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$^\circ$C, it consumes electrical energy at a rate of 480 W. What is its power consumption when its temperature is 150.0$^\circ$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$^\circ$C.

Key Concepts

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.

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.

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.

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.

Transcript

00:01 The things we know from this problem is that we have a tungsten heating element.

00:07 It is connected to a battery with a negligible internal resistance.

00:13 And at a temperature, let's call it ti, which is equal to 80 degrees c, it draws a power, let's call that pi, of 480 watts.

00:25 We can assume that the conditions are such that we can use table 25 .2, and we're also told to use equation 25 .12 with a reference temperature t not of 20 degrees celsius.

00:45 And we want to know at a temperature, let's call that tf, of 150 degrees celsius, what is the power that is consumed by this? heating element.

01:00 So since we're told to use equation 25 .1 .2, i went ahead and wrote it down, and we're also told that we can use table 25 .2.

01:10 So this alpha over here, that coefficient corresponds to the one for tungsten, which is 0 .0045 per degree celsius.

01:28 Now, given that we have to find a power and we know that we have to use an equation with resistance, that leads me to think that i should use this equation here.

01:47 Power is equal to the potential difference squared over r, right? but r is a function of temperature, so let's write that.

02:01 So then my power becomes a function of temperature.

02:10 Now for each case, before i do that, let's rewrite this as v squared is equal to pr.

02:26 So we have this situation where you have you supply some potential difference, and then at pi, at some r .i, at some temperature i, you have p .i...

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