A capacitor is charged until it holds 5.0 J of energy, then...

A capacitor is charged until it holds $5.0 \mathrm{J}$ of energy, then connected across a $10-\mathrm{k} \Omega$ resistor. In $8.6 \mathrm{ms}$, the resistor dissipates 2.0 J. Find the capacitance.

Key Concepts

Exponential Decay in Capacitor Discharge

When a charged capacitor discharges through a resistor, the voltage across the capacitor and the charge stored decrease exponentially with time. This exponential decay relationship is described by equations that involve the time constant RC. Recognizing this decay pattern is important for determining the evolution of the circuit’s energy and current over time.

Energy Dissipation in Resistors

As the capacitor discharges, the energy stored within it is converted into heat through the resistor. The energy dissipated, often calculated by integrating the power (which is voltage squared divided by resistance) over time, demonstrates the conversion of stored electrostatic energy into thermal energy. This concept ties together the principles of capacitor energy storage and resistive dissipation in circuits.

Energy Stored in a Capacitor

This concept pertains to the energy that a capacitor holds when charged. The energy is typically calculated using the formula (1/2) * C * V², where C is the capacitance and V is the voltage across the capacitor. Understanding how energy is stored in a capacitor is essential for analyzing how much energy can be dissipated when the capacitor discharges through a circuit element, such as a resistor.

RC Circuit and Time Constant

An RC circuit consists of a resistor and a capacitor connected in series. A key parameter in these circuits is the time constant, given by the product RC. The time constant represents the time it takes for the capacitor to charge or discharge to about 63% of its maximum value, which is critical for understanding the rate at which the voltage and current decay over time.

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