Chapter 26: Problem 72
A 6.00-\(\mu\)F capacitor that is initially uncharged is connected in series with a 5.00-\(\Omega\) resistor and an emf source with \(\varepsilon =\) 50.0 V and negligible internal resistance. At the instant when the resistor is dissipating electrical energy at a rate of 300 W, how much energy has been stored in the capacitor?
Short Answer
Step by step solution
Understand the Power Dissipation in the Resistor
Calculate the Current Through the Resistor
Determine the Voltage Across the Resistor
Calculate the EMF Source Voltage shared across the Capacitor
Calculate the Energy Stored in the Capacitor
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Ohm's Law
It's vital to understand Ohm's Law when analyzing circuits because it helps determine how voltage and current distribute in a circuit, especially when components like resistors are involved. In this problem, we used Ohm's Law to determine the voltage drop across a resistor given a current of 7.75 A and a resistance of 5.00 \( \Omega \). This calculation is crucial as it helps identify how much of the total voltage from the emf source is utilized by the resistor versus how much goes across the capacitor.
Power Dissipation
In our example, a resistor dissipates 300 W of power while being part of an electrical circuit with a current of 7.75 A and a resistance of 5.00 \( \Omega \). Calculating power dissipation allows us to measure how much energy is being used at a given moment in a circuit, which is crucial for understanding efficiency and preventing overheating.
Capacitor Discharge
In the problem, once the voltage across the resistor is calculated and subtracted from the emf of the source, the remaining voltage (11.25 V) is the voltage across the capacitor. This allows us to calculate the energy stored as \( 3.796 \times 10^{-4} \) J, which indicates how much energy is available for the capacitor to discharge.
Electrical Circuits
In this exercise, we dealt with a simple series circuit comprising a resistor and capacitor connected to an emf source. This setup highlights fundamental principles like Ohm's Law and power dissipation, combined with the concept of controlled capacitance through charge and discharge cycles. Analyzing these components' interactions provides insights into the stored energy and power dynamics in electrical circuits, which is crucial for designing efficient and safe electronic devices.