A 6.00-μF capacitor that is initially uncharged is connected...

A $6.00-\mu \mathrm{F}$ capacitor that is initially uncharged is connected in series with a $5.00-\Omega$ resistor and an emf source with $\mathcal{E}=50.0 \mathrm{V}$ and negligible internal resistance. At the instant when the resistor is dissipating electrical energy at a rate of $250 \mathrm{W},$ how much energy has been stored in the capacitor?

Key Concepts

RC Circuit Transient Response

This concept involves the behavior of an electrical circuit during the charging (or discharging) process of a capacitor when connected in series with a resistor. The voltage across the capacitor and the current in the circuit vary with time in an exponential manner, governed by a differential equation. The transient analysis describes how the capacitor gradually builds up charge, influencing both the current and power dissipation in the resistor.

Time Constant

The time constant of an RC circuit, given by the product R×C, defines the characteristic time over which the capacitor charges or discharges. It provides a measure for the speed of the transient process; after one time constant, the capacitor’s voltage reaches approximately 63% of its final value when charging. This concept is crucial for determining the timing and exponential decay factors in circuit behavior.

Capacitor Energy Storage

This key concept addresses the method for calculating the energy stored in a capacitor once it is charged. The energy is given by the formula U = ½ C V², where C is the capacitance and V is the voltage across the capacitor. Understanding capacitor energy storage is fundamental in analyzing how electrical energy is accumulated over time during the charging process.

Resistor Power Dissipation

The power dissipated by the resistor in an RC circuit is related to the instantaneous current flowing through it and follows the formula P = I²R. In charging circuits, this power decreases exponentially as the capacitor voltage rises and the current falls. This concept is important for relating the rate at which energy is being lost as heat in the resistor to the evolving energy storage in the capacitor.

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