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Ampere's Law is integral in understanding how magnetic fields are produced by electric currents.

It's one of Maxwell’s equations, and it describes the relation between current and magnetism. In essence, Ampere's Law states that for any closed loop path, the sum of the length elements times the magnetic field in the direction of the length element is equal to the permeability of free space times the electric current enclosed in the loop. Mathematically, it's expressed as \[\begin{equation}\bigcirc B \bullet dl = \text{μ0}\times I_{enc}\end{equation}\] where B is the magnetic field, dl is a differential element of the path length, \text{μ0} is the permeability of free space, and \text{I}_{enc} is the enclosed current.

In a practical sense, when we have an inductor with a current changing over time, Ampere's Law can be used to understand how this changing current generates a magnetic field, which then, according to Faraday's Law of Electromagnetic Induction, induces an electromotive force (emf) in the opposite direction, known as the back emf. This is crucial for the operation of an LC circuit, where changes in electric current and magnetic fields are intimately connected.

Kirchhoff's Voltage Law (KVL) is a fundamental principle used in circuit analysis. It's based on the conservation of energy principle applied to electrical circuits. Kirchhoff's Voltage Law states that the algebraic sum of all potential differences (voltages) around any closed loop in a circuit must equal zero.

In more elaborate terms, as you move around a loop, the increases in electric potential (coming from emf sources like batteries or induced emf in inductors) are exactly balanced by the drops in potential (like the voltage across resistors, capacitors, or inductors). Mathematically, the principle is expressed as \[\begin{equation}\sum V = 0\end{equation}\] where V represents the voltage drops or gains.

This law underlies the calculation of the maximum voltage across a capacitor in an LC circuit. Since the energy within the loop is conserved, the voltage developed across the capacitor must equal the voltage drop across the inductor, which allows us to solve for one knowing the other.

Understanding capacitor voltage calculations requires a grasp of what a capacitor is and how it behaves in a circuit. A capacitor is a device that stores charge and energy in the electric field between its plates. The voltage across a capacitor is proportional to the amount of charge it has stored.

The voltage across a capacitor in an LC circuit can be calculated using the formula \[\begin{equation}V_C = \frac{q}{C}\end{equation}\] where V_C is the capacitor voltage, q is the charge on the capacitor, and C is the capacitance.

In an LC oscillating circuit, the capacitor’s voltage and charge vary over time. As the circuit oscillates, energy is transferred back and forth between the capacitor's electric field and the inductor's magnetic field. The maximum capacitor voltage occurs when all the energy is stored in the electric field, which is the point of maximum charge on the capacitor plates.

The phenomenon of inductor back emf (electromotive force) is essential for the operation of LC circuits. An inductor is a coil of wire that resists changes in the current passing through it due to the magnetic field created by the current itself. When the current in an inductor changes, the magnetic field does too, creating a voltage across the inductor that opposes the change in current. This is known as back emf or counter emf.

The induced emf in an inductor can be calculated with the formula \[\begin{equation}V_L = L\frac{di}{dt}\end{equation}\] where V_L is the back emf, L is the inductance, di is the change in current, and dt is the change in time.

This back emf is particularly significant during the initial charging and discharging of the LC circuit, where it can cause oscillations if it's not dissipated. In the context of the textbook exercise, understanding inductor back emf helps to determine the voltage across the inductor at any point in time, which thanks to Kirchhoff's Voltage Law, is the same as the voltage across the capacitor at that instant.