A heating element made of tungsten wire is connected to a...

A heating element made of tungsten wire is connected to a large battery that has negligible internal resistance. When the heating element reaches $80.0^{\circ} \mathrm{C}$, it consumes electrical energy at a rate of $480 \mathrm{~W}$. What is its power consumption when its temperature is $150.0^{\circ} \mathrm{C} ?$ Assume that the temperature coefficient of resistivity has the value given in Table 25.2 and that it is constant over the temperature range in this problem. In Eq. (25.12) take $T_{0}$ to be $20.0^{\circ} \mathrm{C}$.

Key Concepts

Electrical Power Consumption in Resistive Elements

The power dissipated by a resistive element within an electrical circuit is given by the formula P = V²/R when connected to a voltage source. This relationship implies that for a fixed voltage, the power consumption is inversely proportional to the resistance. As the element’s temperature increases and its resistance changes, the rate at which it consumes electrical energy correspondingly changes, impacting its power output.

Ohm’s Law

Ohm’s Law, which states that V = IR (where V is voltage, I is current, and R is resistance), provides the foundation for analyzing electrical circuits. In the context of resistive heating elements, it is used in conjunction with the power formula to understand how changes in resistance due to temperature variations affect the current and consequently the power consumption under a constant-voltage condition.

Temperature Coefficient of Resistivity

The temperature coefficient of resistivity is a material-specific parameter that quantifies how much a material's resistivity changes with temperature. For metals, it is generally positive, meaning that resistivity increases with increasing temperature. This parameter is crucial when dealing with devices like heating elements, as it directly affects the resistance and hence the performance of the device over varying temperatures.

Temperature-Dependent Resistance

Resistance in conductive materials typically changes with temperature, and this relationship can often be approximated by a linear model over moderate temperature ranges. The formula R = R?[1 + ?(T – T?)] relates the resistance at a given temperature T to the known reference resistance R? at temperature T?, with ? being the temperature coefficient of resistivity. This concept is key to understanding how a heating element's performance shifts as it heats up.

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