the resonance frequency in Hertz, L is the inductance in Henrys, and C is the capacitance in Farads.

## 4. What are the applications of an infinite periodic LC circuit?

Infinite periodic LC circuits have many applications in electronics, including in radio and television receivers, oscillators, and filters. They are also used in wireless charging systems, where the circuit's resonance frequency is used to transfer energy wirelessly to charge devices.

## 5. What are the advantages of using an infinite periodic LC circuit?

One advantage of using an infinite periodic LC circuit is that it can store and transfer energy without any loss due to resistance. This makes it useful in applications where energy efficiency is important. Additionally, the resonance frequency of the circuit can be easily adjusted by changing the values of the inductor and capacitor, making it a versatile component in electronic systems.

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