Question

Before bendable tungsten filaments were developed, Thomas Edison used carbon filaments in his light bulbs. Though carbon has a very high melting temperature \(\left(3599^{\circ} \mathrm{C}\right)\) its sublimation rate is high at high temperatures. So carbonfilament bulbs were kept at lower temperatures, thereby rendering them dimmer than later tungsten-based bulbs. A typical carbon-filament bulb requires an average power of \(40 \mathrm{~W}\), when 110 volts is applied across it, and has a filament temperature of \(1800^{\circ} \mathrm{C}\). Carbon, unlike copper, has a negative temperature coefficient of resistivity: \(\alpha=-0.0005^{\circ} \mathrm{C}^{-1}\) Calculate the resistance at room temperature \(\left(20^{\circ} \mathrm{C}\right)\) of this carbon filament.

Short answer

Answer: The resistance of the carbon filament at room temperature is approximately \(571.73 \ \Omega\).

Step-by-step solution

Calculate resistance at the operating temperature

Using the formula for power: \(\text{Power} = \frac{\text{Voltage}^2}{\text{Resistance}}\), we can rearrange to solve for resistance: \(\text{Resistance} = \frac{\text{Voltage}^2}{\text{Power}}\) Now, we can plug in the given power of 40 W and voltage of 110 V: \(\text{Resistance}_{1800^{\circ} \mathrm{C}} = \frac{(110 \mathrm{~V})^2}{40 \mathrm{~W}} = \frac{12100 \mathrm{~V}^2}{40 \mathrm{~W}} = 302.5 \ \Omega\)

Determine the change in temperature

We need to find the change in temperature between the operating temperature and room temperature: \(\Delta T = T_{\text{operating}} - T_{\text{room}} = 1800^{\circ} \mathrm{C} - 20^{\circ} \mathrm{C} = 1780^{\circ} \mathrm{C}\)

Calculate the resistance at room temperature

Using the formula for temperature dependence of resistance: \(R_{20^{\circ} \mathrm{C}} = R_{1800^{\circ} \mathrm{C}} \left[1 - \alpha \cdot \Delta T\right]\) Plugging in the values found in Steps 1 and 2, and given \(\alpha = -0.0005^{\circ} \mathrm{C}^{-1}\): \(R_{20^{\circ} \mathrm{C}} = 302.5 \ \Omega \left[1 - (-0.0005^{\circ} \mathrm{C}^{-1}) \cdot 1780^{\circ} \mathrm{C}\right]\) \(R_{20^{\circ} \mathrm{C}} = 302.5 \ \Omega \left[1 + 0.89\right]\) \(R_{20^{\circ} \mathrm{C}} = 302.5 \ \Omega \cdot 1.89 = 571.725 \ \Omega\) Therefore, the resistance of the carbon filament at room temperature is approximately \(571.73 \ \Omega\).

Key concepts

Electrical Resistance

Electricity is all about the movement of electrons, and resistance is what opposes this movement. Think of it like trying to slide down a slide covered in sticky syrup; the syrup is like resistance—it's what makes it hard to slide down smoothly. In technical terms, electrical resistance measures the degree to which a material hinders the flow of electric current.

Materials with higher resistance make it harder for the current to flow, which is why we don't use something like rubber to make electrical wires; it has too much resistance, and the electrons can't flow through easily. Metals like copper or aluminum, on the other hand, have a much lower resistance and are great for wires. The resistance of a material depends on its composition, length, thickness, and even the temperature, which can cause the resistance to change.

Power Calculation

When you're about to charge your phone, you plug it into the wall without thinking much about it, right? But there's some cool science happening there!

Power calculation is like working out how much energy your phone is getting from that outlet every second. The power (measured in watts) tells us the rate at which energy is used or delivered. We can calculate it by taking the square of the voltage (yes, voltage is a big deal—it's like the pushing force for electrons) and then dividing it by the resistance (

For example, a light bulb that requires 40 watts to light up properly needs a good balance of voltage and resistance to work right. If the voltage is too low or the resistance too high, the light bulb won't shine as brightly, just like how your phone won't charge well with a weak charger.

Temperature Dependence of Resistance

Just like how your fingers get all clumsy when they're cold, materials can act differently when the temperature changes. Specifically, the resistance of materials changes with temperature—a concept crucial in designing all sorts of electronic gadgets.

In most metals, as the temperature goes up, the atoms jiggle around more, which makes it tougher for the electrons to slip through the material, resulting in higher resistance. This is why your phone might get a bit wonky if it overheats—the internal resistance changes might muddle its performance.

However, some materials, like our friend carbon used in the old light bulbs, exhibit negative temperature coefficients of resistivity. That means their resistance actually goes down as the temperature goes up. It's like resistance is moving backward: the warmer the carbon gets, the easier it is for electricity to flow through it. Strange, but true - science is full of surprises!