A certain toaster has a heating element made of Nichrome...

A certain toaster has a heating element made of Nichrome wire. When the toaster is first connected to a $120-\mathrm{V}$ source (and the wire is at a temperature of $20.0^{\circ} \mathrm{C}$ ), the initial current is 1.80 A. However, the current begins to decrease as the heating element warms up. When the toaster reaches its final operating temperature, the current drops to $1.53 \mathrm{A}$. (a) Find the power delivered to the toaster when it is at its operating temperature. (b) What is the final temperature of the heating element?

Key Concepts

Temperature Dependence of Resistance

The temperature dependence of resistance is characterized by the fact that a material's resistance typically changes with its temperature. This behavior is quantified using the temperature coefficient of resistance, which shows how much the resistance increases (or decreases) per degree change in temperature. In materials such as Nichrome, used in heating elements, the resistance increases with temperature, affecting both the current flow and the thermal equilibrium state of the device.

Ohm's Law

Ohm's Law states that the current through a conductor between two points is directly proportional to the voltage across the two points, expressed as V = IR. This principle is fundamental in determining the resistance of an element when the voltage and current are known, and it is particularly useful in analyzing circuits with resistive heating elements, where the resistance can change with conditions such as temperature.

Joule Heating

Joule heating, also known as resistive or ohmic heating, describes the process by which electrical energy is converted into heat energy in a conductor. The power dissipated as heat in a resistor is given by P = VI, or alternatively P = I²R or P = V²/R. This concept explains how a heating element warms up as current flows through it and is crucial for calculating the power delivered to devices like toasters.

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