Question

When a flashlight bulb with a tungsten filament is lit, the applied potential difference is \(3.991 \mathrm{~V}\) and the temperature of the tungsten filament is \(1.110 \cdot 10^{3}{ }^{\circ} \mathrm{C}\). The resistance of the bulb when it is at room temperature \(\left(20.00^{\circ} \mathrm{C}\right)\) and is not lit is \(1.451 \Omega\). What current does the bulb draw when it is lit?

Short answer

To find the current drawn by the flashlight bulb when it is lit, we first need to calculate the resistance of the tungsten filament when lit (\(R_{lit}\)) using the temperature coefficient of resistivity formula: \(R_{lit} = 1.451 \Omega \cdot(1+(4.5 \times 10^{-3}K^{-1})(1089.85 K))\) After calculating the resistance, we use Ohm's Law to find the current: \(I = \frac{3.991 V}{R_{lit}}\) Once the current is calculated, that will give us the current drawn by the bulb when it is lit.

Step-by-step solution

Find the resistance of the tungsten filament when lit

We'll use the temperature coefficient of resistivity formula to find the resistance of the bulb when it is lit. This formula states: \(R_T = R_0(1 + \alpha\Delta T)\) Where: \(R_T\) is the resistance at temperature T, \(R_0\) is the resistance at a reference temperature (in our case, room temperature), \(\alpha\) is the temperature coefficient of resistivity (for tungsten, \(\alpha = 4.5 \times 10^{-3} K^{-1}\)), \(\Delta T\) is the change in temperature, which is the difference between the temperature of the filament when it is lit and at room temperature. First, we need to convert the given temperatures from Celsius to Kelvin: \(T_0 = 20.00^\circ C + 273.15 = 293.15 K\) \(T_{lit} = 1.110 \cdot 10^{3}{ }^{\circ} C + 273.15 = 1.383 \cdot 10^{3}{ } K\) Now, calculate \(\Delta T\): \(\Delta T = T_{lit} - T_0 = 1383 - 293.15 K = 1089.85 K\) Now, we can find the resistance of the tungsten filament when lit, \(R_{lit}\): \(R_{lit} = R_0(1+\alpha\Delta T) = 1.451 \Omega \cdot(1+(4.5 \times 10^{-3}K^{-1})(1089.85 K))\)

Calculate the current drawn by the bulb using Ohm's Law

Ohm's Law states that \(V = IR\), where V is the potential difference, I is the current, and R is the resistance. We can rearrange it to solve for the current: \(I = \frac{V}{R}\) Now we can plug in the given potential difference, \(3.991 V\) and the resistance we calculated in Step 1: \(I = \frac{3.991 V}{R_{lit}}\) Calculate the current, and this will give us the current drawn by the bulb when it is lit.