A small toaster that operates at 120 V has a heating element made from a 4.4-m-long, 0.70-mm-diameter nichrome wire. The resistivity, density, and specific heat of nichrome are, respectively, 1.5 x 10⁻⁶ Ωm, 8400 kg/m³, and 450 J/kg K. If half the heat energy is lost to the air, how long does it take the heating element to warm from 20℃ to 450℃, about the temperature at which it first begins to glow red?

Verified step by step guidance

1. Determine the resistance of the nichrome wire using the formula for resistance: $R=\frac{\rho }{L}.$

Key Concepts

Ohm's Law

Ohm's Law states that the current flowing through a conductor between two points is directly proportional to the voltage across the two points and inversely proportional to the resistance. This relationship is expressed as V = IR, where V is voltage, I is current, and R is resistance. Understanding this law is crucial for analyzing electrical circuits, such as the toaster's heating element, to determine how much current flows at a given voltage.

Heat Transfer and Specific Heat Capacity

Specific heat capacity is the amount of heat required to raise the temperature of a unit mass of a substance by one degree Celsius. In this context, it helps determine how much energy is needed to heat the nichrome wire from 20℃ to 450℃. Additionally, understanding heat transfer mechanisms, including conduction, convection, and radiation, is essential for calculating energy losses to the surrounding air, which affects the heating time.

Resistivity and Resistance

Resistivity is a material property that quantifies how strongly a given material opposes the flow of electric current, expressed in ohm-meters (Ωm). The resistance of a wire can be calculated using the formula R = ρ(L/A), where ρ is resistivity, L is the length of the wire, and A is its cross-sectional area. This concept is vital for determining how much electrical energy is converted into heat in the toaster's heating element, influencing its performance and efficiency.