A 5.00-μF capacitor in series with a 4.00-Ωresistor is charged with a 9.00-V battery for a long

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A 5.00-μF capacitor in series with a 4.00-Ωresistor is...

A $5.00-\mu F$ capacitor in series with a $4.00-\Omega$ resistor is charged with a $9.00-\mathrm{V}$ battery for a long time by closing the switch (position $a$ in the figure). The capacitor is then discharged through an inductor $(L=40.0 \mathrm{mH})$ by closing the switch (position $b$ ) at $t=0$. a) Determine the maximum current through the inductor. b) What is the first time at which the current is at its maximum?

A $5.00-\mu F$ capacitor in series with a $4.00-\Omega$ resistor is charged with a $9.00-\mathrm{V}$ battery for a long time by closing the switch (position $a$ in the figure). The capacitor is then discharged through an inductor $(L=40.0 \mathrm{mH})$ by closing the switch (position $b$ ) at $t=0$.
a) Determine the maximum current through the inductor.
b) What is the first time at which the current is at its maximum?

Key Concepts

Energy Conservation in LC Circuits

In an ideal LC circuit without resistance, the total energy remains constant as it oscillates between the capacitor and the inductor. At the moment when the capacitor has fully discharged, all the initial energy stored in the capacitor is transferred to the inductor as magnetic energy. This conservation principle allows for the determination of maximum current by equating the initial stored energy to (1/2)L I² in the inductor.

Phase and Timing in LC Oscillations

The oscillatory behavior of an LC circuit is characterized by its sinusoidal variation with time. The phase of the oscillation determines when certain quantities, like current or voltage, reach their maximum values. Typically, the maximum current in an LC circuit occurs when the energy transfer is complete, which is at a quarter of the period of oscillation from the initial condition when the capacitor is fully charged.

LC Circuit Oscillations

An LC circuit, comprising an inductor and a capacitor, exhibits oscillatory behavior due to the continuous exchange of energy between the capacitor’s electric field and the inductor’s magnetic field. The angular frequency of these oscillations is determined by the inverse square root of the product of the inductance and capacitance, ? = 1/?(LC). Understanding these oscillations is crucial for analyzing transient and resonant phenomena in circuits.

RC Circuit Charging

This concept involves understanding how a capacitor charges through a resistor when connected to a constant voltage source. Over time, the capacitor gradually accumulates charge until it reaches an equilibrium state where its voltage equals that of the battery. This steady-state charging behavior is determined by the time constant, which is the product of the resistance and the capacitance.

Energy Stored in a Capacitor

The energy stored in a capacitor is a key concept that quantifies the electrical energy held in the electric field created between its plates. The energy is given by the expression (1/2)CV², where C is the capacitance and V is the voltage across the capacitor. This stored energy is available for conversion into other forms, such as kinetic energy in circuit elements.

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