An L C circuit like that in Figure CQ^32.8 consists of a 3.30-H inductor and an 840-pF capacitor

step 1 the capacitor charge is at its maximum value so 5 in this expression is equal to 0 and hence we

An L C circuit like that in Figure CQ^32.8 consists of a...

An $L C$ circuit like that in Figure $\mathrm{CQ}^{32.8}$ consists of a $3.30-\mathrm{H}$ inductor and an $840-\mathrm{pF}$ capacitor that initially carries a $105-\mu \mathrm{C}$ charge. The switch is open for $t<0$ and is then thrown closed at $t=0 .$ Compute the following quantities at $t=2.00 \mathrm{ms} :(\mathrm{a})$ the energy stored in the capacitor, $(\mathrm{b})$ the energy stored in the inductor, and $(\mathrm{c})$ the total energy in the circuit.

Key Concepts

Conservation of Energy in Oscillatory Systems

In an ideal LC circuit, the total energy, which is the sum of the energies stored in the capacitor and inductor, remains constant over time. This conservation of energy principle means that any decrease in the energy of one component results in an equivalent increase in the energy of the other, resulting in perpetual exchange in a lossless system, which is a hallmark of simple harmonic motion.

Energy Stored in an Inductor

An inductor stores energy in its magnetic field when current flows through it. The stored energy is given by the formula E = ½ L I², where L is the inductance and I is the current. In an LC circuit, this magnetic energy plays a critical role during the oscillations, as it complements the energy stored in the capacitor, with energy transforming from one form to the other.

LC Circuit Oscillations

An LC circuit consists of an inductor (L) and a capacitor (C) connected together, creating an oscillatory system where energy continuously transfers between the electric field of the capacitor and the magnetic field of the inductor. The natural behavior of these circuits is to oscillate with a frequency determined by the values of L and C, given by ? = 1/?(LC). This phenomenon is fundamental in many applications, such as tuning and filtering in electronic circuits.

Energy Stored in a Capacitor

The energy stored in a capacitor is associated with the electric field created by the charge separation between its plates. It is determined by the equation E = ½ (q²/C), where q represents the charge on the capacitor and C its capacitance. This concept is important for understanding how electrical energy is stored and later released in an LC circuit during its oscillatory cycle.

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