A 160-μF capacitor charged to 450 V is discharged through a 31.2-k Ωresistor. (a) Find the time

step 1 The first part of this question wants us to find the RC time constant. So the time constant is f

A 160-μF capacitor charged to 450 V is discharged through a...

A $160-\mu \mathrm{F}$ capacitor charged to $450 \mathrm{V}$ is discharged through a $31.2-\mathrm{k} \Omega$ resistor. (a) Find the time constant. (b) Calculate the temperature increase of the resistor, given that its mass is $2.50 \mathrm{g}$ and its specific heat is $1.67 \mathrm{kJ} / \mathrm{kg} \cdot^{\circ} \mathrm{C}$ noting that most of the thermal energy is retained in the short time of the discharge. (c) Calculate the new resistance, assuming it is pure carbon. (d) Does this change in resistance seem significant?

Key Concepts

Thermal Effects on Resistive Materials

Thermal effects on resistance refer to the change in a material’s electrical resistance as its temperature changes. In conductors or semiconductors like carbon, the resistance may vary due to temperature-dependent changes in resistivity. This concept is crucial when evaluating the performance of a resistor after it has dissipated a significant amount of energy, as the temperature rise can alter its resistance value, affecting circuit behavior.

Specific Heat Capacity and Heat Transfer

Specific heat capacity is a measure of the amount of energy required to raise the temperature of a unit mass of a substance by one degree Celsius. In the context of electrical circuits, when a resistor dissipates electrical energy as heat, the specific heat capacity, along with the mass of the resistor, dictates how much its temperature will increase. This relationship is given by the equation ?T = Q/(m·c), where Q is the energy transferred, m is the mass, and c is the specific heat.

RC Time Constant

The RC time constant is a fundamental concept in circuits comprising a resistor and a capacitor. It is defined as the product of the resistance (R) and the capacitance (C) and describes the rate at which the capacitor charges or discharges. In transient circuit analysis, the time constant determines how quickly a capacitor will release its stored energy through the resistor, with the voltage decaying exponentially over time.

Energy Stored in a Capacitor

This concept refers to the energy that can be stored in the electric field of a capacitor. The energy is given by the formula ½CV², where C is the capacitance and V is the voltage across the capacitor. This stored energy becomes significant in transient events, such as discharges, where it is quickly converted into other forms, typically heat, as it is released through circuit components like resistors.

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